View Full Version : The Foundations of a Non-Naive Mathematics
Hi,
Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (include its links).
I'll be glad to get your detailed remarks and insights.
Thank you,
Lama
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Edit (11/8/2004):
Here is a list of my axioms:
Tautology:
x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
(tautology is also known as the opposite of a contradiction).
(EDIT: instead of the above definition, I change Tautology to: The identity of a thing to itself.
It means that in my framework we do not need 'if, then' to define a Tautology)
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
Urelement:(no internal parts)
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.
It means that in my framework we do not need 'if, then' to define a Tautology)
A definition for a point:
A singleton set p that can be defined only by tautology* ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.
(*more detailed explanation of the first two definitions:
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Remark:
In Standard Math we had to write:
Point proposition:
If a content of a set is a singleton and a urelement and has no directions, then it is a point.
Segment propositon:
If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
But since in this framework a Tautology is the identity of a thing to itself,
we do not need an 'if, then' proposition for tautology.
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Let us examine {.} and {._.} definitions by using the symmetry concept:
1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.
It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.
2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'
In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .
In short, in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise (mathworld.wolfram.com/Tautology.html)
As we can see, in my framework '<','>' symbols have a deeper meaning then 'order'.
Actually, in order to talk about 'order' we first need a 'direction')
The axiom of independency:
p and s cannot be defined by each other.
The axiom of complementarity:
p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).
The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.
The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels (lowest bound and highest bound are the ends of some given element, or a collection of more than one element, where beyond them it cannot be found*.)
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*Let us clarify the 'finite' concept in my framework:
In my system I have 4 building-blocks, which are:
{}, {.}, {._.}, {__}
The cardinal of {} is 0.
The cardinal of {.} is one of many.
The cardinal of {._.} is one of many.
The cardinal of {__} is The one.
The bounds of lowest and highest existence (the ends) of these building-blocks
are determined by their cardinality, for example:
(in this example I omitted {.}_AND_{._.} and used only their building-blocks)
The lowest and highest bounds of {.} are cardinals 1 to 1.
The lowest and highest bounds of {._.} are cardinals 1 to 1.
The lowest and highest bounds of {} are cardinals 0 to 0.
The lowest and highest bounds of {__} are cardinals The 1 to The 1.
The lowest and highest bounds of {{.},{._.},{.}} are cardinals 1 to 3.
The cardinals beyond {.} are 0, n>1 and the 1.
The cardinals beyond {._.} are 0, n>1 and the 1.
The cardinals beyond {} are n>0 and The 1.
The cardinals beyond {___} are any cardinal which is not The 1.
The cardinals beyond some n are 0 and any j where j>n.
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The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.
The Axiom of the unreachable strong limit:
No input can be found by {__} which stands for Fullness.
The Axiom of potentiality:
p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}.
The Axiom of phase transition:
a) There is no Urelement between {} and {.}.
b) There is no Urelement between {.} and {._.}.
c) There is no Urelement between {._.} and {__}.
Urelement (http://mathworld.wolfram.com/Urelement.html).
The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.
(By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )
The Axiom of the paradigm-shift:
Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.
Ok, let us examine the duality concept.
In my system:
1) Each element of the Real-Line is both some unique element, and a scale factor of the entire Real-Line.
Strictly speaking, each element has both local and global properties of the Real-Line system.
There is an important graphic model at page 5 of No-Naïve-Math.pdf
that can help you to understand the duality idea.
2) A point is a Real-Line building-block that can be defined only by using =
A segment is a Real-Line building-block that can be defined by using < , > or =
No segment {._.} can be a point {.} exactly as no < or > can be =
It means that no segment can be constructed (defined) by finite or infinitely many points.
Because the Real-Line has at least {._.} and {.} building blocks, we get an absolute/relative system that has also properties of a fractal, because of a simple reason:
A point is a 0-dimension element that is not affected by the “over-all” scale factor.
An interval is a 1-dimension element that is affected by the “over-all” scale factor.
The result is an interaction between two opposite properties of the Real-Line:
1) The relative property can be defined as infinitely many unique intervals along the Real-Line and also in infinitely many different scales of it.
2) The absolute property can be defined as infinitely many points along the Real-Line.
Not one of these properties can satisfy the definition of the Real-Line.
Strictly speaking, the Real-Line is at least an absolute/relative system.
Because of this duality of each element in the Real-Line, the Real-Line has an invariant cardinality over infinitely many different scales of itself.
This self-similarity over infinitely many different scales is the most basic property of a fractal.
matt grime
Jun21-04, 10:00 AM
One presumes you have demonstrated that your new definition of the real line is equivalent to the set of all cauchy sequences of rational numbers "modulo convergence", or the set of dedekind cuts. That is, there is a point where you show how one construction can be used to derive the other. To save me time, what page number of the pdf will that be on?
there is a point where you show how one construction can be used to derive the other
In my theory (please read page 11 of No-Naive-math.pdf)
{.} is a potential {} and {._.} is a potential {__}.
{} AND {__} are the unreachable limits of the Language of Mathematics.
The Language of Mathematics become meaningful only if it uses the products of the interactions between {.} and {._.}.
It means that any meaningful thing in the Language of Mathematics it at least {.} and {._.}.
Strictly speaking, The Language of Mathematics is at least an absolute/relative system.
matt grime
Jun21-04, 11:41 AM
That doesn't answer the question; you could try answering the question, say, I realize that would be setting a precedent for you of course.
So, continue posting at your own risk.
This is the precise answer to your question Matt.
But there is another question according to your response, which is: "Do you understand my answer?"
matt grime
Jun21-04, 01:36 PM
I know it is he, as is www, shemesh and some others, and I don't intend to get into a 'debate'. Unless my screen is broken there still doesn't appear to be a page reference given, from which one presumes I don't understand the answer. Cranks, eh?
Matt please show us that you understand my answer.
matt grime
Jun21-04, 01:41 PM
An answer to the request would look something like: if S is the set of reals as defined by you then <argument> implies it is the set of reals accoridng to the proper definitions, conversely, if you take the proper definitions then <argument> which implies my construction. What you wrote is utter utter rubbish.
What you wrote is utter utter rubbish.
So, you demostrated that you do not understand my answer.
matt grime
Jun21-04, 02:00 PM
your 'answer' did not contain any reference to the definition (any of them) of the real numbers; it was thus not an answer to the question that explicitly asked you to prove your view was equivalent to any of these. but we long since stopped expecting you to understand such things.
and the question only asked you for a reference to a page in the articles.
your 'answer' did not contain any reference to the definition (any of them) of the real numbers
My definition of the Real-Line is better then Dedekind's Cut or Cauchy sequences of rational numbers, because of a simple reason:
Your absolute-only system cannot deal with real complexity because redundancy and uncertainty are not its "first-order" properties, my absolute/relative system can.
In short, standard Math system of the Real-Line is trivial because Dedekind's Cut or Cauchy sequences of rational numbers are trivial (and by using the word "trivial" I do not use the interpretation of current community of mathematicians that use this word for “self-evidence” or “extremely simple” thing).
If you can prove that < or > are =, then and only then my system is a superfluous system.
matt grime
Jun22-04, 04:33 AM
You promise it's superfluous? ok, in the reals x=y iff (x>y)and(x<y) is false.
Does this mean that your set, what ever it is, isn't the real numbers? If you can't produce an equivalence to the proper real numbers.
Note that your set of real numbers doesn't actaully have any numbers in it. It is a set of "global scale factors", which you've not defined, with some other properties and operations and stuff. The set could equally well have bananas in it.
Does this mean that your set, what ever it is, isn't the real numbers? If you can't produce an equivalence to the proper real numbers.
What I say is very simple:
A point is a Real-Line building-block that can be defined only by using =
A segment is a Real-Line building-block that can be defined by using < , > or =
No segment {._.} can be a point {.} exactly as no < or > can be =
No > or < can be constructed by finite or infinitely many =
Conclusion: Real-Line building-blocks are at least {._.} and {.}
In Standard Math the Real-Line building-block is only {.} and this is the reason why it is an absolute-only system.
It is a set of "global scale factors", which you've not defined,
You do not understand the duality idea (where each Real-Line number is both some well-defined element and a scale factor of the entire Real-Line), because you look at it only from an absolute-only (or fixed in your language) point of view.
Warning: There is no return to an absolute-only point of view, after you understand the Real-Line from an absolute/relative point of view.
Please show us an explanation (not a thechnical use of some function) by Standard Math, that can clearly show us why a proper subset of the Real-Line can have the cardinality of the entire Real-Line?
matt grime
Jun22-04, 09:17 AM
Two sets have the same cardinality if there is a bijection between them. That is the explanation because that is the definition. That you don't think cardinality ought to be invariant under bijections is your misunderstanding of how mathematics works.
As you reject proper maths you must define what the reals are without using proper maths, otherwise your argument is vacuous. So until you define from your first principles only how to construct the set of real numbers you are not on solid ground.
You are also wrong to say that no interval may be constructed by using infinitely many equalities, but then I don't suppose you know about o minimal structures and tarski's construction (any subset of a real finite dimensional vector space defined by a finite number of inequalities may be given by a finite set of equalities) , of course we can get a simpler refutation of your position by using an uncountable number of equalities.
Besides you contradict yourself in that post by saying a point is a segment because segments are defined using < > OR = (ie it includes all points), and then saying NO segment is a point.
You've still not defined what 'scale factor' means.
Besides you contradict yourself in that post by saying a point is a segment
No, I say that a point and a segment are different things exactly as '>' or '<' cannot be '=' .
A point is a Real-Line building-block that can be defined only by using '='
A segment is a Real-Line building-block that can be defined by using '<' , '>' or '=' (the use of '=' here is the tautology of a segment to itself and there is nothing here which is related to points).
It means that no '>' or '<' can be constructed by finite or infinitely many '=' .
In short, no segment (interval) can be represented by points and vise versa, and we need at least segments {._.} and points {.} to define the Real-Line (again, no one of them can be defined in the terms of the other).
Two sets have the same cardinality if there is a bijection between them. That is the explanation because that is the definition.
A bijection is the result of your measurement, so how can you use it to explain why a proper subset of some set can have the same cardinality of the entire set?
matt grime
Jun22-04, 11:12 AM
*cough* seeing as a cardinal number is an equivalence class of sets modulo the relation 'there is a bijection between them' then I think we can explain why two sets have the same cardinality by using a bijection.
The other bits of your post demonstrate that your initial explanation was, and still is, in need of rewriting because it is inconsistent when read literally, that you intended to mean something else is immaterial (your illexplained use of 'used to define')
One notes you do not refute the comments about o-minimal structures.
a cardinal number is an equivalence class of sets modulo the relation 'there is a bijection between them'
Now all you have to is to explain to us why there can be a bijection between a set to its proper subset.
The other bits of your post demonstrate that your initial explanation was, and still is, in need of rewriting because it is inconsistent when read literally, that you intended to mean something else is immaterial (your illexplained use of 'used to define')
Look, we are in 'theory development' where people some times need to use their own abilities to understand another points of views. So, please put aside your rigorous well-defined standards and move out of the limits of your spot light from time to time.
Believe me, it will be a good exercise for your brain mussels.
One notes you do not refute the comments about o-minimal structures.
Ordered-minimal structures because the definable subset of R are exactly those that must be there because of the presence of '<'.
I'll be glad if you show me where I can find in Standard Math my fractal point of view of the Real-Line, when I use an o-minimal (<,>) and a point (=) as the minimal (must have) building-blocks of the Real-Line.
You've still not defined what 'scale factor' means
A multiplication operation of each well-defined R number with the entire Real-Line.
matt grime
Jun22-04, 03:38 PM
What one earth do you mean explain how there can be a bijection? I can write one down and demonstrate it's a bijection, eg N to N\{1} given by x -> x+1, so look, by example there can be.
o minimal does not as far as i am aware mean ordered minimal, though there are ordered o minimal structures
a scale factor is a mulitplication operation that multiplies an element of R the R? that makes no sense., you've not described how to do this.
and there is a difference between changing definitions, or rather deciding that which you first examine is not what you want and then defining something else (this happens a lot in developing theories when done properly), and not defining something properly which means all subsequent deductions are invalid.
CrankFan
Jun22-04, 07:11 PM
wow, people on this board sure are cranky... I'm going to go read a book.
CrankFan,
No, in general, people here are not cranky. Only people in this particular subforum, which is for Theory Development -- a.k.a. unsupported pseudoscientific speculation.
- Warren
I can write one down and demonstrate it's a bijection, eg N to N\{1} given by x -> x+1, so look, by example there can be.
When you are asked to explain why there is a bijection between some set to a proper subset of itself, the answer cannot be: "because set A has a bijection with its subset B, that can be written (for example) as f:A -> B"
So, if f:A -> B means a bijection between set A to its proper subset B, and you are asked to explain "Why?", then the answer "because I wrote it" is not an answer.
o minimal does not as far as i am aware mean ordered minimal
http://cowles.econ.yale.edu/conferences/wkshp/lec/steinhorn3.pdf
a scale factor is a mulitplication operation that multiplies an element of R the R? that makes no sense., you've not described how to do this.
Please read pages 5 and 8 (where 5 and 8 are not the acrobat screen number, but my paper number) of my paper here: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
and there is a difference between changing definitions, or rather deciding that which you first examine is not what you want and then defining something else (this happens a lot in developing theories when done properly), and not defining something properly which means all subsequent deductions are invalid.
In post #15 I wrote:
A point is a Real-Line building-block that can be defined only by using =
A segment is a Real-Line building-block that can be defined by using < , > or =
No segment {._.} can be a point {.} exactly as no < or > can be =
No > or < can be constructed by finite or infinitely many =
Conclusion: Real-Line building-blocks are at least {._.} and {.}
Your response to this was: "Besides you contradict yourself in that post by saying a point is a segment ..."
Please explain us how you can come to this conclusion, when you read what is written above (the green part)?
matt grime
Jun23-04, 05:00 AM
if something may be defined by some equality, it is defined by some set of equalities or inequalities.
seems you really don't ever want to learn how maths works then given your inability to understand cardinalities.
if something may be defined by some equality, it is defined by some set of equalities or inequalities.
Is this an explanation to the question: "Why there can be a bijection between a set to its proper subset?" ?? :bugeye:
matt grime
Jun23-04, 08:23 AM
of course not, it is the answer to the other part of the question, the on about defining things with equalities. you've explained though that you don't mean equalities in general when talking about things defined by = hence my comments about you not defining things properly
Matt please give a detailed answer to post #23, thank you.
Please let me write some analogy Matt:
Let us say that blue means a new thing.
You have yellow glasses; therefore you see any blue (new) thing as a green thing (which is not a new thing).
I asked you to take off your yellow glasses before you look at my blue (new) things.
You ignore my request and say:"nothing is new here, don't you see?
what you call blue is nothing but well-known green things!"
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Let us go straight to main points, where my theory is different from the Standard point of View.
1) Through My new point of view, any number is first of all an information-form which is based on at least {._.}_AND_{.}, where {._.}_AND_{.} is the minimal existence of any number which is not 0.
2) In any given quantity which is > 1, each number can be ordered by several internal symmetrical degrees that can be clearly shown here: http://us.share.geocities.com/complementarytheory/ONN.pdf.
3) {._.}_AND_{.} of set N, cannot be put in a bijection with proper subsets of themselves, because of {._.} that exists as an internal building-block of each N member, and I clearly show it in page 3 of http://www.geocities.com/complementarytheory/No-Naive-Math.pdf.
4) Standard Math ignores {._.} because through its point of view, any given {._.} can be defined by {.}.
5) Through my point of view {._.} cannot be defined by {.} .
6) If any number is at least {._.}_AND_{.}, then a bijection between Q or R sets to proper subsets of themselves, can be defined if and only if any Q or R member is both some unique element, and a scale factor of the entire set (in N or Z the scale factor result is always out of the domain of the original pairs, when no one of the pairs is 1,-1 or 0).
7) My new system is consistent (well-defined) and cannot be understood by Standard point of view (yellow glasses).
Can you take off your yellow glasses before we continue?
matt grime
Jun25-04, 09:45 AM
Why do you still think I want to understand your crackpot idiotic ramblings? I only care about correcting your mistaken beliefs about mathematics and your inexact usage of langauge.
Why do you still think I want to understand your crackpot idiotic ramblings? I only care about correcting your mistaken beliefs about mathematics and your inexact usage of langauge.
At last we can see what is Math for you.
Math is your religion Matt, so bye bye to you limited and non-creative mind. :zzz: :zzz: :zzz:
Gokul43201
Jun25-04, 11:02 AM
Wow ! What fun !
matt grime
Jun25-04, 02:32 PM
What on Earth do you mean by 'at last'? And why the abuse of type face and colour? You didn't at any point honestly think I had any interest in what someone who patently has no foraml training in, nor desire to learn about, mathematics had to say about their uneducated version of mathematics? Notice the key one there: you have only ever wanted to put forward your twisted view of things. You are not ramunajan, ok, you are a crank who does not understand the first thing about mathematics and can't seem to realize that the following statements are not consistent.
X is defined by s statement of the form P(X)
Y is defined by statements of the form P(Y), Q(Y), or Z(Y)
nothing that qualifies to be a Y can be an X.
CrankFan
Jun26-04, 07:23 AM
When Matt said:
"I can write one down and demonstrate it's a bijection, eg N to N\{1} given by x -> x+1, so look, by example there can be."
Lama responded:
"When you are asked to explain why there is a bijection between some set to a proper subset of itself, the answer cannot be: "because set A has a bijection with its subset B, that can be written (for example) as f:A -> B"
First f:A->B doesn't imply that f is a bijection.
Second, Matt didn't merely suppose the existence of a bijection, he showed you how to construct it, f(x) = x + 1.
You ignored his perfectly reasonable demonstration and then started complaining, why?
Which confuses me. If you really don't know what a bijection is then why are you wasting your time redefining the foundations of mathematics? Instead, maybe you should bury your head in some math books so that you can learn why, for yourself.
A function that is both one-to-one and onto is a bijection. (**INFORMALLY** it establishes an exhaustive pairing of elements between the domain and range). Now, armed with this definition, you can use it to verify to your own satisfaction that the function matt described above is a bijection.
The beauty of math is that no one has to convince you of anything. Assuming that you're working with the same definitions that everyone else is (which is critical) then the conclusions follow naturally. In your case I don't think you are understanding the basic definitions. For example, it was a major mistake of yours to think that f:A->B alone, implies that f is a bijection.
Basic mistakes like this tell us that you don't understand the subject very well, so why should we take your grandiose claims seriously? when you've shown that you don't even understand the basics? After all, you want to be a guy who rewrites the foundations of mathematics. If I had such a lofty goal, at the very least I'd familiarize myself with the current foundations of mathematics, before characterizing them as "naive".
Your paper is most understandable at the beginning yet shows the same kind of basic errors discussed above as early as page 2. The paper becomes more and more incomprehensible as you develop your own poorly defined and intuitive pseudo-math vocabulary in favor of well defined and established mathematical terms.
it was a major mistake of yours to think that f:A->B alone, implies that f is a bijection.
CracnkFan, you missed the main point of my argument.
I used f:A->B here as the most general notation of Standard Math mapping, where some case of it is a bijection (1-1 and onto).
Your paper is most understandable at the beginning ...
If you do not understand my paper, then you cannot show us any meaningful conclusion about it.
Let us use my glasses analogy again:
Let us say that blue means a new thing.
You have yellow glasses; therefore you see any blue (new) thing as a green thing (which is not a new thing).
I asked you to take off your yellow glasses before you look at my blue (new) things.
You ignore my request and say: "nothing is new here, don't you see?
what you call blue is nothing but well-known green things!"
Let us say that you take off your yellow glasses and then for the first time you can see a natural (by transparent glasses) blue thing, but you cannot understand it because any blue thing can be understood by you only if it is a green thing.
I'll say it clear and load for the first and last time: We are in a Theory Development forum where every concept is not beyond re-examination.
You come to this forum wearing your yellow glasses concepts about the Language of Mathematics, and because of this you prevent from yourself to re-examine so-called "well-defined" things.
Please use your transparent glasses in a theory development forum, and also be aware that in this case no green things can fully help you to understand natural (by transparent glasses) blue things.
If you cannot do that, then please let us not waste our time in a non-dialog.
My main approach about what is called "The Language of Mathematics" cen be found in the front page of my website here: http://www.geocities.com/complementarytheory/CATpage.html
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Let us go straight to main points, where my theory is different from the Standard point of View.
1) Through My new point of view, any number is first of all an information-form which is based on at least {._.}_AND_{.}, where {._.}_AND_{.} is the minimal existence of any number which is not 0.
2) In any given quantity which is > 1, each number can be ordered by several internal symmetrical degrees that can be clearly shown here: http://us.share.geocities.com/complementarytheory/ONN.pdf.
3) {._.}_AND_{.} of set N, cannot be put in a bijection with proper subsets of themselves, because of {._.} that exists as an internal building-block of each N member, and I clearly show it in page 3 of http://www.geocities.com/complementarytheory/No-Naive-Math.pdf.
4) Standard Math ignores {._.} because through its point of view, any given {._.} can be defined by {.}.
5) Through my point of view {._.} cannot be defined by {.} .
6) If any number is at least {._.}_AND_{.}, then a bijection between Q or R sets to proper subsets of themselves, can be defined if and only if any Q or R member is both some unique element, and a scale factor of the entire set (in N or Z the scale factor result is always out of the domain of the original pairs, when no one of the pairs is 1,-1 or 0).
7) My new system is consistent and well-defined, and cannot be understood by Standard point of view (yellow glasses).
8) Standard Math has no answer to the question: "What is a number"?
My theory gives rigor answer to this question by using the information concept, where redundancy and uncertainty are fundamental ("first-order") properties of it.
Standard Math uses only the on_redundancy no_uncertainty information form
as "first-order" property.
9) Furthermore, my theory includes our cognition's ability to define 'a number', as a legal part of it.
CrankFan
Jun26-04, 12:00 PM
"If you do not understand my paper, then you cannot show us any meaningful conclusion about it" -Lama
I can conclude that the paper is incoherent babble.
If you want to formulate a new number system, you're free to do that however as Matt already pointed out if your "reals" aren't equivalent to our reals then why would we want to replace our reals with your "reals"? You've already dropped a few hints that indicate that your "reals" aren't equivalent to our reals. That you would try to pass your "reals" off as The Reals strikes me as incredibly arrogant.
Which gets to the crux of the issue. No one seems to be making a fuss about the claims you've made regarding your proposed number system, but rather the claims you've made about the number system of "standard mathematics" and how your number system is so much better than what it offers. That, and your apparent ignorance on the subject of the number system of standard mathematics.
Matt already pointed out if your "reals" aren't equivalent to our reals...
What are you talking about?
What do you mean by "our reals" ?
Do you think that I am an alien out of space?
We are all living in this beautiful blue planet and share our thoughts with each other (and now we can do it by internet).
In my work I clearly show that what is called "the Language of Mathematics (including its logical reasoning)" is based only on no_redundancy no_uncertainty information form (as its "first-order" property) which is nothing but some private case of infinitely many universes of different information forms.
I take my theory and try to share this deep insight with you.
What I find is a community of religious people that do not want to examine any new idea (that maybe can enrich their world) even if we still talking in the philosophical level.
In my opinion, the Language of Mathematics is maybe the most powerful language that we have, that maybe has a direct influence to our survival on this blue planet.
Therefore we have to be opened to new fundamental ideas about this language and put aside this stupid (and in my opinion, even a dangerous attitude) "our xxx" manifests.
Think big man, let us get together out of our defensive corners and at least open our minds to the possibilities of ideas that can be found in each one of us.
In short, do not kill things too quickly, because the first rule of survival is to find the balance between opposite things.
ahrkron
Jun26-04, 02:53 PM
It is not a matter of politics, survival, or religious behavior.
The problem is, Lama, that you are starting from your own constructs and then trying to pass them as concepts that have a different definition, one that is standard and well agreed upon (or many definitions that can be shown to be equivalent to each other).
Also, you need to learn much more about math's concepts and the proper use of its language. Otherwise, you will continue to put your effort on a crusade that, frankly, is not producing anything worth it.
I don't mean to be rude, but you need to realize that what you are doing is not math or anything close. Not only that, it is also inconsistent, unnecessary, ugly and, to make it worse, badly written. It is not a matter of people being "rigid" or "close-minded", but of you needing to learn more and to develop skills on mathematical thinking and comminucation.
ahrkron, my heart with you, because my work has a strange property.
When you look at it, you see your own real face.
Dear ahrkron,
I do not think that your last post is the best you can do.
So please use your professional mathematical skills, and give your detailed answer to post #34.
Maybe the dialog between ex-xian (which is a moderator of another forum) and me, can halp here, so here it is:
Thanks for the kind words...(this thread is becoming a mush fest :biggrin: ).
Please give your detailed epsilon-delta proof, which is not based on a proof by contradiction, and please add an informal explanation to each formal part of it, thank you.
The definition of a limit is as follows:
lim_x->c f(x) = L (the limit, as x tends to c, of f(x) is L) means that given an ε>0, there exists a δ>0 such that 0<|x-c|<δ implies |f(x) - L|<ε.
The point is that given any epsilon, you can always find a delta that corresponds to that epsilon so that the above implication holds. There is no magic delta that holds for all epsilon.
Here's a very simple example of a proof that the lim_x->2 2x+3 = 7, written in line form rather than paragraph form to make it easier to refer to.
1) Let ε>0
2) We seek a δ such that if 0<|x-2|<δ then |(2x+3)-7|<ε.
3) Let δ=ε/2.
4) Let |x-2|<δ, that is |x-2|<ε/2.
5) Then 2|x-2|<ε,
|2x-4|<ε, and
|(2x+3)-7|<ε.
6)That is |f(x)-L|<ε. So, by definition, lim_x->2 2x+3 = 7. qed
1) An arbitrary epsilon is "chosen." The epsilon is abitraray so no generality is lost, but in no way does it stand for every epsilon greater than zero.
2) Just a statement of what we're trying to do. For a given epsilon, there is one of more corresponding deltas that fulfill the requirments. For example, if ε=1, then a delta of 1/2 would be sufficient.
3) The delta is chosen. This is accomplished by manipulating |f(x)-L|<ε to get it in the form of |x-c|<(an expression that involves ε). In this example:
|(2x+3)-7|<ε
|2x-4|<ε
2|x-2|<ε
|x-2|<ε/2.
This gives the connection between the chosen epsilon and the delta.
4)Statement of definition.
5)Showing definition holds.
6), 7) Closing statements of proof.
Hi ex-xian,
Thank you for the very clear post about epsilon-delta proof.
It is based on a bijection (edit: insted of bijection please read injection) between infinitely many arbitrary ε to their unique δ, where each connection is a unique 1-1 case (there is no magic δ for all ε, as you said).
Well my friend, in my paper I am talking about the logical (general) meaning behind any epsilon-delta proof, which in this case uses ε and δ connection to show how any given interval |x-x0| implies δ=0.
Let us write this logical proof by contradiction:
If |a-b| = δ < all ε > 0 then δ = 0.
Proof:
Let us say that δ > 0
1) δ < all ε > 0
2) δ > 0
Since δ < all ε > 0 and d > 0 then δ<δ that cannot be true, so (1) and (2) cannot both be true.
Therefore, it is true that If (1), then not (2) --> δ = 0, QED (a proof by contradiction).
In my paper I use this proof to show that it is limited to an excluded-middle logical reasoning (exactly as there is no magic δ for all ε).
This is some example of re-examination of fundamental mathematical concepts, and in this particular paper I re-examine:
1) Logical reasoning.
2) Limit.
3) universal quantification.
Please read again http://www.geocities.com/complementarytheory/ed.pdf , thank you.
Thank you,
Lama
kaiser soze
Jul6-04, 03:29 AM
Lama:
Proof:
Let us say that δ > 0
Did you mean "let us say that there exists delta such that delta >0"?, otherwise your statement makes no sense, delta is not a literal.
Kaiser.
matt grime
Jul6-04, 09:49 AM
Originally Posted by Lama
Hi ex-xian,
Thank you for the very clear post about epsilon-delta proof.
It is based on a bijection between infinitely many arbitrary ε to their unique δ
where each connection is a unique 1-1 case (there is no magic δ for all ε, as you said).
Wrong, again, about a piece of mathematics, yet you are about to state something categorically about it. Given an epslion, there is not a unique delta, in fact there need to be uncountably many possible deltas for any given epsilon, hence there is no bijective correspondence. But you've always indicated that you don't actually know what a bijection is so I suppose we shouldn't be too hard on you.
Well Matt you are well-known by your profound and quick conclusions.
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Hi ex-xian,
Thank you for the very clear post about epsilon-delta proof.
It is based on a bijection between infinitely many arbitrary ε to their unique δ
where each connection is a unique 1-1 case (there is no magic δ for all ε, as you said).
No such bijection exists. Although no delta works for every epsilon, for a given epsilon there is an infinite number of possible deltas.
Well my friend, in my paper I am talking about the logical (general) meaning behind any epsilon-delta proof, which in this case uses ε and δ connection to show how any given interval |x-x0| implies δ=0.
That's not what a delta-epsilon proof is supposed to show. I gave the formal defintion in my previous post.
Let us write this logical proof by contradiction:
If |a-b| = δ < all ε > 0 then δ = 0.
Well, again, this isn't what a delta-epsilon proof is for. Also, this is trivially true. The only non-negative number that is less than every postitive number is 0.
A delta-epsilon is used to show the existence of a limit. What limit and function are you working with?
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for a given epsilon there is an infinite number of possible deltas.
Lama:
Thank you for the correction.
By mistake I wrote bijection instead of Injection.
So, if for any given epsilon there is at least one delta, we can say that there is a 1-1 and not onto between an epsilon and a delta (this correction has no influence on my argument in this post).
...in this case we use ε and δ connection to show how any given interval |x-x0| implies δ=0.
this is trivially true
Lama:
It is trvially true according to the framework that you choose to work with.
For example, it is trivialy true if:
1) 'If |a-b| = δ < all ε > 0 then δ = 0' is an hypothesis ('If' is used).
2) A universal quantification can be realted to a collection of inifintely many elements.
Let us write again this logical proof by contradiction:
If |a-b| = δ < all ε > 0 then δ = 0.
Proof:
Let us say that δ > 0
1) δ < all ε > 0
2) δ > 0
Since δ < all ε > 0 and d > 0 then δ<δ that cannot be true, so (1) and (2) cannot both be true.
Therefore, it is true that If (1) , then not (2) --> δ = 0, QED (a proof by contradiction).
In my paper I use this proof to show that it is limited to an excluded-middle logical reasoning (exactly as there is no magic δ to all ε).
This is some example of re-examination of fundamental mathematical concepts, and in this particular paper I re-examine:
1) Logical reasoning.
2) Limit.
3) universal quantification.
Please read again http://www.geocities.com/complementarytheory/ed.pdf
In short, what I want to show here is, that fundamental concepts of the Language of Mathematics, can have different interpretations in different frameworks.
You can say that you are interested only in the common framework and you don't care about any other possible framework.
If this is your basic approach, then it is ok with me, but in this case each one of us is talking to himself, and there is no dialog but two monologs.
And for monologs I do not need this thread.
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Did you mean "let us say that there exists delta such that delta >0" ?
Yes.
kaiser soze
Jul6-04, 01:07 PM
I am sorry, but I could not find the function and the limit in your posts. Delta/Epsilon proofs are typically used in context of a given function and limit.
Kaiser.
In my paper I use this proof by contradiction to show that it is trivial only in an excluded-middle logical reasoning.
My paper is some example of re-examination of fundamental mathematical concepts, and in this particular paper I re-examine:
1) Logical reasoning.
2) Limit.
3) universal quantification.
Please read http://www.geocities.com/complementarytheory/ed.pdf
In short, what I want to show here is, that fundamental concepts of the Language of Mathematics, can have different interpretations in different frameworks.
kaiser soze
Jul6-04, 02:32 PM
Your statements make very little sense. I think you do not really understand what epsilon/delta proofs are used for.
Dear kaiser soze,
Math has many faces, and if you choose to ingore it, then we have no dialog between us.
Russell E. Rierson
Jul7-04, 02:09 AM
Dear kaiser soze,
Math has many faces, and if you choose to ingore it, then we have no dialog between us.
Why not re-derive the foundations of mathematics?
Assume that a universal set exists. Forget about the axiomatic enigmas generated by a "bottom-top" approach. Don't let the dogmatic zealots dampen your spirits with their flatulant barkings. :eek: :eek: :eek:
Start with a top-down approach, using an all inclusive symmetry axiom.
Symmetry forms the basis of truth.
Existence is a definition, a predication, which is why Kant so
vehemently denied that existence is a predicate, but alas, existence
is a definitive constraint as is all definitions. To exist means to have some instantiation. So infinite paradox becomes infinite freedom from definitive
constraint, and reality itself is a equilibrium point.
So by defining a largest possible set, we create a paradox,
because the set of all sets is its own power set; the
cardinality of the set of all sets must be bigger than itself along with the Bertrand Russellian set that does not shave itself. .
The "Ein Sof" is an infinity that cannot be comprehended. For every
set A there is a choice function, f, such that for any non-empty sub
set B of A, f(B) is a member of B, and so we see that there may be
an infinite number of sets B within A, and as such the Banach-Tarski
paradox is created. A single sphere is decomposed and re-assembled
into two spheres, each with the same radius as the original sphere.
So we see that:
[paradox] = not-[paradox]
is a paradox of course!
therefore:
paradox = paradox
is absolutely true.
Alpha = Omega
It is the categorical formulation of the simultaneous, situational,
instantiated contradiction, where deductive invalidity is the product
of the utmost categorical truth of the assumption that if the
antecedent of a true conditional is false, then the consequent of the
conditional is true or false indifferently, and of the categorical
falsehood of the conclusion condequently predicates that if it be not
the case that the consequent of a true conditional is true or false
indifferently, then, it is not the case that the antecedent of the
conditional is false. To pronounce the consequent of a true
conditional as being true or false indifferently is tantamount to
saying modally that where the antecedent of a true conditional is
notoriously false, then the consequent can, or could be, or is
possibly true or false. But it may be worthwhile to see that the
definitive, simultaneous equality of both true, and false, can be
formulated without explicitly including modal terms, which become the
predicating operators, which, for the sake of showing that the
consequent paradoxical conundrum is not straightforwardly resolvable
by appealing to concrete philosophical scruples concerning the
intensionality of predicated modal contexts.
The categorical representation of the propositional anti-logic, in
which deductive invalidity depends on the modality of the truth
conditionals concerning the prerequisite of the contingent assumption
and consequent conclusion. The totally relevant content of the
assumption and conclusion, definitely contains no modal terms. But,
the modality attaches to the fact that the conditional assumption is
quite possibly true, while the conditional conclusion is necessarily
false.
Which leads us to an argumentational representation of a completely
non-bogus modal formulation of the "paradox". Deductive invalidity is
most excellently predicated on the categorical truth of the
modal-term-laden assumption and the definitive categorical falsehood
of the modal-term-laden conclusion. Hence, the assumption is such,
that if the antecedent of a contingently true conditional is false,
then, the consequent of the conclusion can be true is itself quite
simply, ...true. Therefore, the conclusion that if it is not the case
that the consequent of a contingently true conditional can be true,
then it is not the case that the antecedent of the true conditional is
false, is itself quite simply, false.
Architechturally speaking:
An exact computational correspondence, i.e. "one to one and onto", becomes a type of "limit", in the compression of information via powerful generalizations. At the limit of informational compressibility, the physical universe is actually a mathematical universe. Does this limit exist? If it exists, we live in a mathematically designed universe, governed by perfectly harmonious equations. In fact, the design and construction phase are an approach towards an equilibrium point.
Ideal mathematical perfection, creates problems for itself. These problems arise with the introduction of "free will" and sentience to the equational composition. Systemic anomalies[rebellious sentient programs] rear their ugly heads. Yet, without sentient beings possessing the attributes of free will and the ability to make a "choice", the universal system of progressively compositionial calculations could not approach the limit of infinite informational equilibrium, and the system would disintegrate.
Counterbalancing occurs with the sentient "Heisenberg compensation" operators, which must be introduced, to maintain the most optimal trajectory towards a state of perfect equilibrium with maximal efficiency. Nomological covariance and consistent history, is maintained in place of an absolute deterministic exhaustion. A totally closed system. Thus these "guardian" sentient programs must police the timeline, ensuring that it remains paradox free.
A recombinatorial mix of sentient attributes allows for further maintenance of an optimal equational trajectory, as sentinel "messiah programs" are martyred in the war against the rebellious "systemic programming anomalies" .
Don't let the dogmatic zealots dampen your spirits with their flatulant barkings.
Dear Russell E. Rierson, thank you, but I want to add that I have no problem with dogmatic approach of others, if I can use it to develop my work.
Take for example persons like Matt Grime, which in my opinion make here a very good job as the bodyguard of Math.
It took me some time (almost 2 years) to understand that I am talking to a full time job bodyguard, so now I take what I take and I do not care anymore that full time job bodyguards do not want to or can’t understand my work.
So we see that:
[paradox] = not-[paradox]
is a paradox of course!
It depends on the framework that we choose to work with.
In an excluded-middle reasoning a = not_a is nothing but a false statment.
Russell E. Rierson
Jul7-04, 11:14 PM
Dear Russell E. Rierson, thank you, but I want to add that I have no problem with dogmatic approach of others, if I can use it to develop my work.
Take for example persons like Matt Grime, which in my opinion make here a very good job as the bodyguard of Math.
It took me some time (almost 2 years) to understand that I am talking to a full time job bodyguard, so now I take what I take and I do not care anymore that full time job bodyguards do not want to or can’t understand my work.
It depends on the framework that we choose to work with.
In an excluded-middle reasoning a = not_a is nothing but a false statment.
contradiction = not-contradiction is a contradiction
true?
or
false?
There is no question here.
a is not_a is nothing but a false statment in boolean logic, because no identity can be in more than one unique state in boolean logic.
matt grime
Jul8-04, 06:49 AM
when did "is" become a connective in boolean logic?
do you mean "and", which is after all the definition of a paradox, something that is simultaneously true and not true.
By the word 'is' I mean to '='
a = not_a is nothing but a false statmant in boolean logic.
For example:
If the Barber of Seville does not shave himself, then he does not fit to his own self identity, which is:
To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = less_than_all or in other words: all = not_all
If the Barber of Seville shaves himself, then he does not fit to his own self identity, which is:
To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = more_than_all or in other words: all = not_all
Some conclusions:
a) The self identity of the Barber of Seville is based on the false statement all = not_all.
b) Self identity, which is based on a false statement, is no more then a false statement.
c) No false statement is a paradox in excluded-middle reasoning.
d) Therefore Russell's paradox is not defined in excluded-middle reasoning.
In general we can conclude the above about any self-referenced definition, which includes in it all condition.
If an all condition is omitted form a self-referenced definition, then the possibility of self identity as a false statement, is avoided in an excluded-middle reasoning.
Russell E. Rierson
Jul8-04, 08:36 PM
There is no question here.
a is not_a is nothing but a false statment in boolean logic, because no identity can be in more than one unique state in boolean logic.
If the statement contradiction = not-contradiction is a contradiction is false, the statement contradiction = not-contradiction is not a contradiction is true?
If the statement contradiction = not-contradiction is a contradiction is false, the statement contradiction = not-contradiction is not a contradiction is true?
In logic we can say that our true result is a false statemant.
This is the reason why some false reuslt can be found in our logical system.
Only the true stands behind any result.
Also in excluded-middle reasoning any examined concept cannot have more than one unique identity,
so a = not_a cannot be but a false statemant (which is the true reuslt) in this case.
matt grime
Jul9-04, 07:23 AM
eh? so A iff not A is false, so? (note the correct use of iff, sometimes denoted <=>, and not =, since 'equals' is not an operator in boolean logic) what does that have to do with anything? what matters is that if we adopt naive set theory we have a case where A and notA must be true, when it is trivially false, so what?
note the correct use of iff, sometimes denoted <=>, and not =, since 'equals' is not an operator in boolean logic
'=' is used here for the tautology of a = a.
a = not_a is no more than a false statment in excluded-middle reasoning.
A and not_A
As usual, you miss the point.
A and not_A cannot be defined in excluded-middle reasoning, because any examined concept cannot have more than a one unique identity.
Our true result in this case is no more then a false statement, and all the big affords that professional mathematicians like you put in their theories to avoid this "paradox", are no more than a full gas in neutral.
Also read post #48.
No proposition can make a statement about itself...
If we look at this propositoin, we can say that within an excluded-middle reasoning, if a self reference of a proposition changes the propositon, then and only then it cannot be refered to itsef, because in an excluded-middle reasoning, each element has exactly one and only one uniqe identity.
By tautology x = x means: x is itself, otherwise we cannot talk about x.
Now we can ask if a teotology is also recursive, for example: x = x = x = ...
If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.
So, Russel's paradox is like if by teotology we examine if x is not_x or x = not_x , which is no more then a false statement from an exluded-middle point of view.
In an excluded-middle reasoning no false statement is a paradox.
Again:
The element x_AND_not_x cannot be defined in excluded-middle reasoning, because any examined concept cannot have more than a one unique identity.
Therefore Russell's Antinomy is nothing but a false statemant and not a paradox in excluded-middle framework.
Some dialog:
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Lama:
I think that we do not understand each other.
I gave you MY definiton of the limit concept.
Now, please give the standard definition for this concept.
After you give the standard definition, then we shall compare between
the two approaches.
Any way do you agree with http://mathworld.wolfram.com/Limit.html definition?
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kaiser:
off course I agree with this definition. I meant for you to provide the defintion for the limit of S(n), no need delta epsilon at this point. A limit can be defined using epsilon and S(n). At any case, I am not interested in your definitions at the moment. I need to be convinced that you understand and know how to use the fundamental "conventional" mathematical defintions before we can move on to your definitions.
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Lama:
Ok, the main persons in modern Math that are related to the so called rigorous definition of the limit concept are Cauchy and Weierstrass.
Cauchy said:" When some sequence of values that are related one after the other to the same variable, are approaching to some constant, in such a way that they will be distinguished from this constant in any arbitrary smaller sizes that are chosen by us, then we can say that this constant is the limit of these infinitely many values that approaching to it."
Weierstrass took this informal definition and gave this rigorous arithmetical definition:
The sequence S1,S2,S3, … ,Sn, ... is approaching to (limit) S if for any given positive and arbitrary small number (e > 0) we can find a matched place (N) in the sequence, in such a way that the absolute value S-Sn (|S-Sn|) become smaller then any given epsilon, starting from this particular place in the sequence
(|S-Sn| < e for any N < n).
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kaiser:
Very good! now based on the definition you provided, which is a correct mathematical definition please find out the limit of the following sequence:
0.9,0.99,0.999,0.9999,0.99999,....
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Lama:
-------post #190
Now please listen to what I have to say.
First please read http://www.geocities.com/complementarytheory/9999.pdf
(which is also related to your question) before we continue.
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Lama:
-------post #191
I disagree with the intuitions of Weierstrass, Cauchy, Dedekind, Cantor and other great mathematicians that developed the current mathematical methods, which are dealing with the Limit and the Infinity concepts.
And my reason is this:
No collection of infinitely many elements that can be found in infinitely many different scales, can have any link with some given constant, in such a way that it will be considered as a limit of the discussed collection.
In short, Nothing is approaching from the collection to the given constant, as can be clearly seen in my sports car analogy at page 2 of http://www.geocities.com/complementarytheory/ed.pdf
Take each separate position of the car, then compare it to zero state and you can clearly see that nothing is approaching to zero state.
Therefore no such constant can be considered as a limit of the above collection.
It means that if the described collection is A and the limit is B, then the connection between A,B cannot be anything but A_XOR_B (any transformation from A state to B state cannot be but a quantum-like leap).
So here is again post #184:
Since I am not a professional mathematician, my best definition at this stage is:
A Limit is any arbitrary well-defined element, where no collection of well-defined infinitely many elements can reach it.
It means that if A is the collection of infinitely many elements and B is the limit, then we can reach B only if we leap from A to B and vise versa.
By using the word "leap" we mean that we have a phase transition from state A to state B.
There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states.
A collection A is incomplete if infinitely many elements of it cannot reach some given limit, or if no limit is given.
From the above definition we can understand that no collection of infinitely many elements is a complete collection, and therefore no universal quantification can be related to it.
If you disagree with me, then please define a smooth link (without “leaps”) between A,B states.
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Lama:
-------post #192
'Any x’ is not ‘All x’
By inconsistent system we can "prove" what ever we want with no limitations
but then our "proofs" are inconsistent.
A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proven by the current axioms of this system, and we need to add more axioms in order to prove these statements.
So any consistent system is limited by definition and any inconsistent system is not limited by definition.
Let us examine the universal quantification 'all'.
As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements, even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements is an inconsistent idea.
For example:
Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.
In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.
Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf
Form this point of view a universal quantification can be related only to a collection of finitely many elements.
An example: LIM X---> 0, X*[1/X] = 1
In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.
'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.
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kaiser:
If you do not see that the limit of the sequence I provided is 1, then you do not understand what a limit is, and therefore can not agree or disagree with its definition.
In loose terms we can say that a sequence has a limit if it is approaching (but never reaching) some conststant. A sequence does not have a limit, if it is not approaching some constant, for example the sequence 1,2,3,4,... does not have a limit, it disperses to infinity.
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Lama:
1 as the limit of the sequence 0.9,0.99,0.999,0.9999,0.99999,.... is based on an ill intuition about a collection of infinitely many elements that can be found in infinitely many different scales, as can be clearly understood by posts #190,#191,#192.
You can show that 1 is really the limit of sequence 0.9,0.99,0.999,0.9999,0.99999,.... , only if you can prove that there is a smooth link (without "leaps") between this sequence and 1, which is not based on {0.9,0.99,0.999,0.9999,0.99999,.... }_XOR_{1} connection.
Maybe this example can help:
r is circle’s radius.
s' is a dummy variable (http://mathworld.wolfram.com/DummyVariable.html)
a) If r=0 then s'=|{}|=0 --> (no circle can be found) = A
b) If r>0 then s'=|{r}|=1 --> (a circle can be found) = B
The connection between A,B states cannot be but A_XOR_B
Also s' = 0 in case (a) and s' = 1 in case (b), can be described as s'=0_XOR_s'=1.
You can prove that A is the limit of B only if you can show that s'=0_AND_s'=1 --> 1
A collaction of elements, wich can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.
matt grime
Jul21-04, 05:13 AM
"In loose terms we can say that a sequence has a limit if it is approaching (but never reaching) some conststant. "
once more you demonstrate you ignorance of mathematics. take an eventually constant sequence to see why.
Once more you demonstrate your limited ability to understand new interpretation of the Limit concept.
When you understand what I say then you clearly see that "approaching" is based on an ill intuition.
what kaiser is saying is that 1 is the limit, but 1 is not included in the set of .9+.09+.009...
i THINK what Lama is saying is:
intuitively the number this "approaches" is 1, getting infinitely close to but never reaching it. but actually the number it really "approaches" is .999...
in other words you're both saying the same thing
...getting infinitely close...
"getting close" is reasonable.
"getting infinitely close" is not reasonable, because nothing can be closer to something when something is some constant and the "closer" element is one of infinitely many elements that can be found in infinitely many different scales.
matt grime
Jul21-04, 07:25 AM
aah, of course, the proper definition is ill defined because when we use a new one that you can't properly explain we have problems. doesn't that tell you your new interpretation might be wrong rather than mathematics?
this reminds me of your insistence that Cantor's argument is wrong, since it causes some problem with your interpretation of what the word all means. (an interpretation it might be pointed out that cannot be applied to anything other than sets of real numbers.)
since cantor didn't produce this result using your "definitions" (which is debatable, given the paucity of your argument), why this is noteworthy is beyond comprehension.
doesn't that tell you your new interpretation might be wrong rather than mathematics?
There is no objective and fixed thing like 'Mathematics'.
The language of Mathematics deeply and fundamentally can be changed, when new interpretations are given to its most fundamental concepts.
Because you demonstrated time after time that you have no ability but to be a full time job bodyguard of the current interpretations of these concepts, you cannot see beyond your own shadow.
since cantor didn't produce this result using your "definitions"
I clearly and simply showed the ill intuitive approach of Cantor’s diagonal method here:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf
I have to say that this is an old version of some of my ideas, for example:
By the new version that can be found in page 3 of:
http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
I clearly show that N members cannot be mapped with their proper sub-collection of the odd numbers.
Also |N|=|Q| only if we compare between notatations and not the numbers themselves (any number (which is not 0) is an element that cannot be less then {.}_AND_{._.} ).
His ill intuition about the collection of infinitely many elements that can be found in infinitely many different scales, brought him to developed his ill transfinite system.
matt grime
Jul21-04, 09:42 AM
But, crackpot, surely if you change the meanings of the words on the left hnad side, as it were, you must change them on the right hand side? something you refuse to do.
that pdf (the second one mentioned in your previous post at the time of writing) is illuminating in its sheer nonsensical approach to everything. fascinating. it doesn't even define what the real numbers are, or why we muyst define the real numbers at all in order to see why cantor's wrong. and its also not at al logical, but surely you wil dismiss me as a closed minded body guard of mathematics.
you've not actually defined what a function is in your "set theory", so we cannot possibly comment about you assertions, however, in the common models of ZF there is a bijection from the naturals to the even (or odd) natural numbers. you don't disprove this easily proved fact. so you must be assuming some other kind of set theory and a model. in some uncommon models of ZFC the power set of the "naturals" is not uncountable... how does that grab you? it's called the skolem paradox: any set theory with an infinite model has a countable model, or something like it.
anyway, that doesn't affect the fact that the idea that because a map from N to N doesn't take into account a map from R to R is obviously garbage. why on earth must it? i repeat, i don't even need to have the real numbers in my model in order to define a bijection from N to N.
note also that saying the real-line is the real axis does not actaully define the real numbers.
i don't see that the rest of it is worth reading given the gross inaccuracies in the first few pages.
You don't realize that the Limit and Infinity concepts are based on ill intuition in the conventional mathematical system.
And I clearly show it in my last posts of this thread.
Cantor’s diagonal method simply fails if instead of the decimal representation method, we give a single and unique notation to each R member.
And even if there exists a Proof of Cantor without using this decimal representation, then it is based on the ill intuition that an interval can be defined in terms of infinitely many elements over infinitely many different scales.
Also I clearly show in my paper on P(Z*)>Z* that can be found in http://www.geocities.com/complementarytheory/NewDiagonalView.pdf that Cantor Did not prove that P(Z*)>Z* .
Because of this ill intuition of the Limit and Infinity concept, all the methods that are based on it, have to be re-examined, in my opinion.
Because I clearly and simply show that no collection of infinitely many elements can be considered as a complete collection, the universal quantification cannot be related to it, and it has to be switched by an inductive approach for quantification.
Because no segment can be defined in terms of points, and vise verse, the whole idea of the transfinite system simply collapses, and a collection of infinitely many elements become a flexible concept instead of the fixed ill intuitive hard concept of the conventional method.
P(Z*)>Z* but not because of Cantor’s proof, but because of the simple fact that (Infinitely many elements)+1 > (Infinitely many elements).
For years we hear professional mathematicians saying: "Do not interfere between a representation of some mathematical object, and the mathematical object itself".
I agree with this idea and keep it in my mind when I develop my ideas, and then I discover that the standard system is full of methods and theories that do not distinguish between a representation of some mathematical object, and the mathematical object itself.
So your conventional mathematical word is an ill world that has to be replaced by a new system.
Let us examine some different interpretations between my approach and the conventional approach:
If we take care only about the integers, then [3,4]<[2,5], but if we "dive in" to the fractal structure of infinitely many sub-intervals, then [3,4]=[2,5].
Why is [3,4]<[2,5] for integers? I'm not asking to compare the lengths, but rather the intervals themselves. If I highlight from 3 to 4 on the number line in blue and from 2 to 5 in red, which highlight is greater than the other? Normally you could say that one number is greater if it's further down the positive side, but both are further down the positive side than the other in some ways.
Lama:
The main idea behind the integers (unless we choose to change it) is to look on the number line as if it has a one and only one scale factor, which its value is 1 and only 1.
In this case any arbitrary interval cannot be but 1 (or -1 if we take zero's left side).
For example:
...___-2___-1___0___1___2___3___4___5___6___...
...___-2___-1___0___1___2___3___4___5___6___...
3___4 < 2___3___4___5 --> [3,4]<[2,5] by the new approach.
By this approach no proper subset of N can be put in 1-1 correspondence with the entire N, for example N and its odds:
...___1___2___3___4___5___6___7___8___9___... (Entire N)
| | | | |
...___1_______3_______5_______7_______9___... ( Entire Odds)
In the standard way the interval {.__.} is omitted and we get:
... 1 2 3 4 5 6 7 8 9 ... (Entire N)
| | | | | | | | |
... 1 3 5 7 9 11 13 15 17 ... ( Entire Odds)
As we can clearly see, standard math does not find 1-1 map between numbers, but between their represented notations, and we can clearly see that the standard point of view does not distinguish between a number and its represented notation.
Also:
2 <-> 3
5 <-> 4
and in this case (where {._.} is omitted) [3,4] = [2,5] by standard math.
-------------------------------------------------------------------------
When [3,4] and [2,5] are taken as R members then the inifinitely many elements that exist between 3 to 5 and 2 to 5 in infinitely many different scales, can be put in 1-1 and onto, and in this case [3,4]=[2,5] because of the duality of each R member, which is clearly explained here:
http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
-------------------------------------------------------------------------
Infinitely many elements in infinitely many scales have bigger cardinality then infinitely many elements that can be found in a one and only one particular scale (scale 0 is excluded in both cases).
Therefore |N|<|Q|<|R| where each number is at least {.}_AND_{._.} (as can be seen in http://www.geocities.com/complementarytheory/No-Naive-Math.pdf).
I was just curious Lama: What are the problems with the number system that exists today that would make you want to change it? Like they say, "if it aint broke, don't fix it."
Hi Gza,
I was just curious Lama: What are the problems with the number system that exists today that would make you want to change it? Like they say, "if it aint broke, don't fix it."
Today's number system is a quantity-only system, which ignores the internal complexity of the natural numbers (and I do not mean to the differences between primes, non-primes, odds , evens, partitions, permutations, etc..., which are all based on 0_redunduncy_AND_0_uncertainty building-blocks), which are the building-blocks of the entire standard system.
In short, my number system is based on the information concept, where each building-block in it has an internal structure that cannot fully described only by quantitative-only and 0_redunduncy_AND_0_uncertainty approach of the standard system.
The main concept of my new number system is based on the complementary relations that exist between symmetry level and information's clarity-level, and these relations are based on what I call complementary-logic, which is based on included-middle reasoning, and both excluded-middle reasoning and fuzzy logic are limited proper sub-systems of it.
By my system we get these benefits:
1) Each building-block has a unique internal complexity, that can be the basis for infinitely many unique building-blocks, which can be found upon
infinitely many different scales.
2) There are infinitely many unique internal structures that can be found
in some particular scale level.
3) There can be infinitely many complex structures, that are based on (1) and (2) building- blocks.
4) These complex structures are much more accurate models then any model which is based on the quantitative-only standard number system, and some of the reasons are:
a) The structure that is based on the complementary relations between symmetry and information concepts (where redundncy_AND_uncertainty are useful properties of them) is inherent property of my new system, and gives it the ability to understand the deepest principles of any dynamic/structural abstract or non-abstract complex object, without first reducing it to quantitative model (which is inevitable when we use the standard quantitative-only number system).
b) The new natural numbers (which are now taken as topological information's building-blocks) are ordered as Mendeliev-like table, which gives us the ability to define their deep topological connections, even before we use them in some particular model.
These deep topological connections can be used as gateways between so-called different models, and expending our understanding about these explored models.
c) My number system is the first number system, which is based on our cognition’s ability to count, as an inherent property of the abstract concept of a number.
By this research I have found and described how the number concept is based on the interactions between our memory and some abstract or non-abstract elements.
Through this approach our own cognition is included in the development of the Language of Mathematics, and we are no longer observers, but full participators where our own congenital abilities are legitimate parts of the mathematical research itself.
For example:
What is called a function is first of all a reflection of our memory on the explored elements.
A function is the property that gives us the ability to compare things and get conclusions that are based on this comparison.
If something is compared by us to itself, we get the self identity of an element to itself by tautology (x=x).
If more then one element is compared, then we get several information clarity degrees that describe several possible interactions between our memory and the explored objects, and these several possible interactions can be ordered by their internal symmetrical degrees.
In this case multiplication and addition operations are complementary operations, where multiplication can be operated only between identical elements (redundancy_AND_uncertainty > 0) and addition is operated between non-identical elements (redundancy_AND_uncertainty = 0).
Because any function (which is not based on self reference of an element to itself) is a connection between at least two elements, its minimal abstract model cannot be less then a pointless line-segment, which is used as a connector between the examined elements.
In this case no interval (memory) can be described in terms of points (objects) and vise versa, and we get these four independent building-blocks of the language of Mathematics (which now includes the mathematician’s cognition-abilities as a legitimate part of it):
{}, {.}, {._.}, {__}
By this new approach we can build, for example, a totally new Turing-like machine, that can change forever our abilities to deal with complexity which is based in simplicity.
Please look at my website http://www.geocities.com/complementarytheory/CATpage.html if you want to understand more.
So, if we return to your first question, is this a wise thing to get off the evolution process?
Lama; what do you propose we use if we want to talk about quantity?
Lama; what do you propose we use if we want to talk about quantity?
Then continue to use only 0_redundancy_AND_0_uncertainty building-block.
And if you want to avoid any change of the current number system, then ignore the duality of any R member, which can be seen in http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
Also continue to use universal quantification as a deductive concept that can be related to a collection of infinitely many elements.
Also ignore memory/object(s) interactions, as a fundamental must-have condition that standing in the basis of the Number concept.
Also ignore Symmetry/Information complementary relations.
Also ignore {__} (the full-set) which is the opposite of {} (the empty-set).
Also ignore {._.} building-block and continue to use only {.} building-block.
Also ignore Multiplication/Addition complementary relations.
Also ignore Complementary-logic and continue to use only Excluded-middle reasoning.
In short, avoid any possibility of evolution process in the Language of Mathematics.
Are you saying we should never want to talk about quantity?
matt grime
Jul23-04, 08:35 AM
And are you saying that there is no evolution in mathematics? damn. so all those journals in my library are pointless because they dont' use your system. such delusions of grandeur in one so ill-informed!
fortunately you are still wrong about mathematics so we are ok even by your standards (I keep asking you if you're aware of all these topoi where the law of the excluded middle is false and you still don't answer)
And are you saying that there is no evolution in mathematics? damn. so all those journals in my library are pointless because they dont' use your system. such delusions of grandeur in one so ill-informed!
Mutation is the keyword here, where the most fundamental concepts of the language of mathematics have more than one intepretation.
Please show me a journal which clearly gives several interpretations to (for example) Limit and Ifinity concepts in the framework of standard Math.
I keep asking you if you're aware of all these topoi where the law of the excluded middle is false and you still don't answer
Please refresh my memory by show us where can we find a bruch of the language of mathematics, where a function is understood and used as a reflection of our memory on the explored elements.
Are you saying we should never want to talk about quantity?
Do all what you want to do.
Muddler
Jul25-04, 02:53 PM
Hi!
I've been following this whole issue for quite a while and I read most of Lama's papers.
I think the main problem here is (as in most cases) misunderstanding.
It is very likely that I am the one who is misunderstanding everything, still I'd like to try to "mediate" between the parties here:
As far as I understand Lama's new concept, it is a refinement of the current mathematical language.
Quantitative considerations using the "old" (forgive me that expression) number systems should still be possible as a "special case" of a quite more complex and fundamental numbering system, which might be the "organic numbering system" suggested by Lama.
Still I think there is a lot confusion when talking about limits and infinity.
To Lama:
I think you should agree that 1 is the limit of the 0.9, 0.99,0.999,... sequence, because that's different from the assumption that 0.999999..... is equal to 1, which seems to be your real problem (I agree with you there by the way)
To everyone else:
You should try to understand what Lama means by talking about duality of the real-line and its also being a scale factor.
If you look at the intervals [3,4] and [2,5] from a quantitative "set-theory-like" point of view, which takes into account the number of elements of each set, then it should be obvious why [3,4] = [2,5].
Still, when agreeing on using identical scale factors to both intervals, it should be absolutely conform with current mathematics and Lamas theory that [3,4] < [2,5]
A problem neither concept is so far able to answer is what happens at the "leap" from 0.9999999999... to 1, only that Lama states that there is one. :wink:
So here is what I would like to know from each party:
Lama: Tell me about your idea of the "leap" !
Everyone: Tell me why Lama's concept is wrong and/or not useful! His idea of an organic number system is intuitively appealing and the considerations behind his "building blocks" are absolutely logical. So what's wrong??
OK, I think by this post I showed myself to be the absolutest crackpot ever, but hey - at least I want to understand !!!! :tongue2:
So what's wrong??
It is that Organic cannot seem to grasp the consequences of the fact that his number system is not the integers / reals / whatever.
For instance, long ago we had a grand discussion about Cantor's diagonal argument; specifically the proof that the real numbers have greater cardinality than the integers. Not once in the entire discussion, however, did Organic talk about cardinal numbers; he substituted his Organic numbers at every opportunity, and stubbornly insisted he was drawing conclusions about cardinal numbers.
This is a common theme with Organic's threads. No matter what mathematical object he considers, he plugs in his ideas wherever possible, and then stubbornly asserts that whatever conclusions he draws must be true about this mathematical object. He has never shown any comprehension that when he plugs in his own ideas, he is now talking about some new mathematical object that is different from the original.
Furthermore, he frequently shows quite a bit of arrogance by accusing the rest of us of not understanding mathematics because we don't do it his way. This is despite the fact that he has not shown one ounce of understanding of the "normal" way of doing things. For instance, I would be entirely surprised if, tomorrow, he was able to post a correct proof, in the "normal" way, that if f(x) = x then f'(x) = 1.
Dear Muddler,
I can't help it.
Maybe I was born with some strange connections in my head, and as a result, I see different things in the most fundamental concepts of what is called 'The Language of Mathematics'.
Any reasoning system is first of all based on some self-evident cases, which a community of people agrees about them without proving them.
When time passes those self-evident cases become the common source of intuitions for this community, and this community will protect this common source as much as it can, because any change of this common source is like an earth wake effect for this community.
I said it before and I continue to say it, my work re-examines the most fundamental concepts that standing in the basis of this beautiful language, and in this most fundamental level, we are like a naked child that comes to this world with the gift of original points of view, which most of them are still unshaped by any well-defined educational system.
At this most basic stage there is a very gentle interaction between our intuition and our reasoning abilities.
If we ignore this most basic interaction, then we do not give ourselves the chance to re-examine new possible interpretations to fundamental concepts, which can maybe lead us to new discoveries.
At this most gentle level, there is no guarantee that your current well-known knowledge of the re-examined system, is used as the main player in this gentle and most fundamental process.
In short, I hoped to find partners for this gentle journey, and not surprisedly I have found very few people who agreed to put aside their arm of knowledge and to go to this journey like a naked child.
It is that Organic cannot seem to grasp the consequences of the fact that his number system is not the integers / reals / whatever.
I have no problem to say it again and again:
The standard number-system is the case of 0_redundancy_AND_0_uncertainy bulding-blocks, which are proper sub-systems of my system.
This is despite the fact that he has not shown one ounce of understanding of the "normal" way of doing things.
Not correct, I give the "normal" point of view before I air my non-standard view about it, for example: http://www.physicsforums.com/showpost.php?p=261949&postcount=59
Hurkyl, I will be more than glad to get your detailed response about it.
He has never shown any comprehension that when he plugs in his own ideas, he is now talking about some new mathematical object that is different from the original.
Not correct, please read again http://www.physicsforums.com/showpost.php?p=263204&postcount=69
and http://www.physicsforums.com/showpost.php?p=263942&postcount=71
To Lama:
I think you should agree that 1 is the limit of the 0.9, 0.99,0.999,... sequence,
Please give a detailed explanation why, thank you.
Tell me about your idea of the "leap" !
Please also read http://www.physicsforums.com/showpost.php?p=261949&postcount=59 thank you.
The standard number-system is the case of 0_redundancy_AND_0_uncertainy bulding-blocks, which are proper sub-systems of my system.
So what? Let me give you an example of a fallacious argument that I hope you will understand.
In the real numbers, I can always divide by 2.
The integers are part of the real numbers.
Therefore, in the integers, I can always divide by 2.
Not correct, I give the "normal" point of view before I air my non-standard view about it, for example: http://www.physicsforums.com/showpo...49&postcount=59
It's very easy for one to quote things one does not understand.
So what? Let me give you an example of a fallacious argument that I hope you will understand.
My example for sub-proper system is this:
In 3-D system we can move in X,Y,Z directions.
In 2-D system we can move in X,Y directions.
2-D system is a proper sub-system of 3-D system.
It's very easy for one to quote things one does not understand.
WHAT? Do you say that I am a layer that quote things, when he is asked to write his own text to clearly show that he undestand what he is talking about?
Hurkyl, This time you went too far, I call you to apologize !!! :mad:
My example for sub-proper system is this:
In 3-D system we can move in X,Y,Z directions.
In 2-D system we can move in X,Y directions.
2-D system is a proper sub-system of 3-D system.
Let me give you an example of a fallacious argument that I hope you will understand.
A line does not divide a 3-D system into parts.
A 2-D system is a proper sub-system of a 3-D system.
Therefore, a line does not divide a 2-D system into parts.
WHAT? Do you say that I am a layer that quote things, when he is asked to write his own text to clearly show that he undestand what he is talking about?
I say that, when accused of not understanding something, you responded with an example where you quoted that something.
Hurkyl,
I have just discovered your poor personality that will do all she need to do to keep living in flatland, where shadows are everything.
:confused:
Which one of us here has spent any effort trying to understand what the other is saying? You do remember back when I was trying to help you develop your ideas, don't you?
kaiser soze
Jul26-04, 12:44 AM
quote: ".... this community will protect this common source as much as it can, because any change of this common source is like an earth wake effect for these community."
I think this description applied mostly to you. One of the qualities a good scientist, is to be able to move on, when he discovers that his work is flawed or simply has no scientific value no matter how long he/she has been working on it.
I remember one short case when I started to develop the idea of what I call 'Equation Tree' (http://www.geocities.com/complementarytheory/ET.pdf).
The first 9 lines are your definitions, and after these 9 lines, you refused to continue any dialog with me about this subject.
Furthermore, since then most of the time you did your best to shut me off this forum at least 3 times, by using your power as the moderator of mathematics forum.
Most of our dialogues where based on your critique attitude about any idea that I gave, which can be a good thing if it is also balanced by some awareness to the corrections and the development that I achieved by listening to your criticism.
But I am sorry to say that you did not pay any attention to the development process which took place in my system during the years, and your basic attitude was and still is as if nothing happened through these 2-3 years since we know each other.
Your last posts are clearly showing this, better then 1000 witnesses.
So please do not play the sheep, because you are a wolf :devil: behind this :confused: face.
I think this description applied mostly to you. One of the qualities a good scientist, is to be able to move on, when he discovers that his work is flawed or simply has no scientific value no matter how long he/she has been working on it.
Maybe you are not aware about it, but we are in 'theory development forum' where people developing their ideas.
I clearly showed you that I perfectly understand the standard point of view.
You are the one how left in the middle of our dialog, after you realized that despite of this understanding I air my non-standard view.
Now you return for a short post, not to continue an open dialog, but to educate me.
I'm usually pretty good about not interjecting inane comments that have no real purpose in the development of a discussion, but my reading of this post leads me to the conclusion that Lama needs to get a life. I'm not saying this in a malicious way either, it's just that there are much bigger fish to fry than quibbling over something like this.
Hi Gza,
Can you be clearer please?
Dear Lama:
I am afraid that you will not be understood in your life. please be aware to that possibility. You should continue your significant work as Wittgenstein started already in the 20 century.
I am sure that you don’t earn money from doing mathematics.
Yours
Moshek
Dear Moshek,
Thank you very much, I hope to see the begining of a community of persons, which never afraid to re-examine any fundamental old or new concept of the Language of Mathematics, through an open dialog.
For me the language of Mathematics, is first of all an open and continuous dialog.
matt grime
Jul26-04, 08:04 AM
Please show me a journal which clearly gives several interpretations to (for example) Limit and Ifinity concepts in the framework of standard Math.
how about non-standard analysis? why are you incapable of understanding that even if we were to offer something that might be considered a refinement of, say, the limit of a sequence, that we would then choose a different name for it to avoid confusion with something that is well established and useful? and that the new object still owuldn't undo the old definition. moreover that some terms are used for more than one thing, and that context is required to clarify what is meant.
For instance the word "complete" has many meanings in mathematics: the reals are complete in the analytic sense, but not in the algebraic sense.
doron, "limit" is just a definition, it does its job, if you want a new object have one but don't call it a limit. the nature is not important only the ability to express it clearly to others. so saying that 0.999.. is not equal to 1 (in the reals, base 10), for example, demonstrates that you simply do not know the meanings of the the terms involved. that they are not equal is not some philosophical debate about the nature of mathematics and life, it is just a consequence of the axioms and definitions of mathematics.
one need only think of your aguments that tend to start:
if a function is... and so on to realize that you don't actually know anything about mathematics, and aren't prepared to learn.
matt grime
Jul26-04, 08:14 AM
As far as I understand Lama's new concept, it is a refinement of the current mathematical language.
erm, no, doron has repeatedly shown himself to be ignorant of every part of mathematics he has commented on. (try finding out the thread where he tries to hide the fact he doesn't know what a bijection is).
Still I think there is a lot confusion when talking about limits and infinity.
To Lama:
I think you should agree that 1 is the limit of the 0.9, 0.99,0.999,... sequence, because that's different from the assumption that 0.999999..... is equal to 1, which seems to be your real problem (I agree with you there by the way)
do you know what the real numbers are? do you think they are decimal expansions, for instance?
To everyone else:
You should try to understand what Lama means by talking about duality of the real-line and its also being a scale factor.
how can we? he's not ever defined what duality and scale factor mean, nor what he considers to be the real numbers
If you look at the intervals [3,4] and [2,5] from a quantitative "set-theory-like" point of view, which takes into account the number of elements of each set, then it should be obvious why [3,4] = [2,5].
one of the worst uses of an equals sign i've seen.
Still, when agreeing on using identical scale factors to both intervals, it should be absolutely conform with current mathematics and Lamas theory that [3,4] < [2,5]
and now we're using inequalities too! fantastic, care to explain what scale factor means? in you ordering is [1,2]<[1,3] or not? is it even a total ordering, partial ordering? does it have a minimal element? maximal element?
A problem neither concept is so far able to answer is what happens at the "leap" from 0.9999999999... to 1, only that Lama states that there is one. :wink:
what effing leap? there is no leap. appears you think the reals are decimals after all.
So here is what I would like to know from each party:
Lama: Tell me about your idea of the "leap" !
Everyone: Tell me why Lama's concept is wrong and/or not useful! His idea of an organic number system is intuitively appealing and the considerations behind his "building blocks" are absolutely logical. So what's wrong??
OK, I think by this post I showed myself to be the absolutest crackpot ever, but hey - at least I want to understand !!!! :tongue2:
simply over the year(s) doron has produced several garbage papers that have been refined to the current situation, he's cried wolf once too oftem for us to even bother trying to believe it is correct. do you know how long it took him to even offer some definition of what "uncertainty and redundancy" are? i don't mean in a metaphysical sense, purely in how to look at one of his diagrams and decide what corresponds to uncertainty, whatever metaphyiscal meaning we might have there. indeed he's not actually explained why it is that a diagram with such a property corresponds to being uncertain in anysense. how's that for an example?
Well Matt,
Some definition does its job in some framework.
But what do you mean does its job, and what is your framework?
You are talking about the useful technical level of some system, but this is only one way to research things.
At this stage my research is based on the 'Why' question, which is in my opinion the most important question in the first stage, where 'How' questions are used at this stage to support the 'Why' questions.
When time passes there are maybe more 'How' questions then 'Why' questions.
And then there can be again a period of big ‘Why’ questions that maybe can lead us to another session of paradigm-changes and so on.
There is no precise law in this evolution process, and we cannot have paradigm-changes without big ‘Why’ questions.
In my opinion, after more then 2000 years of linear and ‘objective’ approach, we have to go deeper then that and include our modern insights in the most fundamental concepts of this language.
For example: Our cognitive abilities to do Math have to be included in any fundamental understanding of any fundamental development, where concepts like Information, Symmetry, Nonlinearity and Complexity are involved with each other by dynamic and flexible non-trivial processes, which are based on our simplest insights about them.
Again, Mutation is the keyword here, where our self-evident perceptions find deeper levels of reasoning/intuition interactions.
If you concentrate, as you suggesting, only in the tactical/practical side of this language, then we have no meaningful dialog between us, because at this stage I am standing in the strategic ‘Why’ zone of my research, where persons like Muddler and Moshek are opened to it, and you are not.
-------------------------------------------------
As for your last post, you are so closed that you don't understand how you give us a good example of your inability to understand my well-defined ideas.
------------------------------------------------
(try finding out the thread where he tries to hide the fact he doesn't know what a bijection is).
No problem, you asked for it, you get it.
Dear Muddler, read what is below, and see for yourself Matt's dialog level:
Matt finds a mistake of mine, and he replies:
http://www.physicsforums.com/showpost.php?p=250748&postcount=41
The whole dialog between ex-xian and me, where I write, by mistake, Bijection (1-1 and onto) istead of Injection (1-1 and not onto):
http://www.physicsforums.com/showpost.php?p=250819&postcount=42
------------------------------------------------
The sad thing Muddler is that a person like Matt is going to get his Phd. in this november, and then he is going to multiply his attitute by teaching young persons that will have to obey his methodes, if they want their Phd.
matt grime
Jul26-04, 10:35 AM
mathematics is very safe from being affected by my opinions, doron.
and that isn't the case i was thinking of. there was one where we proved something was both injective and surjective and yet you refused to accept this was a bijection. it was in the cardinality stuff, where all the comments of Hurkyl's above apply.
the one you cite could be shrugged off as a careless slip, however it does reveal something deeply symptomatic of your "work".
mathematics is very safe from being affected by my opinions, doron.
Yes I know, Mathematics (by you) is a stuffed and dead thing, the cannont be effected by any living thing.
and that isn't the case i was thinking of. there was one where we proved something was both injective and surjective and yet you refused to accept this was a bijection. it was in the cardinality stuff, where all the comments of Hurkyl's above apply.
Maybe this old paper of mine can help: http://www.geocities.com/complementarytheory/Identity.pdf
matt grime
Jul26-04, 11:03 AM
"Yes I know, Mathematics (by you) is a stuffed and dead thing, the cannont be effected by any living thing."
You really do believe that don't you? can you grasp the difference between a concept and a definition? "the limit concept" you keep referring to is just a definition, it is fixed, but that doesn't stop you, or anyone else, adding to it, or offering a variation upon it. that is how lots of mathematics is researched. there are currently, to my knowledge, 4 definitions of "phantom" in topology. the one we eventually settle on as "correct" will be that which is most useful to us. the others will not cease to be valid, and we will study them, they may just be called different things. and because mathematics is not a stuffed and dead object, i can take the notion of topological phantom maps and use them in module theory, though beligianis uses phantom to mean something else there instead, so we may opt to use a different name. it doesn't matter though, it is just a name, and as such is chosen for convenience.
mathematics is most definitely not dead, and changes all the time, particularly in its emphasis on what is fashionable. you, however, don't know about these things, and don't seem willing to learn. it might behove you to do so.
let us take one example, to paraphrase:
the definition of the natural numbers is wrong because it does not take into account uncertainty and redundancy
in order to take that seriously you'd need to explain what you mean by all the terms there, why you believe this, and back it up with some evidence as to why the natural numbers MUST take these things into account, something you've singularly failed to do.
and to show you some of the variation of "the limit concept" in topological spaces we can say sequences tend to something in terms of neighbourhoods, or we can use ultrafilters, or nets, or we can use an algebraic notion of limit: every vector space is the direct limit of its finite dimensional subspaces for instance, then there are spectral sequences too. have you seen the constructions of certain types of c-star algebras in terms of nets? no, didn't think so.
Muddler
Jul26-04, 11:37 AM
one of the worst uses of an equals sign i've seen.
Sorry.
I was just quoting that from the terms used in post #67.
It surely is confusing to use common mathematical expressions in such a way, verbal descriptions should have been used instead.
Coming again to that "0.9999.... is equal to 1" issue: could you provide a mathematical proof for that, instead of just stating it as a consequence of current axioms and definitions?
I don't want to take it to a "philosphical" level, I'll try to stick to math as long as I am able to follow it, but here I am just asking for a little patience with me...
Thanks!
:redface:
in order to take that seriously you'd need to explain what you mean by all the terms there, why you believe this, and back it up with some evidence as to why the natural numbers MUST take these things into account, something you've singularly failed to do.
If you understand the symmetry/information complementary relations, then my papers are an open book for you.
, it is fixed, (infinity)
No, it is flexible, incomplete and deeply involved with probability.
adding to it, or offering a variation upon it
Because I basically disagree with your standard definition, which is based on an ill intuition, I am choosing to change it to a much more interesting and fruitful thing then your fixed point of view.
You want variations, to keep the dogmatic core of your standard mathematical community.
But I do not care about your dogmatic core, because I am a mutation and not a variation.
Math Is Hard
Jul26-04, 01:41 PM
Coming again to that "0.9999.... is equal to 1" issue: could you provide a mathematical proof for that, instead of just stating it as a consequence of current axioms and definitions?
have you had a chance to peruse this thread?
http://www.physicsforums.com/showthread.php?t=22866
There's a pretty lengthy discussion going on here that you might enjoy.
If we use a structural point of view in this case, then 0.9999... is a single path of a base 10 fractal ( http://www.geocities.com/complementarytheory/9999.pdf pages 3,4 ), that exists upon infinitely many scale levels that cannot reach 1.
Also we can say that 0.999... = 0.9+0.09+0.009+0.0009+... and we can clearly see that this infinitely long addition cannot reach 1.
Therefore 0.999... < 1.
Another example:
Please look at this beautiful Koch Fractal members.cox.net/fractalenc/fr6g6s.577m2.html
Now let us say the there is a 1-1 map between each fractal level of 0.9999... to each different blue level of Koch Fractal.
0.9999... = 1 if and only if we cannot find anymore a 1-1 map between some 0.000...9 to some Koch Fractal blue level.
Since Koch Fractal can be found in infinitely many blue levels and each blue level has a 1-1 map with some 0.000...9 fractal level, then we can conclude that 0.999... < 1.
Also we can say that 0.999... = 1 if and only if the outer contour of this multi-leveled Koch Fractal can be a smooth curve with no sharp edges.
It is clear that the outer contour line is not a smooth contour in any arbitrary examined scale level.
Therefore 0.999... < 1.
From this model you also can understand what is a "leap".
In short, any transition between a non smooth curve to a smooth curve, cannot be done but by a phase transition leap that also can be described by a smooth_XOR_no-smooth connection.
This model is better than any "abstract" mathematical definition, which leads us to "prove" that 0.9999... = 1.
Also by this "proof" we simply ignore infinitely many information forms that can be found in 0.9999... fractal.
Now think how many information forms are ignored by this trivial and sterile approach of standard Math.
zeronem
Jul26-04, 04:45 PM
I only have one recommendation for Lama. I recommend this book for you Lama,
"Where Mathematics Comes From: How the embodied mind brings Mathematics into being", - by George Lakoff and Rafael E. Nunez
hello3719
Jul26-04, 07:17 PM
If we use a structural point of view in this case, then 0.9999... is a one dimensional path of a base 10 fractal, that exists upon infinitely many scale levels that cannot reach 1.
Please define " one dimensional path of a base 10 fractal". There is a big problem here with definitions.
Furthermore, since then most of the time you did your best to shut me off this forum at least 3 times, by using your power as the moderator of mathematics forum.
Not this forum, just the math forum. And I didn't "shut you off" until people started complaining.
I still recal one of our early disagreements; you liked to assert that the cardinal number (aleph_0 - 1) was smaller than the cardinal number aleph_0. Your posts conveyed that you thought you had a proof of this using your idea of cardinality. You refused to accept that your proof (whatever it was) was a proof about your idea of cardinality rather than the "normal" idea of cardinality. This pattern continued with just about every mathematical idea you considered, and ended when I started moderating your posts because you were hijacking threads to talk about your way of doing things when everyone else wanted to talk about the "normal" way.
To Hurkyl,
Instead of pluralism (which is the healthy essence of any non-trivial system) you and other "well-educated" colleagues of the academic system, prefer to protect the dogmatic core of your community, instead of let it be developed by an open dialog.
But form strategic point of view, each one of us doing his job in this evolution process, which means:
You play the current system and I play the mutation of it.
You will do your best to shut me down and I will do my best to survive and flourish.
Dear zeronem,
Thank you for this information.
Please define " one dimensional path of a base 10 fractal". There is a big problem here with definitions.
There is no big problem here if you understand this model:
http://www.geocities.com/complementarytheory/9999.pdf
I'll be glad to get your opinion, thank you.
zeronem
Jul27-04, 02:59 AM
By the way, what is your Mathematical Background Lama? How far have you gotten in your math classes at school?
The answer to these questions will let us know exactly who we are talking to.
Zeronem:
If you talk like this,
it is better for you
not to know
who you are talking with.
Moshek
:wink:
matt grime
Jul27-04, 03:52 AM
"I am a mutation"
well, you said it, Doron.
Muddler, as to the proof that 0.99.. =1 in the reals (base ten expansion), it would appear that Doron is once more using his own definitions to conclude something about ours.
Something that should warn you he's off on one is that he doesn't ever use any of the parts of the definition of the real numbers to prove anything.
Here is a simple proof:
consider the sequences 1,1,1,1,1... and 0.9, 0.99, 0.999, now the "symbol" 0.99999... means the limit of the second. Now, it follows that these are clearly both cauchy sequences, and the difference between them is a cauchy sequence tending to zero, hence, that the sequences represent the same real number. (Which follows from the characterization of the reals as equivalence classes of cauchy sequences of rational numbers).
In fact we have all seen his characterzation of duality and scale factor and such. Now, all of the things he says about the reals are valid about the rationals and hence do not in anyway characterize reals uniquely.
terrabyte
Jul27-04, 04:22 AM
.999... ≠ 1 is one of my favorite topics
so many exceedingly smart people get it wrong. it is quite clear that when functioning under limits you cannot expect your tried and true methods to yield correct results.
.999... is NOT 1. it's simple and logical
matt grime
Jul27-04, 06:01 AM
In the reals, in base 10 representations, 0.999... and 1 are equivalent, and equal in this sense. they may be taken to represent different cauchy sequences, but they are unequivocally the same real number, Terrabyte. Please, offer a reason, mathematically sound, as to why they are not equal.
you and other "well-educated" colleagues of the academic system, prefer to protect the dogmatic core of your community, instead of let it be developed by an open dialog.
As long as you continue to believe this, you will have shut out a great store of knowledge and innovation.
You will do your best to shut me down and I will do my best to survive and flourish.
As long as you reject criticism, you will not flourish. It is just as important, if not more so, to understand one's mistakes than to understand one's successes.
Do you remember long ago when I was working with you? I stated that I felt it important to understand how your ideas related to mathematics, and eventually decided that, among anything I could try to do, helping you understand that relation would be the most beneficial to you. I still think that is true.
Now, all of the things he says about the reals are valid about the rationals and hence do not in anyway characterize reals uniquely.
Please read again http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
You can find there:
1) A point:
A singleton p that can be defined only by tautology ('='), where p has no internal parts.
2) An interval (segment):
A singleton s that can be defined by tautology ('=') or '<' or '>' , where s has no internal parts.
3) A number is both p_AND_s
4) A general representation of an interval (segment) is {._.}.
5) A general representation of a point is {.}.
6) Only {._.} can be affected by a scale factor.
Both Q and R members are effected by using one of them as a scale factor over the entire Real-Line.
The difference between Q and R members is still the inability to define an R member (irrational) by a ratio between at least two integers.
But this is not the new case here.
The new thing is that any given number is now both absolute and relative element, because it is defined by ({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}.
By the way, what is your Mathematical Background Lama? How far have you gotten in your math classes at school?
The answer to these questions will let us know exactly who we are talking to.
I am an autodidact that dealing (with a lot of love and patience) with the Language of Mathematics for more then 20.
As long as you reject criticism...
On the contrary, I seek for criticism (otherwise I would not be hear) and you are maybe one of the best.
But the sad thing is that I used your criticism to develop my system, and you react as if nothing has been changed during the years.
To be more specific: please criticize http://www.physicsforums.com/showpost.php?p=267089&postcount=101
Thank you.
kaiser soze
Jul27-04, 10:16 AM
You receive criticism only when you agree with it. If you really want to learn, you need to be able to accept criticism even if you do not initially agree; this because you acknowlegde that whom ever criticizes you knows more than you do in the acclaimed subject....:wink:
Kaiser.
You receive criticism only when you agree with it...
Please show me an example of how a person can receive criticism, and on the same time he does not agree with it.
In short, 'receive' and 'agree' are synonyms in this case, isn't it?
kaiser soze
Jul27-04, 11:24 AM
I'll try an analogy: when a parent says to his little child "do not speak to strangers", the child should accept this "criticism", even if the reason is unclear at that moment. When this child grows up, it will become clear to him why this "criticism" was given.
Kaiser.
I'll try an analogy: when a parent says to his little child "do not speak to strangers", the child should accept this "criticism", even if the reason is unclear at that moment. When this child grows up, it will become clear to him why this "criticism" was given.
Your analogy does not hold in this case, because we are talking about fundamental things which are the heart bits that give life to the language of Mathematics.
In this most simple level we do not need sophisticated methods but very sensitive intuition/reason gentle interactions, which become our self-evident building-blocks that are used as the milestones of our research.
What I have found is, that instead of developing this gift of self research that exists in most of the young students, the academic system does its best to force its methods on the minds of the students, because there is no time and no money to develop the unique internal skills of each student.
As a result the student is forced to be under the doctrine of some external method which he must agree with it in every step along the "learning" process.
After couple of years under this forcing external race-like method, most of the students are not able to use anymore their natural skills of self learning, and most of them become no more then good technicians of the academic system.
I have found that a lot of doctors and professors of Mathematics simply lost any ability to understand simple things.
My work is on these simple things, and I developed my skills to work and create in this simple level, for more then 20 years.
No external reasons like money or academic title where my motivations and no forcing external race-like methods washed my natural ability to understand simple things.
kaiser soze
Jul27-04, 01:15 PM
Mathematics was developed and taught long before academic institutions existed, yet its fundamental concepts and foundationds are still valid.
Kaiser.
Mathematics was developed and taught long before academic institutions existed.
Yes, it developed by people who where not forced by external methods or exteral reasons or lack of time.
yet its fundamental concepts and foundationds are still valid.
Not all of them.
kaiser soze
Jul27-04, 02:11 PM
The least I can offer you, is to accept and learn from others, who may have more knowledge from you in a given area - that is if you wish to interact with them.
Kaiser.
Dear kaiser soze,
Please show some detailed example, which cearly show why my work failes because I am not listening to persons that have more knowledge in this given area.
kaiser soze
Jul27-04, 03:12 PM
you refuse to accept the simple fact that the LIMIT of the sequence 0.9, 9.99, 0.999, .... is 1.
Kaiser.
Don't you see the simple fact, which can clearly understood by http://www.physicsforums.com/showpost.php?p=267089&postcount=101 that 1 is not the limit of your sequence?
kaiser soze
Jul27-04, 03:39 PM
I rest my case.
terrabyte
Jul27-04, 03:50 PM
In the reals, in base 10 representations, 0.999... and 1 are equivalent, and equal in this sense. they may be taken to represent different cauchy sequences, but they are unequivocally the same real number, Terrabyte. Please, offer a reason, mathematically sound, as to why they are not equal.
you're essentially limiting the structure of your numerical system by this "entity" we call infinity. now, infinity is a useful term, we use it frequently to extend expressions out to incredible precision, but since the term itself is not closed, there lacks a decided "conclusion" for formulas utilizing infinity. thus any formulas that use infinity are relegated to being approximations, albeit really damn good ones.
the limit of a sequence is something that is NEVER reached. hence defined as such 1 IS the limit of .9+.09+.009...
but from that statement above it's logically sound that since it is NEVER reached, the sequence can NEVER be equal to 1.
the flaw is in the system, whether we choose to fix it or ignore it is the question...
ex-xian
Jul27-04, 04:06 PM
Wow...I came over here hoping to find some info on the recent experiment that supposedly disproves the Copenhagen interpretation of QM. Imagine my surprise when I see my friend Dorian (known as Lama here) posting this stuff here. And I was even quoted! I am indeed honored.
If anyone is interested, my dialogues with him are here, http://www.iidb.org/vbb/showthread.php?t=89321
Welcome my dear ex-xian,
I invite you to continue our dialog here or in your forum.
An information of this recent experiment you can find here:
http://drauh.typepad.com/blog/2004/04/
http://www.kathryncramer.com/wblog/archives/000530.html
Since Quantum elements are both wave and particle, no one of these properties can completely disappear, so I do not see any new point in this recent experiment.
the academic system does its best to force its methods on the minds of the students
Do you see the irony in that you are trying to force your ideas on others?
Do you see the irony in that you are trying to force your ideas on others?
Am I an academic institute that gets money for my knowledge and forces people to show that they got it, by using an industry of examinations that if they do not pass them they will not get their diploma?
Can you please show how I force you to agree with me?
By the way, I am still waiting to you at http://www.physicsforums.com/showpost.php?p=267696&postcount=116
ex-xian
Jul27-04, 05:56 PM
Welcome my dear ex-xian,
I invite you to continue our dialog here or in your forum.
An information of this recent experiment you can find here:
http://drauh.typepad.com/blog/2004/04/
http://www.kathryncramer.com/wblog/archives/000530.html
Since Quantum elements are both wave and particle, no one of these properties can completely disappear, so I do not see any new point in this recent experiment.
Then you don't follow what's happening. Apparantly, this is the first time that light has been both a wave and a particle and not either/or.
Anyway, it hasn't been published, it hasn't been repeated. It's very preliminary, yet you insist on making dogmatic, authoritative statements. This is indicitive of your way of rejecting established science out of hand in order to favor your own bizarre theories.
With all due respect, I'll wait for real scientists who have spent decades doing this to decide rather than taking your opinion.
About your theories. I'm getting ready for graduate school next month, which I'm spending all my free time doing actual mathematics. I've responded to your posts again and again. You ignore what I write (the construction of the reals w/o using points--which invalidates your theory), take my words and use your own defintion (the "well-defined" bit), or you totally misunderstand what I write and put your own odd spin on it (my example of a delta-epsilon proof).
ex-xian
Jul27-04, 05:58 PM
Yes, it developed by people who where not forced by external methods or exteral reasons or lack of time.
Not all of them.
You've never even come close to showing this.
ex-xian
Jul27-04, 06:09 PM
you're essentially limiting the structure of your numerical system by this "entity" we call infinity. now, infinity is a useful term, we use it frequently to extend expressions out to incredible precision, but since the term itself is not closed, there lacks a decided "conclusion" for formulas utilizing infinity. thus any formulas that use infinity are relegated to being approximations, albeit really damn good ones.
the limit of a sequence is something that is NEVER reached. hence defined as such 1 IS the limit of .9+.09+.009...
but from that statement above it's logically sound that since it is NEVER reached, the sequence can NEVER be equal to 1.
the flaw is in the system, whether we choose to fix it or ignore it is the question...
There's no flaw in the system. When I was taking my calculus classes, my teachers always made a point to emphasize that when we say a sequence or series equals a number, we really mean that the sequence has limit of that number. Saying "equals" is just shorthand. That's not to say, however, that 0.99... is not equal to 1.
Here's a really simple way of showing that 0.99... = 1 that doesn't involve limits.
Clearly, 1/3 = 0.333...
So 3(1/3) = 3(0.333...)
Thus 1 = 0.999...
This is not the first time that both wave and particle properties of a quantum element, simultaneously appearing in a physical experiment.
When both stils are opened and we check the photons after they passé both of them, we can change gradually the wave picture to a particle picture, and vice versa.
ex-xian
Jul27-04, 06:21 PM
This is not the first time that both wave and particle properties of a quantum element, simultaneously appearing in a physical experiment.
When both stils are opened and we check the photons after they passé both of them, we can change gradually the wave picture to a particle picture, and vice versa.
Thanks, I think everyone knows this. And this is totally not what I'm talking about. You should actually read up on this topic before you try to discuss it anymore.
ex-xian,
I'll be glad to get your opinion on http://www.physicsforums.com/showpost.php?p=267089&postcount=101
ex-xian
Jul27-04, 06:26 PM
ex-xian,
I'll be glad to get your opinion on http://www.physicsforums.com/showpost.php?p=267089&postcount=101
My opinion is that it's juvevile and wrong.
Assume that .99.. < 1. Then 1/3(.999) < 1/3 (1), and .33... < 1/3. A contradiction, therefore .999 >/= 1. By a similiar argument, .99... is not > 1. Therefore .99.. = 1.
Please, instead of citing old posts or referncing your own papers, actually address what I wrote if you want to prove me wrong.
Please give your detailed explanation why do you think it is wrong?
ex-xian
Jul27-04, 06:34 PM
Please give your detailed explanation why do you think it is wrong?
See my edit. Also, I find that you saying that your fractal analogy is better or less abstract than a rigorous mathematical proof to be absurd in the highest degree.
Are you relied upon by employers and applicants alike to certify you know what you claim to know? And you do realize there's a difference between being forced to pass examination to receive a diploma and having ideas forced on you?
Can you please show how I force you to agree with me?
You refuse to participate in a discussion that don't involve your ideas, and when you were allowed to post in the math forum, you attempted several times to take over threads where mathematics was being discussed.
If we use a structural point of view in this case, then 0.9999... is a one dimensional path of a base 10 fractal, that exists upon infinitely many scale levels that cannot reach 1.
0.999... is not a "one dimensional path". You've provided no definition of terms "base 10 fractal", "infinitely many scale levels". And obviously, since there is no definition available for these terms, you obviously cannot know that it "cannot reach 1".
Also we can say that 0.999... = 0.9+0.09+0.009+0.0009+... and we can clearly see that this sequence cannot reach 1.
What sequence? 0.999... is a decimal number. 0.9+0.09+0.009+... is an infinite sum (which equals 1, which can easily be computed from the fact that this is a geometric series) In any case, it is not clear that "this sequence cannot reach 1".
Therefore 0.999... < 1.
As you have not made any valid statements yet, you cannot conclude that 0.999... < 1.
Another example:
Please look at this beautiful Koch Fractal http://members.cox.net/fractalenc/fr6g6s.577m2.html
Now let us say the there is a 1-1 map between each fractal level of 0.9999... to each different blue level of Koch Fractal.
0.999... is not a fractal, and I'm fairly sure there is no such term as "fractal level". Furthermore, you are merely saying (aka assuming) there is a 1-1 map between two things; unless you can show this assumption is true, you cannot be sure that any of your conclusions are true.
0.9999... = 1 if and only if we cannot find anymore a 1-1 map between some 0. ....9 to some Koch Fractal blue level
What is 0. ....9? And you've given no reason why this statement should be true.
Since Koch Fractal can be found in infinitely many blue levels and each blue level has a 1-1 map with some 0. ....9 fractal level, then we can conclude that 0.999... < 1.
Ignoring what comes before "each blue level", I think this actually logically follows from your previous statements. However, you've suggested no reason why your previous statements might be true.
Also we can say that 0.999... = 1 if and only if the outer contour of this multi-leveled Koch Fractal can be a smooth curve with no sharp edges.
You've given no reason why we can say that.
It is clear that the outer contour line is not a smooth contour in any arbitrary examined scale level.
It is true that the boundary of the Koch snowflake is not smooth. You've not given any definition of "scale level".
Therefore 0.999... < 1.
Again, this follows from previous statements, but you've given no reason to think those previous statements are true.
From this model you also can understand what is a "leap".
No, I cannot.
In short, any transition between a non smooth curve to a smooth curve, cannot be done but by a phase transition leap that also can be described by a smooth_XOR_no-smooth connection.
You've not provided definitions for "Transition", "phase transition leap", or "smooth_XOR_no-smooth connection".
This model is better than any "abstract" mathematical definition, which leads us to "prove" that 0.9999... = 1.
Better in what way?
Also by this "proof" we simply ignore infinitely many information forms that can be found in 0.9999... fractal.
Since there is nothing about the real numbers called "infinitely many information forms" nor is 0.999... a fractal, ignoring these is a good thing.
Now think how many information forms are ignored by this trivial and sterile approach of standard Matt (oops, Math).[/quote]
I can't, seeing how you've not said what you mean by "information form".
See my edit. Also, I find that you saying that your fractal analogy is better or less abstract than a rigorous mathematical proof to be absurd in the highest degree.
Your "rigorous" proof depends on excluded-middle black_XOR_white reasoning.
therefore you cannot deal with the complexity of 0.999... case.
Only Included-middle reasoning can deal with the complexity of this fractal.
In short, your "rigorous" proof is nothing but the image of your trivial black_XOR_white reasoning method.
Hurkyl,
also please read this http://www.geocities.com/complementarytheory/9999.pdf to understand why I call 0.999... a path of a fractal.
To what proof are you referring?
Do you really have no simple ability to conclude that the base value expansion method is actually a fractal?
Please look at pages 3,4 of http://www.geocities.com/complementarytheory/9999.pdf
matt grime
Jul28-04, 04:56 AM
the limit of a sequence is something that is NEVER reached. hence defined as such 1 IS the limit of .9+.09+.009...
presumably excpeting say, the constant sequences, and the eventually constant sequences then.
you shold stop, you're just showing up your ignorance.
Assume that .99.. < 1. Then 1/3(.999) < 1/3 (1), and .33... < 1/3. A contradiction...
There is no contradiction here beacue 0.333... < 1/3 for the same reasons that 0.999... < 1, which clearly can be understood here http://p071.ezboard.com/fthelanguageofmathematicsfrm2.showMessage?topicID= 2.topic
Well, then, here's a cute question:
If 0.333... isn't the decimal representation of 1/3, then what is?
A cute answer:
1/3 is not a fractal where 0.333... is a single path of a fractal.
They are not the same number exactly as 0.999... and 1 are not the same number.
When your logical system is based on an Included-Middle reasoning, the internal structural properteis of any mathematical element, cannot be ignored anymore.
matt grime
Jul28-04, 06:53 AM
doron, when you talk about "number" can we absolutely clarify that you mean the real numbers as we understand them in, say, calculus? Yes or No. because we are talking about these real numbers, and it is not clear what you are talking about since you've never offered an alternative explanation, hence our presumption you are using the real numbers as we know them already. You still appear to think that it is important what representative we use for a number, when it isn't: 1 and 2-1 are the same number, yet appear different, the same as 1/2 and 2/4. and if you're about to say something involving your new fangled terms, you must define what they are, and you must stop refering to things as the real numbers, when they aren't.
yet appear different,
Let us take your Idea and also say that you and I are the same person yet appear different, any number is actually zero yet appear different, and so on ,and so on...
By my system 'integral' and 'differential' are complementary properties, which simultaneously preventing/defining each other, and the result is infinitely many numbers that have internal complexity, which is based on symmetry/information complementary relations, that can be found on infinitely many different scales.
Multiplication and addition are their complementary internal operations.
The standard R members are based on one and only one building-block of my new system,
And its common property is: 0_redundancy_AND_0_uncertainty.
This building-block also standing in the basis of the excluded-middle reasoning.
Because of this limitation, a number is based only on the quantity concept, which through it 0.999… and 1 are look the same.
But when the minimal conditions for the existence of a number are both structural and quantitative, then 0.999… and 1 are different things.
matt grime
Jul28-04, 08:44 AM
Oh dear, is that the best you can do? That makes it very easy to prove that what you are talking about aren't the real numbers, that's fine, but just don't say that they are [the real numbers] that's all. For instance, they could be thought of as representing some cauchy sequences, where they are different cauchy sequences, that doesn't stop them representing the same real number, in fact by definition, nothing more nor less, they must represent the same real number.
So, Doron, if it's a spade call it a spade, don't confuse objects like this, it only leads you to speak even more confusingly than you already do. And so all that you conclude applies only to your set of numbers not the real numbers. CF hurkyls post on your cardinal/organic number mistakes about 20 posts back.
You and I are different as human beings, yet if all that were required is a representative of a human being both of us are equivalent, in this arm wavy non-mathematical rubbish.
What you call real numbers are a shadow of my system.
Can you separate between something to its shadow?
CF hurkyls post on your cardinal/organic number mistakes about 20 posts back.
If you mean to these posts http://www.physicsforums.com/showpost.php?p=266256&postcount=77 http://www.physicsforums.com/showpost.php?p=267486&postcount=104 then since N,Q and R members are the shadows of my number system, I can talk about them by using my number system,
and this is exactly what I did here: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf
matt grime
Jul28-04, 10:38 AM
No, you can't, since in your system 0.999... and 1 are not equal, then you may not use them to talk about the real numbers where they are equal. Irrespective of your philosophical objections to that fact.
Just because S<T is a containment of sets does not imply that results about T mean anything about S.
A trivial example: In R there are no non-trivial ideals yet Z possesses many.
Incidentally, what do you mean by shadow? that's not a mathematical term, at least not in this context (it is in optimization). If you insist on using odd terms at least tell other people about them.
hello3719
Jul28-04, 10:47 AM
A cute answer:
1/3 is not a fractal where 0.333... is a single path of a fractal.
They are not the same number exactly as 0.999... and 1 are not the same number.
When your logical system is based on an Included-Middle reasoning, the internal structural properteis of any mathematical element, cannot be ignored anymore.
is 1/2 still equal to 0.500.... ?
Matt, think simple, if I say (and show) that standard system is the shadow of my system it means that I can explain your system by my system but you cannot explain my system by your system, because your system is no more than the quantitative shadow of my system.
is 1/2 still equal to 0.500.... ?
If you take 000... as no information then 1/2=0.5
hello3719
Jul28-04, 10:58 AM
so in your system the division operation will never yield a result with an infinite decimal expansion ?
What about pi ? how would you represent it ?
Well hello3719, there is no 'my system' here, because you can find the whole standard system as some particular case of this system.
For example, let us say that x,y system is a particular case of x,y,z system where z is ignored in this case.
We can do any x,y thing by an x,y,z system, but not vise versa.
My system can yields results with an infinite decimal expansions, for example please look at http://www.geocities.com/complementarytheory/9999.pdf pages 3,4.
Now for pi:
In the standard quantitative-only system, the internal information form of each number is ignored (by the analogy: we ignore z) ; therefore the fractal structure (that can be shown above in pages 3,4) of the decimal method is omitted and as a result, for example 1/3=0.33333...
But if any number is not less than quantitative/structural information form, we have the ability to distinguish between 1/3 and 0.333... by both quantitative and structural properties.
In short, we have more possibilities to do Math, and in the case of pi, each different base value define a new fractal of pi where the quantitative sum of each fractal is less then pi constant.
Also please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf including its entire links.
terrabyte
Jul28-04, 12:30 PM
even in the current system, .333... does not accurately represent the number 1/3.
1/3 is a closed expression. .333... is open ended to and beyond infinity. to make it short, 1/3 is complete and rational, .333... is a continuing process to infinity and as such is irrational.
hello3719
Jul28-04, 12:44 PM
even in the current system, .333... does not accurately represent the number 1/3.
1/3 is a closed expression. .333... is open ended to and beyond infinity. to make it short, 1/3 is complete and rational, .333... is a continuing process to infinity and as such is irrational.
you are not using the usual definition of "irrational", irrational merely means that it cannot be expressed as a ratio of 2 integers.
zeronem
Jul28-04, 12:55 PM
even in the current system, .333... does not accurately represent the number 1/3.
1/3 is a closed expression. .333... is open ended to and beyond infinity. to make it short, 1/3 is complete and rational, .333... is a continuing process to infinity and as such is irrational.
A rational number is a number that can be expressed as the ratio of two integers.
A rational number can be expressed as either a fraction, or as a decimal number. If the decimal representation of a number goes on forever without repeating any pattern, then that number is an irrational number.
In this case, .33333.... has a repeating pattern, and therefore is a rational number.
hello3719
Jul28-04, 12:56 PM
[QUOTE=Lama]But if any number is not less than quantitative/structural information form, we have the ability to distinguish between 1/3 and 0.333... by both quantitative and structural properties.
In short, we have more possibilities to do Math, and in the case of pi, each different base value define a new fractal of pi where the quantitative sum of each fractal is less then pi constant.
[QUOTE]
did you know that the "/" sign only represents the division operation ?
We can't talk about quantitive and structural properties when we compare an operation to its result.
Can you tell me how would you OBTAIN Pi ? You only showed how you would represent it not how you knew it was around 3.14.
terrabyte
Jul28-04, 01:15 PM
you are not using the usual definition of "irrational", irrational merely means that it cannot be expressed as a ratio of 2 integers.
exactly. CANNOT be expressed. 1/3 is a complete expression. .333... is an incomplete expression. it cannot EVER be completed, to infinite digits or beyond.
the LIMIT of the expression .333... is 1/3 which means it will NEVER reach that number, no matter how many digits you extend it out to.
get it yet?
hello3719
Jul28-04, 01:30 PM
exactly. CANNOT be expressed. 1/3 is a complete expression. .333... is an incomplete expression. it cannot EVER be completed, to infinite digits or beyond.
the LIMIT of the expression .333... is 1/3 which means it will NEVER reach that number, no matter how many digits you extend it out to.
get it yet?
LOL. Exactly .333... CAN BE EXPRESSED as a ratio of integers. Seems you have a flaw in your logic.
Pi is the ratio between circle's perimeter and circle's diameter.
We can take this ratio and represent it in infinitely many different quantitative/structural ways ,where '/' is not a division operation but a relation sign.
But no one of the representation is equal to pi constant.
hello3719
Jul28-04, 01:34 PM
Pi is the ratio between circle's perimeter and circle's diameter.
We can take this ratio and represent in infinitely many different quantitative/structural ways ,where '/' is not a division operation but a relation sign.
What do you mean by '/' is a relation sign ? define relation sign
hello3719
Jul28-04, 01:36 PM
Pi is the ratio between circle's perimeter and circle's diameter.
Please devise a method to find it using your more "complete system".
If you think that when I write quantitative/structural I mean quantitative divided by structural then to make it clearer I explained that '/' is used here to say that there is a mutual interactions between the two concepts.
Please devise a method to find it using your more "complete system".
1) No consistent system can be a complete system, and my system is incomplete.
2) Multi leveled parallel/serial Turing machine model, which uses any useful combination of my information building-blocks , instead of using only the binary buiding-block (which standing in the basis of the current Turing machine).
hello3719
Jul28-04, 02:00 PM
If you think that when I write quantitative/structural I mean quantitative divided by structural then to make it clearer I explained that '/' is used here to say that there is a mutual interactions between the two concepts.
no, I meant when you write 1/3 we are talking about a division, and .33... is THE RESULT of the operation. This means that you can't compare the result with the operation.
hello3719
Jul28-04, 02:06 PM
1) No consistent system can be a complete system, and my system is incomplete.
2) Multi leveled parallel/serial Turing machine model, which uses any useful combination of my information building-blocks , instead of using only the binary buiding-block (which standing in the basis of the current Turing machine).
I didn't say that it was complete, i said that you assumed before that our "actual" system is a mere shadow of yours , didn't you ?
If yes, then could you please show me in detailed steps how would you obtain Pi.
1/3 is not just an arithmetical operation of 1 divided by 3, but it is also the Q member 1/3.
The same we can say about pi (which is a R member), for example:
If the diameter is 1 then pi is pi/1.
Now you, can take 1/3 or pi and use the base value expansion method, which is a fractal way to represent numbers.
But because some fractals can be found in infinitely many different scale levels, then from the quantitative point of view they never fully represent 1/3 or pi.
hello3719
Jul28-04, 02:13 PM
1/3 is not just an arithmetical operation of 1 divided by 3, but it is also the Q member 1/3.
The same we can say about pi, for example:
If the diameter is 1 then pi is pi/1.
well the Q set existance is based on the fact that the division operation is admitted in the system using the integers thus giving 1/3 a unique identity.
(saying "1/3 is not just an arithmetical operation of 1 divided by 3, but it is also the Q member 1/3." means nothing at all)
hello3719
Jul28-04, 02:15 PM
The same we can say about pi, for example:
If the diameter is 1 then pi is pi/1.
That's the best you can do ?
Q are called rational numbers because these numbers are defined by the ratio that can be found between at least two integers, where the irrational numbers cannot be defined in this way, so as you see '/' is also used for a ratio.
Therefore (for example) 1/3 means the ration between 1 and 3, which is the rational number 1/3.
hello3719
Jul28-04, 02:20 PM
But because some fractals can be found in infinitely many different scale levels, then from the quantitative point of view they never fully represent 1/3 or pi.
Who said that we want to use fractals to represent number theory concepts ?
The concepts we are talking about use some more basic defintions than the fractals' one.
terrabyte
Jul28-04, 02:23 PM
LOL. Exactly .333... CAN BE EXPRESSED as a ratio of integers. Seems you have a flaw in your logic.
not so.
.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.
even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.
.333... is NOT 1/3
there is no flaw in my logic. just a failure in realization on your part.
hello3719
Jul28-04, 02:24 PM
Q are called rational numbers because these numbers are defined by the ratio that can be found between at least two integers, where the irrational numbers cannot be defined in this way, so as you see '/' is also used for a ratio.
ratio and division is the exact same thing in mathematics, so why are you saying "also" ?
Who said that we want to use fractals to represent number theory concepts ?
The concepts we are talking about use some more basic defintions than the fractals' one.
If you use the base value expansion methods then you use a part of a fractal, whether you want it or not.
In this case you cannot ignore anymore the properties of a fractal.
hello3719
Jul28-04, 02:29 PM
not so.
.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.
even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.
.333... is NOT 1/3
there is no flaw in my logic. just a failure in realization on your part.
OMG. Now this proves that you DON'T know what infinity means neither what rational and irrational means.
"the irrational computational result of dividing 1 by 3." this doesn't make sense at all. I will say it one more time, irrational simply AND ONLY characterizes a number THAT CAN'T be expressed as a ratio of 2 integers. It is a simple as this.
terrabyte
Jul28-04, 02:32 PM
yes. and i've said it a few times as well. .333... IS NOT properly expressed by 1/3
step 1. take the number 6
step 2. Divide it by 3.
step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)
compare results.
notice anything? :|
ratio and division is the exact same thing in mathematics, so why are saying "also" ?
Because 1/3 is also known as the constant symbol of the number that is produced by this division operation.
hello3719
Jul28-04, 02:39 PM
If you use the base value expansion methods then you use a part of a fractal, whether you want it or not.
In this case you cannot ignore anymore the properties of a fractal.
refering to http://mathworld.wolfram.com/Fractal.html
"A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales"
Did you know that "scales" are consequences of the arithmetic operation of division AND NOT VICE-VERSA. This means that you can stop using fractals to explain your ideas and just use more basic concepts.
hello3719
Jul28-04, 02:43 PM
yes. and i've said it a few times as well. .333... IS NOT properly expressed by 1/3
step 1. take the number 6
step 2. Divide it by 3.
step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)
compare results.
notice anything? :|
k we have a problem of communication it seems. To clear things out,
When i write .333... i mean there is an INFINITY of 3's. so by definition of infinity i can conclude that 6 * .333... is equal to 2. So what's the problem ?
Did you know that "scales" are consequences of the arithmetic operation of division AND NOT VICE-VERSA. This means that you can stop using fractals to explain your ideas and just use more basic concepts.
Not correct, we can define scales also by mutiplication.
hello3719
Jul28-04, 02:48 PM
Because 1/3 is also known as the constant symbol of the number that is produced by this division operation.
Logical flaws as always. :frown:
1/3 means in english " we are dividing 1 by 3"
so it doesn't have a double identity, like i said before don't compare the result with operation.
zeronem
Jul28-04, 02:49 PM
not so.
.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.
Like I said again, an irrational number has no repeated pattern of integers in decimal notation of a ratio.
1/3 has a repeating pattern of 3's, which makes it rational. 1/7 has a repeating pattern of .142857.... which makes it rational. However Pi, being the ratio of the circle's circumference to diamter, is an irrational number because it has no repeating pattern of integers in decimal notation.
\frac {1}{3} = .\overline {3}
\frac {1}{7} = .\overline {142857}
hello3719
Jul28-04, 02:51 PM
Not correct, we can define scales also by mutiplication.
well sure we can, since division is just going backwards in the multiplication.
Still a scale ISN'T as basic of a definition as is division or multiplication nor , as a consequence, is fractal.
You didn't prove anything.
Logical flaws as always.
1/3 means in english " we are dividing 1 by 3"
so it doesn't have a double identity, like i said before don't compare the result with operation.
Let us make it simpler.
I choose @ as the result of 1/3 so 1/3 = @.
But If you use the base value expansion method, then you use a part of a fractal, whether you want it or not.
In this case you cannot ignore anymore the properties of a fractal.
And these properties can be cleary understood by:
-----------------------------------------------------------------
If we use a structural point of view in this case, then 0.9999... is a single path of a base 10 fractal ( http://www.geocities.com/complementarytheory/9999.pdf pages 3,4 ), that exists upon infinitely many scale levels that cannot reach 1.
Also we can say that 0.999... = 0.9+0.09+0.009+0.0009+... and we can clearly see that this infinitely long addition cannot reach 1.
Therefore 0.999... < 1.
Another example:
Please look at this beautiful Koch Fractal members.cox.net/fractalenc/fr6g6s.577m2.html
Now let us say the there is a 1-1 map between each fractal level of 0.9999... to each different blue level of Koch Fractal.
0.9999... = 1 if and only if we cannot find anymore a 1-1 map between some 0.000...9 to some Koch Fractal blue level.
Since Koch Fractal can be found in infinitely many blue levels and each blue level has a 1-1 map with some 0.000...9 fractal level, then we can conclude that 0.999... < 1.
Also we can say that 0.999... = 1 if and only if the outer contour of this multi-leveled Koch Fractal can be a smooth curve with no sharp edges.
It is clear that the outer contour line is not a smooth contour in any arbitrary examined scale level.
Therefore 0.999... < 1.
From this model you also can understand what is a "leap".
In short, any transition between a non smooth curve to a smooth curve, cannot be done but by a phase transition leap that also can be described by a smooth_XOR_no-smooth connection.
This model is better than any "abstract" mathematical definition, which leads us to "prove" that 0.9999... = 1.
Also by this "proof" we simply ignore infinitely many information forms that can be found in 0.9999... fractal.
Now think how many information forms are ignored by this trivial and sterile approach of standard Math.
well sure we can, since division is just going backwards in the multiplication.
Still a scale ISN'T as basic of a definition as is division or multiplication nor , as a consequence, is fractal.
You didn't prove anything.
By these two operations we get the same result, which is a fractal.
Therefore the fractal property is the invariant and these operations are only tools to explore it.
To cealrly see and understand this fractal please look at http://www.geocities.com/complementarytheory/No-Naive-Math.pdf page 5.
ex-xian
Jul28-04, 03:36 PM
yes. and i've said it a few times as well. .333... IS NOT properly expressed by 1/3
step 1. take the number 6
step 2. Divide it by 3.
step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)
compare results.
notice anything? :|
step 1. Take the number 1.
step 2. Divide it by 3.
step 3. Keep dividing
...
...
...
step 4,653. Keep dividing
...
...
...
step 3039209823752820. Keep dividing.
Notice anything?
ex-xian
Jul28-04, 03:38 PM
Your "rigorous" proof depends on excluded-middle black_XOR_white reasoning.
therefore you cannot deal with the complexity of 0.999... case.
Only Included-middle reasoning can deal with the complexity of this fractal.
In short, your "rigorous" proof is nothing but the image of your trivial black_XOR_white reasoning method.
I used a proof by contradiction which you attempted (and failed) to do. I used "<" which you've tried to use again and again.
You reply with meaningless gibberish.
Will you actually address what is posted rather than ramble in you private littel world of non-math.
ex-xian
Jul28-04, 03:41 PM
A cute answer:
1/3 is not a fractal where 0.333... is a single path of a fractal.
They are not the same number exactly as 0.999... and 1 are not the same number.
When your logical system is based on an Included-Middle reasoning, the internal structural properteis of any mathematical element, cannot be ignored anymore.
You've already established that you can't do analysis. Surely you can do long division and see how inane this statement is.
ex-xian
Jul28-04, 03:50 PM
not so.
.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.
So you're just going to use a unique definition of irrational number? That's fine, but don't expect anyone who actually does math to understand you (or take you seriously).
even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.
This is just silly. If there are an infinite number of 3's you can't add anymore to the end.
.333... is NOT 1/3
there is no flaw in my logic. just a failure in realization on your part.
You're right, there's no flaw in your logic, you just don't understand what irrational and rational numbers are.
I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.
If there's a 3 in every position to the right of the decimal place, then where, pray tell, are you going to add one more digit?
terrabyte
Jul28-04, 04:13 PM
i'm going to add 1 more digit wherever you "stop" at of course.
otherwise how do you expect to convey the "meaning" of your number to me?
1. we derive numerical meaning from the differences in number digits. 1 is clearly different from 2. 1.1 is slightly different from 1.2. the differences in these digits allow quantities to have meaning.
2. the quantity .333... has no meaning until it is brought into the realm of known quantities. hence it has to be definable within the boundaries of numbers within proximity to it AND be distiguishable as such. this may seem like a pithy statement with no meaning but hold on...
3. the number .333... for this number to be exactly 1/3 it must have infinite digits of 3. Infinite in the sense that they're unending, not in the sense that they're greater than all numbers.
4. define a number that is slightly less than this number. will .333...2 work? not really, that number is closer to 1/3 than .333... or maybe .333... where the number of digits 3 is (Infinity-1) <getting into cardinality with that but meh whatever>.
5. as you can see, because of the nature of digits stringing out to infinity it is impossible within the current system to define the number in relation to other numbers within proximity.
terrabyte
Jul28-04, 04:15 PM
which is one more reason why it's irrational, not merely in the sense that it cannot truly be expressed as a ratio of two integers, but because the quantity is unplacable on the number line.
ex-xian
Jul28-04, 04:25 PM
i'm going to add 1 more digit wherever you "stop" at of course.
otherwise how do you expect to convey the "meaning" of your number to me?
1. we derive numerical meaning from the differences in number digits. 1 is clearly different from 2. 1.1 is slightly different from 1.2. the differences in these digits allow quantities to have meaning.
2. the quantity .333... has no meaning until it is brought into the realm of known quantities. hence it has to be definable within the boundaries of numbers within proximity to it AND be distiguishable as such. this may seem like a pithy statement with no meaning but hold on...
3. the number .333... for this number to be exactly 1/3 it must have infinite digits of 3. Infinite in the sense that they're unending, not in the sense that they're greater than all numbers.
4. define a number that is slightly less than this number. will .333...2 work? not really, that number is closer to 1/3 than .333... or maybe .333... where the number of digits 3 is (Infinity-1) <getting into cardinality with that but meh whatever>.
5. as you can see, because of the nature of digits stringing out to infinity it is impossible within the current system to define the number in relation to other numbers within proximity.
You really don't understand math do you? That's just sad.
i'm going to add 1 more digit wherever you "stop" at of course.
Stop what?
3. the number .333... for this number to be exactly 1/3 it must have infinite digits of 3. Infinite in the sense that they're unending, not in the sense that they're greater than all numbers.
There is no (right) end to .333...
terrabyte
Jul28-04, 04:48 PM
thus it will NEVER equal 1/3
thus it will NEVER equal 1/3
Can you exhibit a number between 0.333.... and 1/3?
terrabyte
Jul28-04, 04:58 PM
.333... is not a rational number.
can you exhibit a number slightly greater than or less than .333... ?
Yes, lots.
0.3 < 0.333... < 0.4
0.33 < 0.333... < 0.34
0.333 < 0.333... < 0.334
0.3333 < 0.333... < 0.3334
...
Pick any upper bound for "slightly", and I can find one of these inequalities such that the smaller and larger are slightly different than 0.333...
What is the point of this exercise?
terrabyte
Jul28-04, 05:11 PM
k expand those out to infinity. infinite precision, so-to-speak
all of those numbers cease to have distinction
terrabyte
Jul28-04, 05:14 PM
.333...2 and .333... and .333...4 are all the same number.
there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"
that's the purpose of the exercise
ex-xian
Jul28-04, 05:18 PM
.333...2 and .333... and .333...4 are all the same number.
there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"
that's the purpose of the exercise
.333...2 and .333...4 aren't numbers. They're products of your misunderstandings. If "..." is means inifinitly many, you can't have infinitely many 3's and then add a 2 or a 4.
.333...2 and .333... and .333...4 are all the same number.
No.... one of those is a number; the other two are gibberish.
Anyways, you've evaded my response. None of the approximations I intend to make have an infinite number of nonzero terms, however if you pick any positive value as an allowable tolerance for error, one of my approximations will be within this tolerance.
It seems you are trying to ask me to tell you what the "next" decimal number is, but there is no such thing. (If there was a "next" number, then what happens if I take their midpoint?)
And I'll ask again, what does all of this have to do with whether 1/3 = 0.333...?
terrabyte
Jul28-04, 05:21 PM
then you can't define a number "slightly" greater than or "slightly" less than that number, and it has no meaning.
hence irrational
then you can't define a number "slightly" greater than or "slightly" less than that number, and it has no meaning.
You said that 1.1 is slightly different than 1.2.
0.333 is quite a bit closer to 0.333... than 1.1 is to 1.2.
Were you wrong earlier when you said 1.1 is slightly different than 1.2?
And I'll ask again, what does this have to do with 0.333... = 1/3?
zeronem
Jul28-04, 05:33 PM
With your logic terrabyte, you would equally assert that 1/7 = .142857....... is a irrational number. Once again this proves your mis-understanding in Math.
You fail to realize that some rational numbers particularly 1/3 and 1/7 in there decimal notation run through a pattern of integers continuously. However irrational numbers have no significant pattern in there decimal notation.
1/3 = .3333333333333333333333333333333333333333333333333 33333333333333333
1/7 = .1428571428571428571428571428571428571428571428571 42857142857142857
Pi = 3.141592654,
There is no pattern in Pi, therefore it is irrational and such a number cannot be described as a ratio of two integers. We have only found fractions that closely resemble Pi that our rational. like 22/7 = 3 + 1/7
terrabyte
Jul28-04, 05:34 PM
No.... one of those is a number; the other two are gibberish.
Anyways, you've evaded my response. None of the approximations I intend to make have an infinite number of nonzero terms, however if you pick any positive value as an allowable tolerance for error, one of my approximations will be within this tolerance.
It seems you are trying to ask me to tell you what the "next" decimal number is, but there is no such thing. (If there was a "next" number, then what happens if I take their midpoint?)
And I'll ask again, what does all of this have to do with whether 1/3 = 0.333...?
the distinction or uniqueness of numbers has infinite precision, meaning no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.
that distinction is obliterated by using an infinite string of digits to express a quantity.
case in point our little exercise to determine the closest numbers to .333... the exercize would have worked for any number, i was just using it to illustrate that when we take digits and extend them out to an arbitrar "infinity" we lose the distinction that determines WHAT numbers actually are.
without incrementation, the ability to add MORE digits to further define value, you hit a logical dead end.
this is completely due to the structuring of infinity
you ask how this has to do with .333... and 1/3 well it goes way beyond those two numbers and their properties and into the intrinsic values and quantities of ALL numbers, and the limitations of our mathematical and numerical system.
the distinction or uniqueness of numbers has infinite precision, meaning no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.
that distinction is obliterated by using an infinite string of digits to express a quantity.
Would you care to prove your point? Give me two decimals representing different quantities, and I'll find a decimal representing a number between them. (One way is I could simply add them and divide by 2)
terrabyte
Jul28-04, 05:50 PM
that's the point.
when expanded out to infinite digits you can NOT create different quantities. there is no longer any room for incrementation.
we just did the exercise expanding .32 <.333... <.34 | .332 < .333... < .334
you ran into the same roadblock
hello3719
Jul28-04, 05:58 PM
how old are you terrabyte ? It seems you don't understand anything.
hello3719
Jul28-04, 06:02 PM
.333...2 and .333... and .333...4 are all the same number.
there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"
that's the purpose of the exercise
Still don't understand the definition of infinity, very sad.
terrabyte
Jul28-04, 06:02 PM
how old are you to make comments that have no basis on what's being debated?
where is your facts to back up this "argument" that i don't understand anything?
there is no longer any room for incrementation.
Which is fine, because the real numbers don't have incrementation.
You claimed:
the distinction or uniqueness of numbers has infinite precision, meaning no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.
that distinction is obliterated by using an infinite string of digits to express a quantity.
I ask you to present two decimals representing different quantities for which I cannot find a decimal between them.
terrabyte
Jul28-04, 06:03 PM
instead of hollow "very sad" remarks how about you explain what YOU understand infinity to be and how it fits into the picture.
if you have nothing relevant to say, by all means, keep quiet
hello3719
Jul28-04, 06:05 PM
how old are you to make comments that have no basis on what's being debated?
where is your facts to back up this "argument" that i don't understand anything?
You can't even accept the definition of irrational neither infinity. Second of all mathematics isn't a question of "debate". This is not a politics forum.
We sure seem to have a lot of arrogant, ignorant kids on the forum all of a sudden. Are you all friends? Why are you here?
- Warren
hello3719
Jul28-04, 06:09 PM
you think that .333...4 is a number where there's an infinite of 3's. Tell me how can you add a 4 if there is a NON-ENDING quantity of 3's?
terrabyte
Jul28-04, 06:13 PM
real numbers DO have incrementation AND distinction.
every digit 0-9 that is different from one in the same location of another number DEFINES difference.
this is intrinsic AND integral to the structure and utilization of the number system.
when you put a limit on the ability of that system to function of course discrepancies will arise. which is exactly what "Infinity" is. a Limit.
don't come back and say it ISN'T, hello3759 (i know that's going to be your next pithy argument) because it most certainly IS. if it isn't define me a number greater than it.
there isn't one is there?
real numbers DO have incrementation
On what grounds do you suggest that? Remember your own statement:
no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.
If there was incrementation, then what comes between a number and its increment?
hello3719
Jul28-04, 06:20 PM
don't come back and say it ISN'T, hello3759 (i know that's going to be your next pithy argument) because it most certainly IS. if it isn't define me a number greater than it.
there isn't one is there?
Greater than .333.... ? no problem
why not 0.4?
terrabyte
Jul28-04, 06:20 PM
a smaller increment.
always
terrabyte
Jul28-04, 06:20 PM
greater than infinity
hello3719
Jul28-04, 06:22 PM
greater than infinity
infinity isn't a number
terrabyte
Jul28-04, 06:24 PM
semantics
say something that has meaning, Hello3719, PLEASE
hello3719
Jul28-04, 06:27 PM
why doesn't it has meaning ?
it's true it isn't a number unless you can prove it
terrabyte
Jul28-04, 06:30 PM
fine, you stick to that then it's absolutely impossible for you to utilize it as a number.
those are your guidelines.
now express the quantity of how many digits .333... has with values of 3?
ex-xian
Jul28-04, 06:30 PM
terrabyte, how much formal mathematics have you actually had? You seem to be a bright, but ignorant, person.
when expanded out to infinite digits you can NOT create different quantities. there is no longer any room for incrementation.
Could you explain this?
Clearly, 0.4 < 0.5 and 0.3983984 < 0.4 and .0000000000003 < 0.00000000000000003, right?
In the case of 0.333.... and 0.5454... how is not clear that the latter is greater than the former?
If this isn't what you meant, could you elaborate?
Also, is 1/2 = .5 = .5000....?
How do you technically define rational and irrational numbers?
ex-xian
Jul28-04, 06:32 PM
fine, you stick to that then it's absolutely impossible for you to utilize it as a number.
those are your guidelines.
now express the quantity of how many digits .333... has with values of 3?
It has as many digits as there are natural numbers.
terrabyte
Jul28-04, 06:33 PM
It has as many digits as there are natural numbers.
fine, how many natural numbers are there then?
you're just digging a deeper hole :D
hello3719
Jul28-04, 06:34 PM
fine, you stick to that then it's absolutely impossible for you to utilize it as a number.
those are your guidelines.
now express the quantity of how many digits .333... has with values of 3?
by notation i am using " ... " to mean never ending quantity of 3's.
why would you say that "never ending" is a number ?
Keep focus.
ex-xian
Jul28-04, 06:35 PM
fine, how many natural numbers are there then?
you're just digging a deeper hole :D
Countably infinitly many....this is going in a cirlce, and I don't see your point.
how many natural numbers are there then?
The natural numbers have cardinality \aleph_0 and order type \omega.
terrabyte
Jul28-04, 06:37 PM
terrabyte, how much formal mathematics have you actually had? You seem to be a bright, but ignorant, person.
Could you explain this?
Clearly, 0.4 < 0.5 and 0.3983984 < 0.4 and .0000000000003 < 0.00000000000000003, right?
In the case of 0.333.... and 0.5454... how is not clear that the latter is greater than the former?
If this isn't what you meant, could you elaborate?
Also, is 1/2 = .5 = .5000....?
How do you technically define rational and irrational numbers?
rational is expressable EXACTLY as a ratio of two integers. in the case of 1/3 the computation is NOT exact, since it never ends up to and beyond infinity.
thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.
as it stands there IS NO rational decimal form for .333...
and trailing digits 0 such as .50000 <-- have no significance unless used as incremental spacers for digits with value such as .500001 <--
thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.
By definition, 1/3 is the solution to 3 * x = 1.
3 * .333.... = .999.... = 1
Thus .333.... = 1/3.
terrabyte
Jul28-04, 06:39 PM
Countably infinitly many....this is going in a cirlce, and I don't see your point.
my point is. it makes no sense to say "infinity is not a number" when it serves as a convenient <yet weightless> argument in a dialog yet proceed to USE it as a quantity in further discussions.
hypocrisy is sometimes deemed the worst of vices...
hello3719
Jul28-04, 06:42 PM
rational is expressable EXACTLY as a ratio of two integers. in the case of 1/3 the computation is NOT exact, since it never ends up to and beyond infinity.
thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.
as it stands there IS NO rational decimal form for .333...
and trailing digits 0 such as .50000 <-- have no significance unless used as incremental spacers for digits with value such as .500001 <--
Could you please explain what "EXACT" means in mathematics ?
as i said before , "..." in 0.333... means that there is an infinity of 3's.
Do the division, it is a NON EVER ENDING division where we get a 3 in each step.
ex-xian
Jul28-04, 06:45 PM
rational is expressable EXACTLY as a ratio of two integers. in the case of 1/3 the computation is NOT exact, since it never ends up to and beyond infinity.
What do think a ratio is? 1/2 means 1 divided by 2 and 1/3 means 1 divided by 3. Why does the operation of division bother you? Why does division produce something "bad" when you have 1/3 and something "not bad" when you have 1/2? You seem to have a problem with infinity. Why? If you think there's something wrong with infinitly many digits, then the burden is upon you to demonstrate why.
thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.
as it stands there IS NO rational decimal form for .333...
Can you give a formal proof?
and trailing digits 0 such as .50000 <-- have no significance unless used as incremental spacers for digits with value such as .500001 <--
Why not? What is special about the digit 0 there isn't there for 3?
Also, you ignored my other questions. I'll repeat them.
How much mathematical training have you actually had?
Clearly, 0.4 < 0.5 and 0.3983984 < 0.4 and .0000000000003 < .00000000000000003, right?
In the case of 0.333.... and 0.5454... how is not clear that the latter is greater than the former?
terrabyte has been banned. He used to call himself ram1024, ram2048, ram4096, etc. We have banned this person three times already, yet he still does not seem to understand that he is not welcome here, and nor are his pointless threads.
If any of you see activity that you suspect is due to the same individual, please let the staff know so we can deal with it.
- Warren
ex-xian
Jul28-04, 06:52 PM
my point is. it makes no sense to say "infinity is not a number" when it serves as a convenient <yet weightless> argument in a dialog yet proceed to USE it as a quantity in further discussions.
When call something a number, we mean that it is a natural number, an/a irr/rational number, a complex number, a quaternion, etc. Infinity is none of these, but it's still a quantity. Why is this a problem?
hypocrisy is sometimes deemed the worst of vices...
Intersting that you accuse on hypocricy while you were the person complaining about "immature" behavior.
Can you exhibit a number between 0.333.... and 1/3?
This is a very good question.
My answer is:
Because of the duality of any R member (which clearly can be shown here http://www.geocities.com/complementarytheory/No-Naive-Math.pdf in page 5) at least the entire infinite fractal representarions of 0.333... cen be found between fractal 0.333... and constant 1/3.
To make it cealer, if a=1 and b is any posivite R member < 1 and > 0 , then fractal b*0.333... can be found infinitely many times between a*0.333... and a*1/3.
By definition, 1/3 is the solution to 3 * x = 1
3 * .333.... = .999.... = 1
Thus .333.... = 1/3.
If x is a fractal then your definition does not hold exactly as x/0 = 1 does not hold.
x holds only if x=1/3.
If you say that 1/3 is not a number but an operation between two numbers, then we can do this:
1/3 = @, therefore 3 * x = 1 iff x = @.
Form these examples we can learn (in my opinion) that there must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
I used a proof by contradiction which you attempted (and failed) to do. I used "<" which you've tried to use again and again.
You reply with meaningless gibberish.
Will you actually address what is posted rather than ramble in you private little world of non-math.
1) the contradiction that you find and you use in your proof, depends on the logical reasoning that you use; therefore your proof is no more then a 'must-have' result of the basic laws of excluded-middle reasoning, which standing in the basis of your reasoning method.
In short, there is no one and only one universal law that leads us to find a one and only one possible result.
2) My logical reasoning is based on an included-middle reasoning, where the contradiction concept does not exist because two opposites are simultaneously preventing/defining their middle domain.
Therefore I cannot fail to produce a proof by contradiction in an included-middle reasoning framework.
The Included-middle reasoning framework and also its relation to an excluded-middle reasoning, is clearly and simply shown here: http://www.geocities.com/complementarytheory/CompLogic.pdf
3) If you read carefully http://www.geocities.com/complementarytheory/No-Naive-Math.pdf then I think that you will understand what is the meaning of ‘<’ or ‘>’ in my framework.
4) I think that you will not be able to understand my system, if you continue to use your basic aggressive attitude, which can be shown by the expressions that you use in your replies.
I am here for communication, not for war.
Something about understanding:
In my opinion, to understand something is to be simultaneously in and out of the framework of the explored thing.
It means that no-thing can be really understood only within its framework.
I think that this insight standing in the basis of any good scientific approach, because from one hand it gives us the motivation to find more general frameworks, and on the other hand we know that this is a 'never ending story' of “built-in” evolution process.
hello3719
Jul29-04, 07:44 AM
Ok, then please write down all the axioms you are using.
It is too general, please be more specific, thank you.
Ok, then write down all the axioms you are using.
Why are you so aggresive? is it a hard thing for you to say 'please'? :wink:
matt grime
Jul29-04, 07:59 AM
what on earth does it mean to 'in the framework' of sometihng in order to understand it? i understand how a cd player works, but i'm not in any sense in a framework of a cd player... the mind boggles.
Direct experience baby, not just theory.
matt grime
Jul29-04, 09:28 AM
oh, so how about theoretical physics such as black holes or string theory? why must a theory be experienced directly? well, obviously it mustn't, but that's your problem. of course if you must have direct experience then you can't possibly use the real numbers or any such since they can't be directly experienced (they don't exist, in any physical sense, and we all know how subjective you can get).
Matt,
Physics has two legs, the theoretical leg and the experimental leg, and they both complement each other.
Only by these two legs we can say that we understand something.
When we deal with ‘theory_only’ system then ‘in’ and ‘out’ frameworks are the inductive/deductive interactions of our ideas, or our local/global points of view.
Matt,
I thought about something and I'll be glad to know your opinion.
It goes like this:
When two violins in the same room are tuned with each other, if we play on one of them we find that the strings of the other violin are also vibrate.
Now let as say that intuition is our tuned instrument, and if a person expresses its intuitions by developing a way of thinking, the people that embrace this way of thinking probably share the same intuitions.
On the basis of these common intuitions a community can be established.
Let us say that this community is the first organization that deals with some part of the human knowledge, so in these early stages this community has no comparators on this part of the human knowledge.
Quickly this community becomes the most developed organization, which holds this part of the human knowledge, and other parts of human civilization look at this organization as the one and only one possible intuition which standing in the basis of a one and only one way (school) of thought (and I am not talking about variations, which are actually different brunches of the same way of thought, or the same school of thought if you like).
2500 years are passing and this school of thought survives because of two main reasons:
1) This way of thought was fitting to the needs of the human civilization along these 'slow' (linear) years.
2) Any other alternative intuitions (if they where at all) where put aside because:
a) They where not useful in their time.
b) And if they where useful and also a real alternative to the current school, then the current school used its power and money to block this alternative intuition by forcing its educational methods on the public.
We have to understand that intuitions cannot be learned, but a lot of external power can distort them until they lost their ability to be the source of a new school of thought.
The 120 century is the time where our civilization moved from linear time to non-linear time.
In this time the power of few holds the destiny of our civilization, and most of their power is based on the technical abilities that where developed by this school of thought, that was established 2500 years ago.
But our technical achievements, which are not balanced by another ways of thought, are like a government with no opposite.
We have learned that evolution needs diversity; otherwise we quickly get a dead planet.
The field of evolution in our non-linear time splits to "hardwhere" and "softwhere" parallel paths, where the hardwhere side is our technology and the softwhere side is our morality.
We can clearly see that there is no balance between the levels of these two paths, and this lack of balance in a non-linear time can quickly lead us to a dead-end street.
Therefore I think that we have to do the best we can to find the balance between our morality level and our technical abilities.
The first place that binds both paths is the language of mathematics.
In my opinion people how learn this powerful language, must first of all to develop their moral abilities by opening themselves to another intuitions which are not their intuitions and let them flourish in their communities.
By this way we develop our tolerance and learn how to live side by side, and if other intuitions are better then our intuition in this period of time, we do our best to help them flourish instead of trying our best to shut them down.
And we have the motivation to do that because we understand that we are all in the same boat.
My intuitions and ideas about the language of mathematics are different then the standard school of mathematics.
But in my opinion the most important difference, which I think fits to our non-linear time (more then the standard school) is that I include the mathematician cognition's ability to develop Math as a part of the mathematical research.
By this self-reference attitude I hope to develop the gateway that can connect between our moral abilities to our technical abilities.
And for that I need you help.
What do you think?
ex-xian
Jul29-04, 03:17 PM
It is too general, please be more specific, thank you.
It's not too general at all. I can list the field axioms and the least upper bound axiom, etc, that give tells how the real numbers work.
He's just asking you to list the axioms you're using. And if you do so, please try to it concisly and precisely.
ex-xian
Jul29-04, 03:23 PM
2) My logical reasoning is based on an included-middle reasoning, where the contradiction concept does not exist because two opposites are simultaneously preventing/defining each other.
Therefore I cannot fail to produce a proof by contradiction in an included-middle reasoning framework.
Don't you see how ridiculous this is? You say that contradictions do not exist in your "framework," but you cannot fail to prdocue a proof by contradiction. If contradictions don't exist, then a proof by contradiction cannot work! The proof by contradiction rests on A or not A being a tautology. If you don't hold to this, then you can't do proof by contradiction. You can post a "proof" that isnt' really one, and call it by "contradiction" which it wouldn't be, but don't use math terms unless you mean them. You'll just be harder to understand than you are usually.
3) If you read carefully http://www.geocities.com/complementarytheory/No-Naive-Math.pdf then I think that you will understand what is the meaning of ‘<’ or ‘>’ in my framework.
I think I expressed how arrogant and tiring it is for you to give out reading assignments. Are trying to get more hits on your website or something? Just post it for god's sake.
arildno
Jul29-04, 04:12 PM
It's not too general at all. I can list the field axioms and the least upper bound axiom, etc, that give tells how the real numbers work.
He's just asking you to list the axioms you're using. And if you do so, please try to it concisly and precisely.
I think such a list will never come forth.
Usually, the inability to produce such a list, is indicative of the fact that the claimant is an arrogant ignoramus spouting incoherent blather; however we can't rule out completely the possibility that Lama does not belong in that august company.
Don't you see how ridiculous this is? You say that contradictions do not exist in your "framework," but you cannot fail to prdocue a proof by contradiction. If contradictions don't exist, then a proof by contradiction cannot work!
You missed the point because of your aggressive approach about my work.
Because contradiction does not exist in my framework, my ideas are not based on it at all, instead they are based on the complementary approach, where two opposites are simultaneously preventing/defining their middle domain.
Whole my work is based on this 'school of thought', where contradiction is the basis of your 'school of thought' of excluded-middle reasoning.
We are in two different worlds, and because of your aggressive attitude you still do not understand this, and still continue to examine my work from your 'school of thought' point of view.
So, the two simple questions that I want to ask you are:
1) Why are you so aggressive about my work?
2) Can you put aside your aggressive attitude before we continue?
He's just asking you to list the axioms you're using
Because my work examines several fundamental concepts of the language of mathematics, I need first to know what in what fundamental concept you are interested.
From your post I understand the you wish to see the list of the axioms that are related to the real numbers, so here they are:
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
The axiom of independency:
p and s cannot be defined by each other.
The axiom of complementarity:
p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).
The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.
The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.
The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.
The Axiom of the unreachable strong limit:
No input can be found by {__} which stands for Fullness.
The Axiom of potentiality:
p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}.
The Axiom of phase transition:
a) There is no Urelement between {} and {.}.
b) There is no Urelement between {.} and {._.}.
c) There is no Urelement between {._.} and {__}.
Urelement (http://mathworld.wolfram.com/Urelement.html).
The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.
Tautology means x is itself or x=x.
Singleton set is http://mathworld.wolfram.com/SingletonSet.html .
Multiset is http://mathworld.wolfram.com/Multiset.html .
Set is http://mathworld.wolfram.com/Set.html .
(By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )
The Axiom of the paradigm-shift:
Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.
Let us stop here to get your remarks.
matt grime
Jul30-04, 07:38 AM
Here's a simple comment: you have not shown that the object which we call the real numbers satisfies any of those axioms. Nor do those axioms imply that the real numbers even exist (in a mathematical sense). Nor have you demonstrated that from these axioms alone can you construct anything, never mind something that is our real numbers.
Here's a simple comment: you have not shown that the object which we call the real numbers satisfies any of those axioms. Nor do those axioms imply that the real numbers even exist (in a mathematical sense). Nor have you demonstrated that from these axioms alone can you construct anything, never mind something that is our real numbers.
The real numbers are the shadow of my system.
You can still use Dedekind's sentence of the continuum that is based on the axiom of 'least upper bound', if you wish.
Before you continue Matt, please read http://www.physicsforums.com/showpost.php?p=270261&postcount=255
Thank you.
matt grime
Jul30-04, 10:14 AM
You have not shown that there is any object that satisfies your axioms, and certainly the real numbers don't do it. you've failed to define almost all of the terms you use (the axiom of duality is, for instance, unintelligible).
so, start from the beginning, your "numbers" are a set S satisfying .... whatever your axioms are, prove that a model of this exists (you've not done that) then demonstrate what it is that you mean by "shadow" since it is not clear at all what that means. You might occasionally want to listen to a mathematician without being deaf to the objections: after all you want the dialogue.
what it is that you mean by "shadow"
You can still use Dedekind's sentence of the continuum (L < c < R) that is based on the axiom of 'least upper bound' ( http://mcraefamily.com/MathHelp/CalculusLimitUpperBound.htm ), if you wish.
the axiom of duality is, for instance, unintelligible
Did you read http://www.physicsforums.com/showpost.php?p=270261&postcount=255 ?
Also please look at http://www.geocities.com/complementarytheory/No-Naive-Math.pdf starting from page 5.
matt grime
Jul30-04, 10:57 AM
the pdf does not define what any of those terms mean on page 5, i've read the other link to your post, not interested in commenting here, or there, and you've still not proven that there is a model for you axioms. all we know is that the real numbers aren't it, being only a shadow (something still not adequately defined), so what are they? that do you?
I wrote 'starting from page 5'.
Also I see that you chose to ignore my post to you at http://www.physicsforums.com/showpost.php?p=270261&postcount=255
So, let me put it this way:
There is no way to explain intuitions, it means that we already have them or not.
You do not share with me any common intuitions that are related to fundamental concepts of the Language of Mathematics, therefore you cannot understand my axioms.
On the contrary I can understand your axioms because they are based on intuitions that are less deep than my intuitions, and this is the reason why I am talking about a paradigm shift in the langage of Mathematics.
You will never understand me, and that is a clear fact after almost 2 years of dialogs between us.
So I want to ask you:
I'll be glad to continue our dialog, but why do you continue our dialog?
matt grime
Jul30-04, 12:17 PM
does that reply imply that the terms on page 5 are defined on another page in the pdf? care to point out which one?
i understand that you're a crank, doron, and taking time out to answer yout trivial posts doesn't adversely affect my day, and after all it only takes for good men to do nothing for the crackpots to advance their theory, to paraphrase someone. better to debunk the garbage than let it fester.
I understand that you're a crank
So why do you continue your dialog with me if I am a crank, which his work cannot be understood by you?
better to debunk the garbage than let it fester.
How can you do that if you have no ability to understand even the most trivial thing of my work?
So, I am right about you.
You are no more then a full_time_job_sanitarian of the current school of thought of the Language of Mathematics.
Let me point out one problem. (More later)
The Axiom of the weak limit:
No input can be found by {} which stands for Emptiness.
The Axiom of the strong limit:
No input can be found by {__} which stands for Fullness.
These are the only axioms that involve the terms "no input", "found", "{}", and "{__}". Thus, the only theorems you can prove about these three terms must be proven with only these two axioms.
Some examples of why this is a problem:
You cannot prove {} and {__} are different.
You cannot prove anything can be found by anything.
You cannot prove the existance of any input.
1) axioms are not proven
2) |{}|=0 , |{_}|=1 (The 1 continuum)
Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (please start from page 10 until the end of my article, if you wish to understand my work).
If you're adding, as axioms, that |{}| = 0 and |{__}| = 1, then we can now prove:
If 0 and 1 are different, then {} and {__} are different. (Assuming that | | is supposed to be a logical function)
You still can't prove any of the things I said you couldn't prove, though.
Again, axioms are not proven
|{}|=0 , |{__}|=1 (The 1 continuum)
Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (please start from page 10 until the end of my article, if you wish to understand my work).
Yes, but things may only be proven from axioms.
(Incidentally, any axiom proves itself)
Do you want to understand my work or not?
If yes then please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (please start from page 10 until the end of my article).
http://www.geocities.com/complement...-Naive-Math.pdf doesn't work.
Does it have more axioms than the ones you listed in #260?
terrabyte has been banned. He used to call himself ram1024, ram2048, ram4096, etc. We have banned this person three times already, yet he still does not seem to understand that he is not welcome here, and nor are his pointless threads.
If any of you see activity that you suspect is due to the same individual, please let the staff know so we can deal with it.
- Warren
Warren, i get the feeling that you don't like me... :uhh:
for the record, i wasn't banned three times i was banned once for all three accounts. for "spamming" which i wasn't. but that's neither here nor there.
doesn't work.
My first response to "doesn't work" was:
(If you cannot understand what are the unreachable limits of any information system (including the Language of Mathematics) and how I use the reachable information forms, which exist between these unreachable limits, to create Math that is based on an Included-Middle reasoning, then dear Hurkyl I cannot help you.)
Then I realized that it is technically doesn't work, so sorry about my first response and here it is again the link of my work: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (please start from page 10 until the end of my article).
Please read a post of mine to Matt, to understand more http://www.physicsforums.com/showpost.php?p=270261&postcount=255
Thank you.
Here is again a list of my axioms, which are related to R:
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
The axiom of independency:
p and s cannot be defined by each other.
The axiom of complementarity:
p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).
The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.
The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.
The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.
The Axiom of the unreachable strong limit:
No input can be found by {__} which stands for Fullness.
The Axiom of potentiality:
p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}.
The Axiom of phase transition:
a) There is no Urelement between {} and {.}.
b) There is no Urelement between {.} and {._.}.
c) There is no Urelement between {._.} and {__}.
Urelement (http://mathworld.wolfram.com/Urelement.html).
The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.
Tautology means x is itself or x=x.
Singleton set is http://mathworld.wolfram.com/SingletonSet.html .
Multiset is http://mathworld.wolfram.com/Multiset.html .
Set is http://mathworld.wolfram.com/Set.html .
(By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )
The Axiom of the paradigm-shift:
Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.
Let us stop here to get your remarks.
arildno
Jul31-04, 01:50 PM
"A definition for a point:
A singleton p that can be defined only by tautology ('='), where p has no internal parts."
1. What is a singleton?
2. What is "="?
3. What is an "internal part"?
4. What is an external part?
1. What is a singleton?
2. What is "="?
1. Singleton set http://mathworld.wolfram.com/SingletonSet.html
2. Tautology (x is itself or x=x).
3. What is an "no internal parts"?
4. What is an external part?
3. Non divisible or Urelement (http://mathworld.wolfram.com/Urelement.html).
4. I did not use these words.
(By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )
A point with no internal parts does not exist.
You missed the point. :wink:
I said "no internal parts".
Which means, it is non-divisible.
hello3719
Jul31-04, 05:45 PM
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') or '<' or '>', where s has no internal parts.
A number is consider as a point in your system right ?
If yes, then by the 1st definition it would follow that there can only exist one number,because in our system, numbers are constructed using previous numbers (ex. 1 + 1 = 2) Unless i misunderstood the internal parts thing.
A number is consider as a point in your system right ?
A number in my system is |{}| or p_AND_s , where p_AND_s is at least Multiset_AND_Set.
hello3719
Jul31-04, 08:22 PM
A number in my system is |{}| or p_AND_s
what is |{}| ?
matt grime
Aug2-04, 04:04 AM
But, Doron, the only people to agree with anything you've said are also cranks (though moshek is borderline), so let me have my fun. Remember you're the one who asked me to come here and comment on your work. Tough if you don't like it.
Matt, there is a time (after more then a year) when you have to realize that you have no ability to give any useful comment about my work, because you demonstrate time after time that its most basic principles are beyond your basic intuitions that you have about the language of Mathematics and its reasoning.
I can show you the most beautiful things in the world, but if you are blind to them, then it is a waste of time for both of us.
As I said before (http://www.physicsforums.com/showpost.php?p=270261&postcount=255), in my opinion your school of thought is based on the shadow of the intuitions of my school of thought.
If you insist and give comments to things that you cannot see (understand) then this is your problem.
Let us think about Mandelbrot set http://aleph0.clarku.edu/~djoyce/julia/julia.html .
The set itself is the black areas, where no information can be found.
This black area is the invariant or the constant side of Mandelbrot set, but the other side of it is its border area, where the interesting information is created when Mandelbrot set goes to infinity.
No one of these sides can be ignored if we want to understand what is a Mandelbrot set.
The same approach has to be used if we want to understand what is R collection.
Any R member is a unique (invariant and constant) element in the collection, but on the same time each constant is a scale factor of the entire R collection.
It means that the entire R collection exists between two opposite states (minus is the mirror -not the oppisie- of plus side).
In one state, when 0 is the scale factor, no R member except 0 can be found.
On the other state No R member can be found when we reach oo (as clearly can be shown here: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf ).
Furthermore, because of this duality of any R member, we get a system which is both absolute (when a single scale is examined) and relative (where the same place of the real line is examined simultaneously on several different scales).
Another example:
Pi = the relations between the perimeter and the diameter of a circle.
Pi is invariant in any arbitrary given scale, but when several scale levels are simultaneously compared, we can clearly see that each circle has a different curvature.
If our system is a circle, then if we want to understand what is a circle, then both its invariant and its variant properties cannot be ignored.
(We also have to be aware to the fact the no circle can be found when Diameter or Perimeter = 0, or Diameter or Perimeter = oo.
In short, our basic approach is to find the gateways between opposite properties, and the best way to do it, is by an including-middle logical reasoning (http://www.geocities.com/complementarytheory/CompLogic.pdf).
You claim that standard mathematics is a "shadow" of your system.
This suggests that you can take any concept or any proof in your system, and simply look at its shadow to get something in standard mathematics.
To state this again, you should be able to provide a standard mathematical description of some concept by taking your concept and looking at its shadow.
You should be able to provide proof that would be acceptable in standard mathematics simply by taking your proofs and looknig at the shadow.
You keep claiming that standard mathematics is merely a shadow of your system, but you never demonstrate any understanding of the shadow.
You keep claiming that standard mathematics is merely a shadow of your system, but you never demonstrate any understanding of the shadow.
If someone developing a x,y,z system, he first must understand the a x,y system.
If you understand my work (and you did not demonstrate this even once), then you can clearly see that I understand the foundations of the standard x,y system.
On the contrary all we need is, for example, to read your reply in http://www.physicsforums.com/showthread.php?p=273568#post273568 in order to understand that you have no sense of symmetry to simple mathematical ideas.
Please demonstrate some fundamental mathematical idea, which can clearly show that I do not understand (again, not disagree with, but do not understand) its standard interpretation.
Matt , I ask now your explantion to that :
"But, Doron, the only people to agree with anything you've said are also cranks (though moshek is borderline), so let me have my fun. Remember you're the one who asked me to come here and comment on your work. Tough if you don't like it."
Moshek :eek:
matt grime
Aug3-04, 04:15 AM
YOu ask for comments, Doron, and because they aren't what you want to hear you dismiss them. You know nothing of mathematics as you repeatedly demonstrate, and I will keep pointing out where you make gross errors in your representations of what mathematics is, or does. You make lots of mistakes and say things that are out right lies abot mathematics; I will not stand for that.
You are a seek person Matt, and I am going to call your university to tell them that, if you continue to bother me.
Tom Mattson
Aug3-04, 06:15 AM
This is a Forum, Lama. You have no right to tell anyone not to respond to your posts, and you certainly have no right to threaten reprisals against anyone who does.
I'll start with binary logic.
You've had trouble understanding quantification.
"For all q, P(q)"
"There exists z such that P(z)"
(Where, in both cases, P is some logical proposition, and I've specifically used letters other than x for the dummy variables to indicate that x is not special)
You've had trouble understanding proof by contradiction.
From "If P then false"
We conclude "not P"
This is a Forum, Lama. You have no right to tell anyone not to respond to your posts, and you certainly have no right to threaten reprisals against anyone who does.
Do you allow a parson like Matt to write what ever he likes, including to hurt another persons that are posting in my thread (http://www.physicsforums.com/showpost.php?p=274454&postcount=292), without any limits to his vulgarity?
Is there any tool in this forum where I can block persons like Matt?
I'll start with binary logic.
1) My reasoning is not binary logic, because binary logic is a trivial logical reasoning, when you look at it from the point of view of an included-middle logical reasoning, as you clearly can see in http://www.geocities.com/complementarytheory/CompLogic.pdf.
And I developed my included-middle logical reasoning system, after I understood the triviality of the binary logic.
2) The triviality of a proof by contradiction and the trivial use of a universal quantification as something that can be related to a collection of infinitely many elements, are clearly demonstrated here: http://www.geocities.com/complementarytheory/Anyx.pdf
Tom Mattson
Aug3-04, 08:58 AM
Do you allow a parson like Matt to write what ever he likes, including to hurt another persons that are posting in my thread (http://www.physicsforums.com/showpost.php?p=274454&postcount=292), without any limits to his vulgarity?
It looks like you have a different definition of "vulgar" than I do, because I don't consider calling someone a crank to fall in that category.
Is there any tool in this forum where I can block persons like Matt?
Yes, you can use the "Ignore" feature. It will take Matt's posts right off your screen. In fact you can use that feature to make all voices of disagreement evaporate from your screen. Wouldn't that be wonderful?
Tom Mattson,
It looks like you have a different definition of "vulgar" than I do, because I don't consider calling someone a crank to fall in that category.
1) I do not care when Matt call me a crank (sometimes you can find 'crakpot' or 'idiot' in his "rich" vocabulary), but when he add other persons as cranks just because they understand my work, then for me he went too far.
2) Matt did not write any meaningful comments on my work, because he did not show any ability to understand it.
When time passes, he becomes more and more aggressive and instead of write about my work, he writes about me, which is irrelevant.
Please see by yourself his basic tune in this thread.
On the contrary you can see persons that can understand my work, for example: http://www.physicsforums.com/showpost.php?p=266203&postcount=76 and also have a high-quality ability to communicate and explain themselves.
you can use the "Ignore" feature
In this forum I do not have any abilities except of open a new thread or add and edit my posts only in "theory development", i even have no ability to get or send private messages, or to see public profiles or change my profile.
In fact you can use that feature to make all voices of disagreement evaporate from your screen
This is a cynic and unnecessary response from a person that his job is to be a super mentor of a public forum.
Can you give us the reason why you clearly take Matt's side?
Tom Mattson
Aug3-04, 01:48 PM
1) I do not care when Matt call me a crank (sometimes you can find 'crakpot' or 'idiot' in his "rich" vocabulary), but when he add other persons as cranks just because they understand my work, then for me he went too far.
I don't think Matt does call Moshek a crank because he understands your work. I think he calls him a crank for independent reasons.
http://physicsforums.com/showthread.php?t=23442
2) Matt did not write any meaningful comments on my work, because he did not show any ability to understand it.
Funny, every single real mathematician at every message board to which you post, says the same about you.
When time passes, he becomes more and more aggressive and instead of write about my work, he writes about me, which is irrelevant.
Please see by yourself his basic tune in this thread.
I see both his tune, and your tune. You do the same thing to him that he does to you. But even so, you cannot expect him to accede to your request that he not post in your threads anymore. To put a finer point on it, they aren't even "your" threads.
And you certainly have no place to call his university over it.
In this forum I do not have any abilities except of open a new thread or add and edit my posts only in "theory development", i even have no ability to get or send private messages, or to see public profiles or change my profile.
Well then I guess you'll have to ignore Matt the old fashioned way.
This is a cynic and unnecessary response from a person that his job is to be a super mentor of a public forum.
Can you give us the reason why you clearly take Matt's side?
Oh, brother.
I don't clearly take Matt's side. If I did, then I'd be deleting your ad hominem arguments against him, while leaving his in place. But as it is, I pretty much leave the two of you alone. I only stepped in here because you threatened to interfere with his personal life.
Tom Mattson,
Thank you for your clear answer.
Please demonstrate some fundamental mathematical idea, which can clearly show that I do not understand (again, not disagree with, but do not understand) its standard interpretation.
How about a limit. You've tried to do a proof with one, but you just embarass yourself.
Why, because I clearly show (http://www.geocities.com/complementarytheory/Anyx.pdf) that nothing is approaching to some constant, if our collection is infinitely many elements in infinitely many different scales?
Please see my axioms if you want to understand my system:
http://www.physicsforums.com/showpost.php?p=272134&postcount=278
Also please read this:
http://www.physicsforums.com/showpost.php?p=273614&postcount=289
1) My reasoning is not binary logic, because binary logic is a trivial logical reasoning, when you look at it from the point of view of an included-middle logical reasoning, as you clearly can see in http://www.geocities.com/complementarytheory/CompLogic.pdf.
And I developed my included-middle logical reasoning system, after I understood the triviality of the binary logic.
2) The triviality of a proof by contradiction and the trivial use of a universal quantification as something that can be related to a collection of infinitely many elements, are clearly demonstrated here: http://www.geocities.com/complementarytheory/Anyx.pdf
If it was so trivial, you should be able to understand it, né? You should be able to, for instance, give a proof by contradiction that any mathematician would accept as valid... no matter what your opinions on the subject are.
In an included-middle logical reasoning there is no contradiction (http://www.geocities.com/complementarytheory/CompLogic.pdf).
kaiser soze
Aug3-04, 04:26 PM
if your system contains the "standard" system, then you should have contradictions.
Kaiser.
In an included-middle logical reasoning there is no contradiction (http://www.geocities.com/complementarytheory/CompLogic.pdf).
Then why have you tried to a proof, that involved a limit, by contradiction?
Besides, this isn't the point. You claimed to have a solid understanding of "standard" mathematics. I challenge you to present a proof, any proof, that demonstrates your so-called understanding.
if your system contains the "standard" system, then you should have contradictions.
Some analogy:
Let us say that to get a glue we need a combinations of matirial A and matirial B.
A has its unique properties, which is not a property of a glue.
B has its unique properties, which is not a property of a glue.
When we combine between A and B we get property C, which is the glue, and then we are using the Glue for our perpos.
My logical system is C.
And A and B are Boolean Logic and Fuzzy Logic, which their own properties are not used, when we get C state.
For better understanding please look at http://www.geocities.com/complementarytheory/CompLogic.pdf pages 1-3.
Thank you.
If boolean logic was a part of your system, then you should be able to demonstrate an understanding of boolean logic.
Then why have you tried to a proof, that involved a limit, by contradiction?
I used a proof by contradiction only to show the standard Math point of view on the limit concept and how it is using a universal quantification on a collection of infinitely many elements, which are existing in infinitely many different scale level, which is something that I disagree with.
And I clearly demonstrated why a universal quantification cannot be related to this kind of a collection(http://www.geocities.com/complementarytheory/Anyx.pdf)
I challenge you to present a proof, any proof that demonstrates your so-called understanding.
No problem, please read this:
http://www.geocities.com/complementarytheory/PZstar.pdf
Okay, for any lurkers, here's the sequence of events. Lama/Dorian/Organic/Math Crank states that his "system" subsumes "standard" mathematics. Okay, fair enough. It is pointed out that if this is the case, he should be extremely knowledgable and proficient in the "standard mathematics." Lama asks for any topic at all so he can demonstrate his knowlege.
To quote,
Please demonstrate some fundamental mathematical idea, which can clearly show that I do not understand (again, not disagree with, but do not understand) its standard interpretation.
Upon being asked to demonstrates his knowledge about ideas, he then proceeds to explain why he disagrees with the issues, something he specifically stated wasn't his intention in the above quoted post.
There is some back and forth, some of us pointing out his inconsitency, to which he replies with this gobbeldygook.
Some analogy:
Let us say that to get a glue we need a combinations of matirial A and matirial B.
A has its unique properties, which is not a property of a glue.
B has its unique properties, which is not a property of a glue.
When we combine between A and B we get property C, which is the glue, and then we are using the Glue for our perpos.
My logical system is C.
And A and B are Boolean Logic and Fuzzy Logic, which their own properties are not used, when we get C state.
For better understanding please look at http://www.geocities.com/complementarytheory/CompLogic.pdf pages 1-3.
Thank you.
What the hell does this have to do with proving that you actually know math? Explain a limit (w/o copying and pasting or posting links), do a proof of a limit by defintion, do an actual proof by contradiction (w/o copying and pasting or posting links), do anything.
You keep avoiding the issue. We all know that you disagree with it all--nobody cares. You challenged us to come up with something about which we thought you were ignorant and you were going to prove us wrong. So far, you've failed at proving anything (but you should be used to that by now).
CrankFan
Aug3-04, 05:36 PM
Matt did not write any meaningful comments on my work, because he did not show any ability to understand it.
Funny, every single real mathematician at every message board to which you post, says the same about you.
Well, obviously Tom: No real mathematician has the acuity to comprehend the brilliance of Lama's work.
Let this serve as a notice to all you real mathematicians. Lama has dropped hints indicating that the conflict which emerges as a result of his struggle against the Bodyguards-of-mathTM to expound on his important .... uhm .... hallucinations, on this virtually unknown web forum may well end in a cataclysmic event whereby mathematicians fall from grace and a are replaced by a golden age, ruled by.... uhm... incompetent kooks like Lama.
Anyway, since he's obviously onto the Bodyguards-of-mathTM, at the very least I suggest that you guys no longer do the secret handshake in public.
[
I used a proof by contradiction only to show the standard Math point of view on the limit concept and how it is using a universal quantification on a collection of infinitely many elements, which are existing in infinitely many different scale level, which is something that I disagree with.
And I clearly demonstrated why a universal quantification cannot be related to this kind of a collection(http://www.geocities.com/complementarytheory/Anyx.pdf)
Nothing you've ever written has ever been clear and nothing you've tried to do has demonstrated anything.
No problem, please read this:
http://www.geocities.com/complementarytheory/PZstar.pdf
Big surprise. You repost something that you've already posted under a different name (I'm assuming that you included the corrections that everyone gave you in the other thread?)
ETA: Can you define your symbols? I've them all mean different things in abstract algebra, analysis, and set theory, and it's not at all clear what you're doing.
It appears that you're trying to show that a set cannot have the same cardinality of it's power set, but some of your symbolism is confusing.
Dear ex-xian,
What exactly is your problem, can't you understand my axiomatic system?
http://www.physicsforums.com/showpost.php?p=272134&postcount=278
Cant you understand my original point of view here:
http://www.geocities.com/complementarytheory/PZstar.pdf
Is this the reason that you so aggressive and jumpy?
To get a better picture of this paper please look at:
http://www.geocities.com/complementarytheory/Identity.pdf including all of its links, thank you.
Dear ex-xian,
What exactly is your problem, can't you understand my axiomatic system?
http://www.physicsforums.com/showpost.php?p=272134&postcount=278
Cant you understand my original point of view here:
http://www.geocities.com/complementarytheory/PZstar.pdf
Is this the reason that you so aggressive and jumpy?
Dear Lama/Organic/Dorian/Math Crank/Person badly in need of a math class,
What exactly is your problem, can't you understand questions that are asked and you've agreed to answer?
http://www.physicsforums.com/showpost.php?p=275215&postcount=312
Can't you understand my original question and remember your promise to demonstarte your knowledge of "standard" math?
http://www.physicsforums.com/showpost.php?p=275118&postcount=303
Is this the reason that you are so defensive and appear so ignorant?
CrankFan,
Can you write more to the point?
For example please look at http://www.physicsforums.com/showpost.php?p=272134&postcount=278
Is this the reason that you are so defensive and appear so ignorant?
If you refuse to read my papers, where I clearly show that I understand standard Math before I air my view about it, then it is your problem.
For example read my dialog with kaiser soze:
Some dialog:
----------------------------------------------------------------------------------
Doron:
I think that we do not understand each other.
I gave you MY definiton of the limit concept.
Now, please give the standard definition for this concept.
After you give the standard definition, then we shall compare between
the two approaches.
Any way do you agree with http://mathworld.wolfram.com/Limit.html definition?
----------------------------------------------------------------------------------
kaiser:
off course I agree with this definition. I meant for you to provide the defintion for the limit of S(n), no need delta epsilon at this point. A limit can be defined using epsilon and S(n). At any case, I am not interested in your definitions at the moment. I need to be convinced that you understand and know how to use the fundamental "conventional" mathematical defintions before we can move on to your definitions.
----------------------------------------------------------------------------------
Doron:
Ok, the main persons in modern Math that are related to the so called rigorous definition of the limit concept are Cauchy and Weierstrass.
Cauchy said:" When some sequence of values that are related one after the other to the same variable, are approaching to some constant, in such a way that they will be distinguished from this constant in any arbitrary smaller sizes that are chosen by us, then we can say that this constant is the limit of these infinitely many values that approaching to it."
Weierstrass took this informal definition and gave this rigorous arithmetical definition:
The sequence S1,S2,S3, … ,Sn, ... is approaching to (limit) S if for any given positive and arbitrary small number (e > 0) we can find a matched place (N) in the sequence, in such a way that the absolute value S-Sn (|S-Sn|) become smaller then any given epsilon, starting from this particular place in the sequence
(|S-Sn| < e for any N < n).
----------------------------------------------------------------------------------
kaiser:
Very good! now based on the definition you provided, which is a correct mathematical definition please find out the limit of the following sequence:
0.9,0.99,0.999,0.9999,0.99999,....
----------------------------------------------------------------------------------
Doron:
-------post #190
Now please listen to what I have to say.
First please read http://www.geocities.com/complementarytheory/9999.pdf
(which is also related to your question) before we continue.
----------------------------------------------------------------------------------
Doron:
-------post #191
I disagree with the intuitions of Weierstrass, Cauchy, Dedekind, Cantor and other great mathematicians that developed the current mathematical methods, which are dealing with the Limit and the Infinity concepts.
And my reason is this:
No collection of infinitely many elements that can be found in infinitely many different scales, can have any link with some given constant, in such a way that it will be considered as a limit of the discussed collection.
In short, Nothing is approaching from the collection to the given constant, as can be clearly seen in my sports car analogy at page 2 of http://www.geocities.com/complementarytheory/ed.pdf
Take each separate position of the car, then compare it to zero state and you can clearly see that nothing is approaching to zero state.
Therefore no such constant can be considered as a limit of the above collection.
It means that if the described collection is A and the limit is B, then the connection between A,B cannot be anything but A_XOR_B.
So here is again post #184:
Since I am not a professional mathematician, my best definition at this stage is:
A Limit is any arbitrary well-defined element, where no collection of well-defined infinitely many elements can reach it.
It means that if A is the collection of infinitely many elements and B is the limit, then we can reach B only if we leap from A to B and vise versa.
By using the word "leap" we mean that we have a phase transition from state A to state B.
There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states.
A collection A is incomplete if infinitely many elements of it cannot reach some given limit, or if no limit is given.
From the above definition we can understand that no collection of infinitely many elements is a complete collection, and therefore no universal quantification can be related to it.
If you disagree with me, then please define a smooth link (without “leaps”) between A,B states.
----------------------------------------------------------------------------------
Doron:
-------post #192
'Any x’ is not ‘All x’
By inconsistent system we can "prove" what ever we want with no limitations
but then our "proofs" are inconsistent.
A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proven by the current axioms of this system, and we need to add more axioms in order to prove these statements.
So any consistent system is limited by definition and any inconsistent system is not limited by definition.
Let us examine the universal quantification 'all'.
As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements, even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements is an inconsistent idea.
For example:
Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.
In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.
Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf
Form this point of view a universal quantification can be related only to a collection of finitely many elements.
An example: LIM X---> 0, X*(1/X) = 1
In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.
'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.
----------------------------------------------------------------------------------
kaiser:
If you do not see that the limit of the sequence I provided is 1, then you do not understand what a limit is, and therefore can not agree or disagree with its definition.
In loose terms we can say that a sequence has a limit if it is approaching (but never reaching) some conststant. A sequence does not have a limit, if it is not approaching some constant, for example the sequence 1,2,3,4,... does not have a limit, it disperses to infinity.
----------------------------------------------------------------------------------
Doron:
1 as the limit of the sequence 0.9,0.99,0.999,0.9999,0.99999,.... is based on an ill intuition about a collection of infinitely many elements that can be found in infinitely many different scales, as can be clearly understood by posts #190,#191,#192.
You can show that 1 is really the limit of sequence 0.9,0.99,0.999,0.9999,0.99999,.... , only if you can prove that there is a smooth link (without "leaps") between this sequence and 1, which is not based on {0.9,0.99,0.999,0.9999,0.99999,.... }_XOR_{1} connection.
Maybe this example can help:
r is circle’s radius.
s' is a dummy variable (http://mathworld.wolfram.com/DummyVariable.html)
a) If r=0 then s'=|{}|=0 --> (no circle can be found) = A
b) If r>0 then s'=|{r}|=1 --> (a circle can be found) = B
The connection between A,B states cannot be but A_XOR_B
Also s' = 0 in case (a) and s' = 1 in case (b), can be described as s'=0_XOR_s'=1.
You can prove that A is the limit of B only if you can show that s'=0_AND_s'=1 --> 1
A collaction of elements, wich can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.
There's something I've been trying to stress for a long time...
If you're not willing to discuss standard mathematics, you shouldn't bring it up.
If you're not willing to discuss standard mathematics, you shouldn't bring it up.
Please explain what do you mean exactly.
hello3719
Aug3-04, 07:11 PM
Lama, are you a Godel wanabe ? Doesn't mean that if his ideas were accepted around 20 years after his death that yours are of the same type. Until now you didn't even show how to obtain pi in your system. And could you please define a number in english without using symbols ?
kaiser soze
Aug4-04, 12:42 AM
Lama, by your analogy, your system does not contain "our" system, which is it then?
Kaiser.
Lama, by your analogy, your system does not contain "our" system, which is it then?
Kaiser.
By my analogy the whole is more then its parts.
Any this is why I call my natural numbers 'Organic Natural Numbers':
http://www.geocities.com/complementarytheory/ONN1.pdf
http://www.geocities.com/complementarytheory/ONN2.pdf
http://www.geocities.com/complementarytheory/ONN3.pdf
kaiser soze
Aug4-04, 02:17 AM
Lama, as long as you are happy with it - just be aware that what you are doing does not even remotely resemble mathematics...
Kaiser.
Lama, as long as you are happy with it - just be aware that what you are doing does not even remotely resemble mathematics...
Kaiser.
There is no objective and totally external thing to us, which we can call it Mathematics:
Math, in my opinion, is first of all a rigorous agreement that based on language.
Symmetry is maybe the best tool that can be used to measure simplicity, where simplicity is the best platform for stable agreement.
In Any agreement we must be aware to the fact that no model of simplicity is simplicity itself.
This awareness to the difference between x-model and x-itself is the first condition for any stable agreement, because it gives it the ability to be changed.
The Axiom of the paradigm-shift:
Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.
Until now you didn't even show how to obtain pi in your system...
Pi = the relations between the perimeter and the diameter of a circle.
Pi is invariant in any arbitrary given scale, but when several scale levels are simultaneously compared, we can clearly see that each circle has a different curvature.
If our system is a circle, then if we want to understand what is a circle, then both its invariant and its variant properties cannot be ignored.
(We also have to be aware to the fact the no circle can be found when Diameter or Perimeter = 0, or Diameter or Perimeter = oo.
In short, our basic approach is to find the gateways between opposite properties, and the best way to do it, is by an including-middle logical reasoning (http://www.geocities.com/complement...y/CompLogic.pdf).
Pi by the standard system:
Pi is also considered as some exact and fixed place in the Real-Line, which is determinated by the exact and fixed positions of 0 and 1 (the length of the radius).
...And could you please define a number in english without using symbols ?
A number is any intercation between our memory and some abstract or non-abstract element(s), where pre-interaction is |{}|(=the cardinal of the Empty set).
kaiser soze
Aug4-04, 04:43 AM
Lama: "Math, in my opinion, is first of all a rigorous agreement that based on language."
Obviously, this is what YOU think mathematics is.
"Source: The American Heritage® Dictionary of the English Language, Fourth Edition
Copyright © 2000 by Houghton Mifflin Company.
Published by Houghton Mifflin Company. All rights reserved.
mathematics
\Math`e*mat"ics\, n. [F. math['e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See Mathematic, and -ics.] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations.
Note: Mathematics embraces three departments, namely: 1. Arithmetic. 2. Geometry, including Trigonometry and Conic Sections. 3. Analysis, in which letters are used, including Algebra, Analytical Geometry, and Calculus. Each of these divisions is divided into pure or abstract, which considers magnitude or quantity abstractly, without relation to matter; and mixed or applied, which treats of magnitude as subsisting in material bodies, and is consequently interwoven with physical considerations."
--> not that I fully agree with the above defintion but obviously your "mathematics" is something else than what most of us here think mathematics IS.
Kaiser.
matt grime
Aug4-04, 04:50 AM
Why would you wish to contact my university? Where have I made any claims that I was in any way representing my unviersity. I've never even named my University in threads to you have I? I certainly don't post from my university email address in forums exactly because I am not in anyway representing them.
Incidentally you might wish to look at www.crank.net and see the legal argument as to the meaning of crank or crackpot.
And I keep telling you that my interest is in making sure you do not spout anything that is false about mathematics.
There are several rebuttals of your (mathematical) claims by me and others, none of which you've addressed.
Moreover, a good example of me correcting your opinions of mathematics must be where I'vew pointed out repeatedly that not all of mathematics is done using "excluded middle reasoning" (though your deduction that it is uses that which you won't allow us), there are plenty of theories that use different logical systems.
Of course if you think it is deficient then you'd better get rid of the computer you use to post this with since (almost) every computer language is just predicate or propositional calculus in disguise.
Here's another one, well, one we've had before:
you think mathematics is not good enough in some sense, and so you start talking about your axioms the real numbers satisfy, but they are not sufficient to define a set of anything, let alone something like the real numbers.
When you say real numbers, you cannot mean R as we know it because that is something constructed by the mathematics that you think is about to collapse because it is wrong. So what are you referring to when you say the real numbers?
You've also not explained what you mean in these axioms for the "reals" what you mean by things like: a point is that which can only be defined by =" and so on, after all the point 0 is the complement of (-inf,0)u(0,inf), moreover by tarski's construct any set defined by a finite number of inequalities may be defined by equalities, but that of course is in the real numbes as we know them (complete totally ordered field), what your version of the reals are is a mystery even now. I told you that bit of mathematics you briefly acknowledged it but didn't explain yourself, or even correct your "axioms", not that we'd expect you to.
--> your "mathematics" is something else than what most of us here think mathematics IS.
Dont you see that also your example is first of all an interpretation, which is based on agreement between a group of people.
This is exactly the reason why I say that my point of view is a paradigm-shift of this agreement.
kaiser soze
Aug4-04, 05:07 AM
So you agree that your "mathematics" is not ours, since you are making out your own "agreement" as you go along. The problem is that no one seems to agree with you. This is exactly what you are doing wrong, you continue to use the same terms we use without proving that (the terms you are using) they are the same (as ours).
Kaiser.
You know what Matt?
Since you cannot see the connections between my axiomatic system and the standard system, let us put aside any standard system defititions, and let us look only on my axiomatic system.
Here it is again, and this time try to look at it as a new Axiomatic system (unless you find some connections to another and already known Axiomatic system):
Here is again a list of my axioms, which are related to R:
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
The axiom of independency:
p and s cannot be defined by each other.
The axiom of complementarity:
p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).
The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.
The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.
The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.
The Axiom of the unreachable strong limit:
No input can be found by {__} which stands for Fullness.
The Axiom of potentiality:
p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}.
The Axiom of phase transition:
a) There is no Urelement between {} and {.}.
b) There is no Urelement between {.} and {._.}.
c) There is no Urelement between {._.} and {__}.
Urelement (http://mathworld.wolfram.com/Urelement.html).
The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.
Tautology means x is itself or x=x.
Singleton set is http://mathworld.wolfram.com/SingletonSet.html .
Multiset is http://mathworld.wolfram.com/Multiset.html .
Set is http://mathworld.wolfram.com/Set.html .
(By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )
The Axiom of the paradigm-shift:
Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.
Let us stop here to get your remarks.
So you agree that your "mathematics" is not ours, since you are making out your own "agreement" as you go along. The problem is that no one seems to agree with you. This is exactly what you are doing wrong, you continue to use the same terms we use without proving that (the terms you are using) they are the same (as ours).
Kaiser.
Dear Kaizer soze, the difference between you and me is in the level of intuition/reasoning gentle interactions, where Axioms come form.
This level cannot be learned by any external method, which means, that you understand directly from within you something, or you don't.
My intuition/reasoning internal interaction goes deeper, in my opinion, then the common intuition/reasoning internal interaction.
Please read the following to get some perspective about what I wrote above:
When two violins in the same room are tuned with each other, if we play on one of them we find that the strings of the other violin are also vibrate.
Now let as say that intuition is our tuned instrument, and if a person expresses its intuitions by developing a way of thinking, the people that embrace this way of thinking probably share the same intuitions.
On the basis of these common intuitions a community can be established.
Let us say that this community is the first organization that deals with some part of the human knowledge, so in these early stages this community has no comparators on this part of the human knowledge.
Quickly this community becomes the most developed organization, which holds this part of the human knowledge, and other parts of human civilization look at this organization as the one and only one possible intuition which standing in the basis of a one and only one way (school) of thought (and I am not talking about variations, which are actually different brunches of the same way of thought, or the same school of thought if you like).
2500 years are passing and this school of thought survives because of two main reasons:
1) This way of thought was fitting to the needs of the human civilization along these 'slow' (linear) years.
2) Any other alternative intuitions (if they where at all) where put aside because:
a) They where not useful in their time.
b) And if they where useful and also a real alternative to the current school, then the current school used its power and money to block this alternative intuition by forcing its educational methods on the public.
We have to understand that intuitions cannot be learned, but a lot of external power can distort them until they lost their ability to be the source of a new school of thought.
The 120 century is the time where our civilization moved from linear time to non-linear time.
In this time the power of few holds the destiny of our civilization, and most of their power is based on the technical abilities that where developed by this school of thought, that was established 2500 years ago.
But our technical achievements, which are not balanced by another ways of thought, are like a government with no opposite.
We have learned that evolution needs diversity; otherwise we quickly get a dead planet.
The field of evolution in our non-linear time splits to "hardwhere" and "softwhere" parallel paths, where the hardwhere side is our technology and the softwhere side is our morality.
We can clearly see that there is no balance between the levels of these two paths, and this lack of balance in a non-linear time can quickly lead us to a dead-end street.
Therefore I think that we have to do the best we can to find the balance between our morality level and our technical abilities.
The first place that binds both paths is the language of mathematics.
In my opinion people how learn this powerful language, must first of all to develop their moral abilities by opening themselves to another intuitions which are not their intuitions and let them flourish in their communities.
By this way we develop our tolerance and learn how to live side by side, and if other intuitions are better then our intuition in this period of time, we do our best to help them flourish instead of trying our best to shut them down.
And we have the motivation to do that because we understand that we are all in the same boat.
My intuitions and ideas about the language of mathematics are different then the standard school of mathematics.
But in my opinion the most important difference, which I think fits to our non-linear time (more then the standard school) is that I include the mathematician cognition's ability to develop Math as a part of the mathematical research.
By this self-reference attitude I hope to develop the gateway that can connect between our moral abilities to our technical abilities.
matt grime
Aug4-04, 05:30 AM
"Here is again a list of my axioms, which are related to R:"
so is this supposed to define R, or is R already defined, if so explain what YOU think it is, and explain please what related to means in this context.
"A singleton set p that can be defined only by tautology ('='), where p has no internal parts."
this makes little sense, so post an example.
"p and s cannot be defined by each other"
again makes no sense, so post an exmaple of "p" and demonstrate that it does not define "s", that is prove that no definition exists, or are you not using "cannot" to mean that?
"Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set."
number? what number? multiset not been defined?
"The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection."
what is a scale factor?
"The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels."
what's a number? i mean that sincerely. since you've not offered an alternative definition of the set of numbers, one cannot conclude anything since the only ones we know about are those defined by a system that you reject.
"There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them."
that is not an axiom, that is your opinion, and one with no backing. It's a niuce sentiment, but does not in itself add anything to an axiomatic system.
"Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system."
what is "well defined" in this context? you've not offered here any way of defining what is and what isn't a set. this has no place in the "axioms" above.
You've still not said what R is. What do you mean by R. YOU, not US.
What shall we do to communicate with each other?
I think that we have to go step by stap so first let us look only on my two first definitions:
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
-------------------------------------------------------------------------
Tautology means x is itself or x=x.
Singleton set is http://mathworld.wolfram.com/SingletonSet.html .
Multiset is http://mathworld.wolfram.com/Multiset.html .
Set is http://mathworld.wolfram.com/Set.html .
No internal parts means: Indivisible.
Before we move on, do you acknowledge by using the standard definitions for sets and multisets that your theory must include all standard theorems about such things? Theorems including \aleph_0 + 1 = \aleph_0, infinite sets with different cardinalities, infinite sets have the same cardinality as some of their subsets, and that there can be a 1-1 bijection between an infinite set and some of its subsets.
(of course, several of those statements mean exactly the same thing)
kaiser soze
Aug4-04, 05:58 AM
Lama,
I will say it again, what you are doing is not mathematics as we know it - mathematics has nothing to do with intuitions, moreover, sometimes our intuitions prevent us from understanding mathematical facts.
Kaiser.
Before we move on, do you acknowledge by using the standard definitions for sets and multisets that your theory must include all standard theorems about such things? Theorems including , infinite sets with different cardinalities, infinite sets have the same cardinality as some of their subsets, and that there can be a 1-1 bijection between an infinite set and some of its subsets.
(of course, several of those statements mean exactly the same thing)
No I take only the most basic definitions of a Set and a Multiset:
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
Lama,
I will say it again, what you are doing is not mathematics as we know it - mathematics has nothing to do with intuitions, moreover, sometimes our intuitions prevent us from understanding mathematical facts.
Kaiser.
I am talikng about intuitions/reasoning interactions.
Please read carefully again what I wrote to you in:
http://www.physicsforums.com/showpost.php?p=275808&postcount=332
matt grime
Aug4-04, 06:24 AM
"A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored"
so you're doing your version of naive set theory then.
you haven' posted an example of a point being defined according to your axioms, nor have you said what the set R is in your opinion.
Once more yo'ure ignoring the mathematics.
kaiser soze
Aug4-04, 06:31 AM
Lama, a computer has no intution, yet it is perfectly able to perform calculations based on mathematical reasoning and constructs. Moreover, every computer is guaranteed to have the same result for a given mathamtical calculation.
Kaiser.
so you're doing your version of naive set theory then.
No Matt this is the standard basic definition for a Set.
See for yourself : http://mathworld.wolfram.com/Set.html.
At this stage of our dialog let us put aside R Collection.
Please let us concentrate only on:
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
-------------------------------------------------------------------------
Tautology means x is itself or x=x.
Singleton set is http://mathworld.wolfram.com/SingletonSet.html .
Multiset is http://mathworld.wolfram.com/Multiset.html .
Set is http://mathworld.wolfram.com/Set.html .
No internal parts means: Indivisible.
Lama, a computer has no intution, yet it is perfectly able to perform calculations based on mathematical reasoning and constructs. Moreover, every computer is guaranteed to have the same result for a given mathamtical calculation.
Kaiser.
A computer is nothing but a blind electro mechanic tool, which is no more than a dynamic mirror of your intuition/reasoning interactions reasoning.
In short, it is the image of your own intuition/reasoning interactions.
matt grime
Aug4-04, 06:44 AM
we cannot really put aside R, since we are talking about numbers, and elements of R and intervals, and it is the R that is the key thing in the errors in your ideas.
give an example of a singleton set then defined only by '=', demonstrating that it cannot be defined by some other means, whatever those means might be, moreover, explain what it means for a set to have no internal parts, ie a set to be indivisible. do you mean "is not the empty set, and if written as the disjoint union of two other sets then one of them is the empty set"? If so, why not say so more clearly rather than inventing new meanings for words?
NB you really ought to stop relying on mathworld and treating it as if its general explanations are actually the definitions of the things you don't understand.
we cannot really put aside R
You are right, but before we get R I want to be sure that you understand my first two definitions, so at this stage, ask your questions only about them, thank you.
kaiser soze
Aug4-04, 06:58 AM
Lama,
Intuition is individual - mathematics is universal.
Kaiser.
matt grime
Aug4-04, 07:04 AM
I have asked questions. I make it three times in three posts:
quoting myself:
give an example of a singleton set then defined only by '=', demonstrating that it cannot be defined by some other means, whatever those means might be (something that you need to explain as well), moreover, explain what it means for a set to have no internal parts, ie a set to be indivisible. do you mean "is not the empty set, and if written as the disjoint union of two other sets then one of them is the empty set"? If so, why not say so more clearly rather than inventing new meanings for words?
Lama,
Intuition is individual - mathematics is universal.
Kaiser.
Mathematics is the interactions between the individual/universal, local/global, intuition/reasoning, induction/deduction, integral/differential, 0/1, Emptiness/Fullness, ...
In short, it is based on an Included-middle framework, which is based on a dialog between at least two opposite concepts.
Thats why it is first of all a language.
give an example of a singleton set then defined only by '='
{.} is a clear and simple example of it.
matt grime
Aug4-04, 07:21 AM
But how is that defined by '=', and only by '=', where apparently you mean that '=' means tautology. Moreover, what is that set? The set which contains a full stop? Moreover, why is it not also "defined" as {{.}u{,}}n{{.}u{:}} where I assign arbitrary and distinct meanings to the symbols . , and :
But how is that defined by '=', and only by '=', where apparently you mean that '=' means tautology. Moreover, what is that set? The set which contains a full stop? Moreover, why is it not also "defined" as {{.}u{,}}n{{.}u{:}} where I assign arbitrary and distinct meanings to the symbols . , and :
The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
By these two axioms, the result cannot be but {.}
matt grime
Aug4-04, 07:35 AM
You've still not explained what {.} means, but we'll presume that it is an arbitrary set of cardinality 1, let us take an example: we can all agree on the finite cardinals, and their existence. So {1} is a set with one element. Demonstrate that this set can ONLY be defined by a tautology, further, explain how the axiom of abstract/representation relations is used in the implication you claim (it is not obvious), moreoever, define what an internal part is. also explain why {1,2}n{1,3} does not also define a singleton set, or is that a tautology? If so please explain what you think tautology is, and give an example defining some set that is not a tautologous statement, and which must therefore be a non-singleton set.
Why do you ignore the name of the axiom?
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
Tautology means x is itself or x=x.
Singleton set is http://mathworld.wolfram.com/SingletonSet.html .
Multiset is http://mathworld.wolfram.com/Multiset.html .
Set is http://mathworld.wolfram.com/Set.html .
No internal parts means a Urelement (http://mathworld.wolfram.com/Urelement.html).
also explain why {1,2}n{1,3} does not also define a singleton set
At this first stage we cannot cannot talk about number > 0, beause in my system we need at leaset two types of Urelements to define a number, which is not |{}|.
matt grime
Aug4-04, 07:58 AM
Yes, look at your own axiom. You must now prove that the set {.}, whatever that may be can ONLY be defined by a tautologous statement. However it is not clear what it means for something to be defined as a tautology, perhaps you would care to explain.
Fine, you don't like numbers, just pretend the symbols 1,2,3 are distinct objects, that they are numbers is not important, I was just trying to offer some example with elements we could be fairly sure existed.
However it is not clear what it means for something to be defined as a tautology
Tautology means x is itself or x=x.
At this stage, by the definition of a point, all we have is {.}={.}
I was just trying to offer some example with elements we could be fairly sure existed
We will get them, please be patient.
{.} is only one of two different building-blocks that we need before we can define a number, which is not |{}|.
Do you have something to say before we continue to the next definition?
kaiser soze
Aug4-04, 08:30 AM
Lama:Mathematics is the interactions between the individual/universal, local/global, intuition/reasoning, induction/deduction, integral/differential, 0/1, Emptiness/Fullness, ...
Prove it! In mathematics, we have axioms, definitions and proofs.
Kaiser.
Prove it! In mathematics, we have axioms, definitions and proofs.
Kaiser.
Please tell us how do we get our Axioms and definitions?
kaiser soze
Aug4-04, 08:58 AM
How? by thinking about them, then stating them in a coherent fashion.
Kaiser.
hello3719
Aug4-04, 09:15 AM
Pi = the relations between the perimeter and the diameter of a circle.
Well give me a way to find an approximation to this number using your system.
matt grime
Aug4-04, 09:24 AM
Tautology means x is itself or x=x.
but this is self referential, or at least inconsistent because you defined '=' to mean tautology. but what is x? is it a proposition or a set, or something else? since you've not bothered to say we can only guess, try saying a singleton set p is defined tautologically when.. and then use some statement about p.
since we're talking about sets, you are saying that a singleton set is defined by p=p, where p is a set. but every set is equal to itself.
Matt,
Again:
The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
By these two axioms, the result (our singleton) cannot be but {.}
hello3719
Aug4-04, 09:59 AM
The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
By these two axioms, the result (our singleton) cannot be but {.}
Not true. The first axiom as you call it is very subjective so you can conclude that the singleton can only be {.} but it isn't a universal conclusion.
Well give me a way to find an approximation to this number using your system.
You can continue to use the base value expansion method, but now you can use more than one information form for each given quantity along the decimals,
and also you have the knowledge of your unique path in the fractal, which is standing in the basis of some particular base value, for example please look at http://www.geocities.com/complementarytheory/9999.pdf pages 3,4 and also in http://www.geocities.com/complementarytheory/ONN3.pdf page 20.
Thank you.
matt grime
Aug4-04, 10:07 AM
That isn't a proof, where is there any deduction? the first "axiom" is not used at all, and you've not proven that {.}, whatever that might be, cannot be defined by some other "rule", principally because you've not said what it means to "define" something. You may say we are going round in circles, but you haven't actually said anything logically sound yet. You see that word you keep writing in BOLD? well, you've never actually used that part of the axiom or shown it to be satisfied by a singleton set.
So, the conclusion we can reach is that singelton sets as we know them from your links to wolfram are not what you mean, unless you can show that the singleton set as we know it satisfies that axiom. Thus we are left to conclude that only sets which are defined only be this alleged "tautology" are permissible in your theory. Otherwise, once more, you are assuming that sound mathematical objects must satisfy your axioms without proof.
This begs the question of what exactly a tautology is still. you keep saying "x is itself, or x=x" but don't say what type of thing "x" is.
Not true. The first axiom as you call it is very subjective so you can conclude that the singleton can only be {.} but it isn't a universal conclusion.
Yes I agree with you, but there is another definition which complementing the picture which is:
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
-------------------------------------------------------------------------
Tautology means x is itself or x=x.
Singleton set is http://mathworld.wolfram.com/SingletonSet.html .
Multiset is http://mathworld.wolfram.com/Multiset.html .
Set is http://mathworld.wolfram.com/Set.html .
No internal parts means: Indivisible.
That isn't a proof...
Axioms are not proven.
matt grime
Aug4-04, 10:20 AM
You keep repeating this as though the definition of a singelton set from wolfram is equivalent to yours, and i keep asking you t prove this and you keep posting the same non-argument. and you still don't say what kind of thing x is, it must be a proposition, so what proposition is it that allows you to conclude {.} is a a singelton set is a point in your system? You can't mix and match, come on, what is the proposition which is tautologous?
That isn't a proof...
Axioms are not proven.
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
matt grime
Aug4-04, 10:29 AM
I'm not asking you to prove an axiom I'm asking you to show that a singleton set (as understood by wolfram) satisfies your axioms, that is can be proven to only be "defined" by a tautology, what this alleged tautology is, even if your singelton sets aren't those as defined by wolfram, and to explain what you mean by "define" what is a definition, what is the allowable syntax and so on. Tautologies are applied to propositions, so what is that proposition?
matt grime
Aug4-04, 10:35 AM
and we're now onto something strange because we must now decide what you mean by a "point"
let us pretend we accept your notions, continue.
matt grime
Aug4-04, 10:36 AM
actually forget it, i'm bored for today, until you can explain what the proposition is that allows you to say {.} is a point, what kind of thing . is for that matter, and so on, I'll be off doing some real maths.
Tautology:
x implies x (An example: seuppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
(tautology is also known as the opposite of a contradiction).
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
Urelement:
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.
{.} is both a Singleton set and a Urelement.
A definition for a point:
A singleton set p that can be defined only(*) by tautology ('='), where p has no internal parts.
(*) only by tautology means: the minimal possible existence of a non-empty set.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
Lama:Mathematics is the interactions between the individual/universal, local/global, intuition/reasoning, induction/deduction, integral/differential, 0/1, Emptiness/Fullness, ...
Prove it! In mathematics, we have axioms, definitions and proofs.
Kaiser.
Please tell us how do we get our Axioms and definitions?
How? by thinking about them, then stating them in a coherent fashion.
Kaiser.
And what internal properties do you use during the thinking process?
Can you be sure that you do not use both intuition and reasoning?
If you ask me then you cannot get far without using also you intuition, in order to develop a meaningful thing in Math.
------------------------------------------------------------------------
kaiser soze
Aug4-04, 01:08 PM
Lama, you are naive to think that mathematics is based on intuition. You can develop perferctly valid (and maybe applicable) mathematical constructs even with the absolute non-intuitive foundations as long as your constructs are consistent. There are even complete mathematical systems developed with no intuition at all - they were developed by software.
Kaiser.
Lama, you are naive to think that mathematics is based on intuition. You can develop perferctly valid (and maybe applicable) mathematical constructs even with the absolute non-intuitive foundations as long as your constructs are consistent. There are even complete mathematical systems developed with no intuition at all - they were developed by software.
Kaiser.
No software can develop anything without some user that starting the whole process by entering the consistent axiomatic system.
Furthermore, the whole process is closed under the axiomatic system (that was first developed by some user) and can be understood only by the group of people who knows the rules.
Please show me a one case where a computer invented its own consistant axiomatic system, which is not trivial.
And if you can find such a system it means that your reasoning itself is a trivial thing if a "blind" thing like a computer can develop it.
So, what exactly is your point here?
Please correct me if I did not understand you, but as I see it, your approach leading me to understand that the language of Mathematics is no more then a mechanic process, where creativity is not needed.
you are naive to think that mathematics is based on intuition
On both intuition_AND_reasoning
Please do not omit again reasoning.
Tautology:
x implies x (An example: seuppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
(tautology is also known as the opposite of a contradiction).
So when you write "x = x" as an example of a tautology, what you really mean is "x -> x" or "if x then x?"
Also, you state that a point can only be defined by a tautology. How does {p, x} n {p, q} not define the point p (where "n" is an intersection)?
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
Urelement:
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.
{.} is both a Singleton set and a Urelement.
By the definitions you given, nothing can be a singelton set and an urelement. A singleton set has "exactly one element" and an urelement "contains no elements."
A definition for a point:
A singleton set p that can be defined only(*) by tautology ('='), where p has no internal parts.
(*) only by tautology means: the minimal possible existence of a non-empty set.
Since a set with one and only one element is "the minimal possible existence of a non-empty set," you're saying that a point is a single set that is a singleton set. Wouldn't it just be easier to say a point is a set with exactly one element instead of resorting to big words to try to make yourself sound smarter?
So when you write "x = x" as an example of a tautology, what you really mean is "x -> x" or "if x then x?"
Also, you state that a point can only be defined by a tautology. How does {p, x} n {p, q} not define the point p (where "n" is an intersection)?
Very simple, x is a self-evident true.
By the definitions you given, nothing can be a singelton set and an urelement. A singleton set has "exactly one element" and an urelement "contains no elements."
"exactly one element that contains no elements" is an indivisible one element.
If it is only a one element (a singleton) but not also a urelement, then it can be also 1 xor 2 xor 3 xor ... where each one of them is divisible, and in this case we cannot get our self-evident_true indivisible {.} .
Since a set with one and only one element is "the minimal possible existence of a non-empty set," you're saying that a point is a single set that is a singleton set. Wouldn't it just be easier to say a point is a set with exactly one element instead of resorting to big words to try to make yourself sound smarter?
"exactly one element that contains no elements" is an indivisible one element.
And if it can be defind only by self-evident true (which means that no other property like '<' or '>' can be related to it) then we cannot get anything but {.}, and we have our point.
<Meaningless gibberish>
Congratulations. You've again managed to not answer a question that's been posted.
No dear ex-xian,
Your aggressive attitude simply blocking your ability to see (understand) fine things.
No dear ex-xian,
Your aggressive attitude simply blocking your ability to see (understand) fine things.
Whatever. I realize it's much easier to accuse those who disagree with you of having an insurmountable bias, but it doesn't make anyone take you seriously. Everytime anyone finds problems with your posts you just change the subject, post something unrelated, or ignore the questions. It's a pattern you've developed and if you can't see it, I really pity you.
I am sorry for you ex-xian, I gave you my full answer to your questions.
You are the one who wrote back "<Meaningless gibberish>" instead of clearly show why you disagree with my answers to your previous post.
"<Meaningless gibberish>" is not a basis for any meaningful dialog between two persons, so please be more serious when you reply if you really want to understand someone's point of view, thank you.
kaiser soze
Aug5-04, 03:13 AM
Lama:So, what exactly is your point here?
We are drifting off mathematical grounds. My point is that mathematics is universal, it does not rely on individual intuition. Sometimes intuition helps us understand and discover mathematical constructs, and sometimes it prevents us from understanding/discovering them.
If you think mathematics depends on intuition then prove that it does not exist without it - you will find this an impossible task.
Kaiser.
My point is that mathematics is universal
Please define 'universal', because maybe we have different interpretations about this word.
My definition for universal is:
The common source of abstract or non-abstract opposites.
This common source is the invariant super-symmetry, which is the gateway of interactions between opposites in non-destructive ways.
And in my opinion, the Language of Mathematics is the art of non-destructive interactions.
Please read http://www.geocities.com/complementarytheory/RTD.pdf to understand my point of view, thank you.
matt grime
Aug5-04, 04:22 AM
Doron, you'll be pleased to know that I@ve figured out some of my confusion. I haven't read back to check since my last post, and I suspect anything you've posted will be made unnecessary by this:
So, a "point", p, is a set (singleton), that can only be defined by a tautology. Ok, I'll accept that as a definition, fine. You generally denote a "point" as {.}, which is by definition something that can only be defined by a tautology. Now, the question is can you provide me with a model of this that we can understand, ie some mathematical object that we all can recognize. And could you then give me the alleged tautology that defines it and can be defined no other way. It's all very well having a statement along the lines X is a degree three extension of Q that is the splitting field for x^2-2, but no such X actually exists, as can be proven. So, is there actually anything that we can understand that is a "point"? Presumably, since it arose there, points in R are defined only by a tautology, and are singleton sets, so show me how, say, sqrt(2) is defined by a tautology.
In fact just give me an example of what it means to be defined by a tautology, since a tautology involves propositions, and objects in general aren't propositions.
Fist let us write again our last post:
Tautology:
x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
(tautology is also known as the opposite of a contradiction).
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
Urelement:(no internal parts)
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
-----------------------------------------------------------------------------------------------
Now let us move to the next step in order to define what is a number in my system.
First let us examine a well-known relation between mathematical objects and their representations.
=>> is ‘represented by’
|{}|=>>0 ; |{{}}|=>>|{0}|=>>1 ; |{{},{{}}}|=>>|{0,{0}}|=>>|{0,1}|=>>2 ;
|{{},{{},{{}}}}|=>>|{0,{0,{0}}}|=>>|{0,1,2}|=>>3 ; …
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)
By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}.
Now we have the minimal building-blocks that allows us to define the standard R members.
(edit:
A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.)
The axiom of independency:
p and s cannot be defined by each other.
By the above axiom {.} and {._.} are independed building blocks.
The axiom of complementarity:
p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).
By the above axiom we define the basic property of the middle domain between {.} and {._.}
The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
The above axiom allows us to:
1) To define the internal structure of standard R members.
2) To define the internal structures of my new number system.
The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.
The above axiom allows us to construct a collection of R members and also a collection of my new number system.
First, let us see how we use my method to construct a collection of R members.
R members are constructed like this:
1) First let us examine how we represent a number by my system:
=>> is ‘represented by’
a) |{}|=>>0
b) There is 1-1 and onto between ‘0’ and the left point of {._.} and we get {‘0’_.}
c) |{{}}|=>>|{0}|=>>1
e) There is 1-1 and onto between ‘1’ and the right point of {._.} and we get {‘0’_’1’}
In short, {.} is the initial place of R collection, which is represented by ‘0’, where {‘0’_.} is the initial place of the second place of R collection, which is represented by ‘1’, and we get our first two must-have building-blocks of R collection.
2) When we get {‘0’_’1’} we have our two must-have numbers, which are ‘0’ and _’1’.
Be aware that ‘0’ is the representation of {.} where ‘1’ is the representation of {._.}.
3) If we get {.}_AND_{._.}, then and only then we have the minimal must-have information to construct the entire R collection because:
a) We have ‘0’ AND _’1’ that give us the to basic scale factors 0 and _1.
b) We also have our initial domain _1, which standing in the basis of any arbitrary scale factor that is determined by the ratio between the initial domain _1 and another segment that is smaller or bigger than the initial domain _1 , for example:
0 = .
1 = 0______1
2 = 0____________2
3 = 0___________________3
.5 = 0__.5
pi = 0______________________pi
The negative numbers are the left mirror image of the above numbers.
There is no division in my number system because both {.} and {._.} are indivisible by definition.
In short, any segment is an independent element, that clearly can be shown in the above 2-D representation.
If we use a 1-D representation, we get the standard Real-line representation, but then we can understand that division is only an illusion of an overlap of independent elements when they are put on top of each other in a 1-D representation, for example:
0__.5 __1_____2_____3__pi
(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.
Since in my system nothing is divisible, then '/' stands for a ratio between at least any given two (indivisible) numbers.
-----------------------------------------------------------------------------------
Let us stop here (before we continue to my new number system) to get your remarks.
matt grime
Aug6-04, 06:51 AM
I know what a tautology is, however, please give an example of a *set* defined by a tautology, and a definition of a set that is not a tautology, so that we might understand what these look like. A tautology is a proposition that is always true regardless of the truth values of its components. How does that have anything to do with a set? How do we know that these are not vacuous statements?
What differentiates a general "singleton set" from one that is also a "point"? Give me an example of a singleton set that is not a point, give me an example of a singleton set that is a point.
Simple requests you keep ignoring? The best you've done is offer a circular argument, saying a point is {.} where {.} is a point. Well, what is {.}? Oh, the minimal structure etc. prove that such a thing exists from the axioms, or is it an axiom that such a thing exists? What is the reason behind this axiom if it is such?
Obviously your definition of an interval does not agree with the proper one, hence any conclusions you draw from your reasoning are not applicable to real mathematics
Obviously your definition of an interval does not agree with the proper one, hence any conclusions you draw from your reasoning are not applicable to real mathematics
By my definition of an interval we define the indivisible singleton set, which is not defined only by tautology, and then you can understand what is {.} and also what is {._.}.
Please keep reading all of my previous post (until the end of it) if you want to understand how my axiomatic system is related to R.
matt grime
Aug6-04, 09:36 AM
But, Doron, an interval (meaning a connected subset of R) is not a singleton set, but that is besides the point. What is a tautology which defines your "interval", I'm not saying it's the only thing defining it, but we've still not seen the proposition which is tautologous. Pick any one of the objects in your theory. What is the tautology that defines it? Recall a tautology is a proposition.
a tautology is a proposition
A tautology is first of all a self-avident true, that can be understood if you understand the axiom.
Here it is again:
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)
--------------------------------------------------------------------------------
Matt, please tell me if this post helps to understand my axiomatic system, thank you:
As I showed in the previous post, each number which is not 0 is at least a representation of {.}_AND_{._.}.
Also each {._.} has 3 basic states which are: '<' for left-right direction, '>' for right-left direction, '<>' for no-direction.
Let us write again The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
Let us examie this part: "...where p_AND_s is at least Multiset_AND_Set."
We know that the elements of a non-empty "normal" set, which its cardinality > 1,
cannot be identical.
But the elements of a multiset, which its cardinality > 1, can be identical.
If p_AND_s is at least Multiset_AND_Set, then any given element which its cardinality > 1 has several variations that can be found between '<>' to '<' or '>'.
For example, let us represent the variations of cardinals 2,3,4:
Let Redundancy be more then one copy of the same value can be found.
Let Uncertainty be more than one unique value can be found.
Let XOR be #
Let a=0,b=1,c=2,d=3 then we get:
b b
# #
{a, a, {a, b}
. . . .
| | | |
|__|_ |__|
| |
{x,x} {{x},x}
c c c
# # #
b b b b b
# # # # #
{a, a, a,} {a, a, c} {a, b, b}
. . . . . . . . .
| | | | | | | | |
| | | |__|_ | |__|_ |
| | | | | | |
|__|__|_ |_____| |_____|
| | |
| | |
{{x,x,x} {{x,x},x} {{x},x},x}
Uncertainty
<-Redundancy->^
d d d d |
# # # # |
c c c c |
# # # # |
b b b b |
# # # # |
{a, a, a, a} V {a, b, c, d}
. . . . . . . .
| | | | | | | |
| | | | |__| | |
| | | | | | | <--(Standard Math language uses only
| | | | |_____| | this no-redundancy_
| | | | | | no-uncertainty_symmetry)
|__|__|__|_ |________|
| |
={x,x,x,x} ={{{{x},x},x},x}
============>>>
Uncertainty
<-Redundancy->^
d d d d | d d d d
# # # # | # # # #
c c c c | c c c c
# # # # | # # # #
b b b b | b b b b b b b b b b
# # # # | # # # # # # # # # #
{a, a, a, a} V {a, a, a, a} {a, b, a, a} {a, a, a, a}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
{x,x,x,x} {{x,x},x,x} {{{x},x},x,x} {{x,x},{x,x}}
c c c
# # #
b b b b b b b
# # # # # # #
{a, b, a, a} {a, b, a, b} {a, a, a, d} {a, a, c, d}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
{{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x} {{{x,x},x},x}
a, b, c, d}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this
|_____| | no-redundancy_no-uncertainty_symmetry)
| |
|________|
|
{{{{x},x},x},x}
Also please pay attantion that the last form is the standard R members 0,1,2,3:
0 = .
1 = 0______1
2 = 0____________2
3 = 0___________________3
And the standrard R is nothing but the above 2-D representation
in a 1-D representation:
0______1______2______3
And because no R member is both Multiset_AND_Set, I call it: The "shadow" of my new number system.
--------------------------------------------------------------------------------
matt grime
Aug6-04, 11:23 AM
No, your post is not in the least helpful as once more you fail to offer an example of a tautology that defines something be it uniquely or otherwise. That is all we require, an example of the kind of tautology that you are using to define these points. Note, a tautology is NOT a self evident truth, a tautology might be very unevident.
The definition of interval is most unenlightening too. What does it mean to look at something from a direction? But this is still besides the point, what tautologous proposition are you using to define anything. Just post one example of it, moreover, what logic system are you using? You reject the usual one in mathematics, so you need to carefully define that too.
I'm happy for you to offer that definition of a point from before, and of an interval, however you need to prove that they exist within your framework and provide models of them.
Your definitions are somewhat circular. We cannot deduce the existence of the reals without having the "points" and we cannot define the points without having the set of reals available. Do you understand that criticism?
You've not defined what R is. Got it? you must construct from first princpals the set of YOUR real numbers.
Only after that can you define points in this set. you have it backwards.
Our Reals cannot exist for you since they are a CONSTRUCTION of the system that you dismiss.
Lama's "tautologies":
1. = (Here, it seems to be some sort of equivalence relation.)
2. x implies x (don't we have any longer: only if x then x?)
3. The opposite of contradiction
4. Self-evidently true
(So combining 3. and 4., we might possibly infer that a contradiction is a non-self-evidently true statement?)
Ok Matt, I'll try again (and again, and again ... until you will understand me):
Tautology:
x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
(tautology is also known as the opposite of a contradiction).
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
Urelement:(no internal parts)
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.
----------------------------------------------------------------------------------
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.
(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)
-----------------------------------------------------------------------------------
Now, let us examine the point and the segment definitions by using the symmetry concept:
1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.
It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.
2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'
In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .
In short, in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise (http://mathworld.wolfram.com/Tautology.html)
A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
I hope that now I am understood.
Lama's "tautologies":
1. = (Here, it seems to be some sort of equivalence relation.)
2. x implies x (don't we have any longer: only if x then x?)
3. The opposite of contradiction
4. Self-evidently true
(So combining 3. and 4., we might possibly infer that a contradiction is a non-self-evidently true statement?)
Please see for yourself in:
http://en.wikipedia.org/wiki/Tautology.
Please see for yourself in:
http://en.wikipedia.org/wiki/Tautology.
The point everyone is making is that a tautology is a proposition--even the link you cite says this. When you claim that a point is defined by tautology, what is the proposition that you're referring to?
The point everyone is making is that a tautology is a proposition--even the link you cite says this. When you claim that a point is defined by tautology, what is the proposition that you're referring to?
Dear ex-xien,
Please read all of:
http://www.physicsforums.com/showpost.php?p=278285&postcount=390
And then please give your remarks, thank you.
Dear ex-xien,
Please read all of:
http://www.physicsforums.com/showpost.php?p=278285&postcount=390
And then please give your remarks, thank you.
Ok...That still doesn't answer my question.
You state,
A definition for a point:
A singleton set p that can be defined only(*) by tautology ('='), where p has no internal parts.
(*) only by tautology means: the minimal possible existence of a non-empty set.
All I'm asking is what proposition is tautologous. Your own link makes it clear that a tautology is a statement.
Are you going to redefine this word too?
Do you understand that in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise ?(http://mathworld.wolfram.com/Tautology.html)
To understand this you have to read http://www.physicsforums.com/showpost.php?p=278285&postcount=390
until the end of it, thank you.
I erased this part ( (*) only by tautology means: the minimal possible existence of a non-empty set) to avoid understanding problems.
A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
Do you understand that in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise ?(http://mathworld.wolfram.com/Tautology.html)
No, I don't. What is the conclusion and what is the premise? That's what I've been getting at. I don't see a proposition in {.} or {_}.
To understand this you have to read http://www.physicsforums.com/showpost.php?p=278285&postcount=390
until the end of it, thank you.
I've read it already, thanks.
A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
If you're going to adjust your definitions to these, it's a bit clearer. Why don't you just define a point as a set with exactly one element. This seems to be the concept you're dancing around.
Why don't you just define a point as a set with exactly one element.
Because only by this definition we cannot distinguish between a point and a segment, which are two different elements that are not defined by each other.
Please read this again:
A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
This is exactly the basic idea that makes the whole change between the standard point of view and my point of view.
If you're going to adjust your definitions to these, it's a bit clearer. Why don't you just define a point as a set with exactly one element. This seems to be the concept you're dancing around.
I don't think that would be satisfactory anyways... Organic is focused on telling us what points and segments are, but he's not telling what we know about / can do with them.
In the usual context, defining something in terms of sets would be fine and dandy, but past experience, and his own admission, has shown that what Organic means by "set" does not coincide with what mathematicians mean by "set".
And back to Organic:
A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
The same problem I mentioned last time applies here. The axioms you give still have the defect that you can't combine them to prove things.
Let me give you an example of an accepted selection of axioms; these are the usual axioms of incidence from Euclidean geometry:
If A and B are points, then there is a line L such that A and B both lie on L.
If L and M are distinct lines, then there is at most one point P that lies on both L and M.
There are points, A, B, and C, that do not all lie on the same line.
Notice that there are 3 statements above, yet they only define three terms: "point", "line", and "lies on".
Furthermore, two of these are constructive: the last one "makes" some points and tells you something about them, and if you have some points, the first one lets you "make" some lines.
Finally, the middle axiom is a restriction.
Look at your axioms:
A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
You have only two axioms, yet you are defining at least four terms: "point", "segment", "indivisible finite content of a non-empty set", and "directions".
Furthermore, they tell you very little. They don't tell you how "make" points and "make" segments, and they don't many any useful restrictions about what you can make.
addendum:
by "make", I don't mean the act of physically making things. I merely mean that you can prove they exist, and have certain properties.
Because only by this definition we cannot distinguish between a point and a segment, which are two different elements that are not defined by each other.
Please read this again:
A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
This is exactly the basic idea that makes the whole change between the standard point of view and my point of view.
I can understand what you mean by point except for the "has no direction" part. So the only difference in a point and a segment is that a segment has direction?
What does that even mean? What does "direction" mean in this sense?
To understand this please first read all of:
http://www.physicsforums.com/showpost.php?p=276964&postcount=383
and only then all of:
As I showed in post #383, each number which is not 0 is at least a representation of {.}_AND_{._.}.
Also each {._.} has 3 basic states which are: '<' for left-right direction, '>' for right-left direction, '<>' for no-direction.
Let us write again The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
Let us examie this part: "...where p_AND_s is at least Multiset_AND_Set."
We know that the elements of a non-empty "normal" set, which its cardinality > 1,
cannot be identical.
But the elements of a multiset, which its cardinality > 1, can be identical.
If p_AND_s is at least Multiset_AND_Set, then any given element which its cardinality > 1 has several variations that can be found between '<>' to '<' or '>'.
For example, let us represent the variations of cardinals(*) 2,3,4:
Let Redundancy be more then one copy of the same value can be found.
Let Uncertainty be more than one unique value can be found.
Let XOR be #
Let a=0,b=1,c=2,d=3 then we get:
b b
# #
{a, a, {a, b}
. . . .
| | | |
|__|_ |__|
| |
{x,x} {{x},x}
c c c
# # #
b b b b b
# # # # #
{a, a, a,} {a, a, c} {a, b, b}
. . . . . . . . .
| | | | | | | | |
| | | |__|_ | |__|_ |
| | | | | | |
|__|__|_ |_____| |_____|
| | |
| | |
{{x,x,x} {{x,x},x} {{x},x},x}
Uncertainty
<-Redundancy->^
d d d d |
# # # # |
c c c c |
# # # # |
b b b b |
# # # # |
{a, a, a, a} V {a, b, c, d}
. . . . . . . .
| | | | | | | |
| | | | |__| | |
| | | | | | | <--(Standard Math language uses only
| | | | |_____| | this no-redundancy_
| | | | | | no-uncertainty_symmetry)
|__|__|__|_ |________|
| |
={x,x,x,x} ={{{{x},x},x},x}
============>>>
Uncertainty
<-Redundancy->^
d d d d | d d d d
# # # # | # # # #
c c c c | c c c c
# # # # | # # # #
b b b b | b b b b b b b b b b
# # # # | # # # # # # # # # #
{a, a, a, a} V {a, a, a, a} {a, b, a, a} {a, a, a, a}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
{x,x,x,x} {{x,x},x,x} {{{x},x},x,x} {{x,x},{x,x}}
c c c
# # #
b b b b b b b
# # # # # # #
{a, b, a, a} {a, b, a, b} {a, a, a, d} {a, a, c, d}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
{{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x} {{{x,x},x},x}
a, b, c, d}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this
|_____| | no-redundancy_no-uncertainty_symmetry)
| |
|________|
|
{{{{x},x},x},x}
Also please pay attantion that the last form is the standard R members 0,1,2,3:
0 = .
1 = 0______1
2 = 0____________2
3 = 0___________________3
And the standrard R is nothing but the above 2-D representation
in a 1-D representation:
0______1______2______3
And because no R member is both Multiset_AND_Set, I call it: The "shadow" of my new number system.
--------------------------------------------------------------------------------
(*) Please pay attention that we are not talking about the natural numbers 2,3,4 but the cardinals 2,3,4.
It means that our Organic Natural Numbers are actually a general representation of information-trees, where any finite quantity of names of R members can be described by them, for example:
Instead of a=0,b=1,c=2,d=3 we can use a=0,b=.5,c=3,d=pi and then we use the same information-trees above.
I called these general information-trees 'Organic Natural Numbers' because:
1) These information-trees of cardinals are always having a structure, which is based on N members.
2) They can be used as natural (not forced) and general representation for any interaction between complementary states, which simultaneously preventing/defining their middle domain.
3) Because no R member is divisible by my system, it has its own organic (complete) unique and independent self existence.
I merely mean that you can prove they exist, and have certain properties.
Thank you dear Hurkyl for your reply, but I am afraid that you did not see the rest of my axiomatic system and what I can do with it, so for a better picture of it please read all of it until the end of it, at lease twice before you reply about my axiomatic system, that can be found here:
The axioms of Complementary Set Theory:
http://www.geocities.com/complementarytheory/My-first-axioms.pdf
Thank you,
Lama
oops, sorry: I have found some mistakes in the above pdf file and I fixed them.
Please see for yourself in:
http://en.wikipedia.org/wiki/Tautology.
Nowhere have you shown any understanding of the actual meaning of tautology in the sense used by logicians.
I know perfectly well what a tautology is; you don't seem to understand it.
Nowhere have you shown any understanding of the actual meaning of tautology in the sense used by logicians.
I know perfectly well what a tautology is; you don't seem to understand it.
Please first read at least tiwce all of http://www.geocities.com/complementarytheory/My-first-axioms.pdf including all of its links, and only then you can raply meaningful remarks on my work.
Please first read at least tiwce all of http://www.geocities.com/complementarytheory/My-first-axioms.pdf including all of its links, and only then you can raply meaningful remarks on my work.
Please read at least 3 times my questions about where the premise and conclusion is on your "tautology", read at least 5 times what wikipedia actually says about tautology, post a reponse explaining what the premise and conclusion is in your "tautology", proofread it 2 times, then present it for us to read.
Only then can anyone make meaningful remarks on your "work." THat is, you have to actually respond to what people ask, rather than give reading assignments or reposting the same stuff over and over.
Lama:
As yet, you have not convinced anyone that your first axiom is not total gibberish.
Instead of wasting virtual space with tons of cooked-up definitions from your own fancy, please focus only on elucidation of your first axiom.
If you think it cannot be "understood" out of context with the rest, then it is not an axiom at all, but just gibberish.
How do complex numbers fit into your new theory? I see only talk of real numbers here. I have read through all 28 pages of this thread and I have had time to look at all of your files on your website. Please explain i=sqrt(-1) in terms of your new notation. Please do not refer me to one of your pdf files, as I know that there is no reference to the complex numbers.
Dear oreopoj,
When we have {},{.}_AND_{._.},{__} then we can construct any information form that we like.
Complex numbers are based on R part + Z part, so all they are is no more then a technical extension that help us to solve equations where numbers like √-1 are involved.
My research about the language of Mathematics is exactly in the opposite direction, which means:
Instead of searching unsolved problems within the standard framework (based on 'How' questions) I went back to the most fundamental concepts of this beautiful language and used most of the time 'Why' questions, that helped me to develop my non-standard framework.
Please read at least 3 times my questions about where the premise and conclusion is on your "tautology", read at least 5 times what wikipedia actually says about tautology, post a reponse explaining what the premise and conclusion is in your "tautology", proofread it 2 times, then present it for us to read.
Only then can anyone make meaningful remarks on your "work." THat is, you have to actually respond to what people ask, rather than give reading assignments or reposting the same stuff over and over.
Dear ex-xian, all the answers to your questions are in http://www.geocities.com/complementarytheory/My-first-axioms.pdf
All you need is to read and (I hope) understand my work.
Good luck.
Lama:
As yet, you have not convinced anyone that your first axiom is not total gibberish.
Instead of wasting virtual space with tons of cooked-up definitions from your own fancy, please focus only on elucidation of your first axiom.
If you think it cannot be "understood" out of context with the rest, then it is not an axiom at all, but just gibberish.
Why do you need more people around you to be sure that my work is (by you) just gibberish?
At least respect yourself and say it clear: "I, arildno, think that your work is nothing but a gibberish!".
Believe me it will look much better then using the "not convinced anyone" style.
Because my framework is a paradigm-shift in the Language of Mathematics,
you have no choice but to understand it by the most fine intuition/reasoning
interactions abilities that you have.
Naturally well educated mathematicians are the first persons that have the biggest problems to understand a paradigm-shift because of more then a one reason, for example:
1) The foundation of their own work can be changed for better but also for worse, but in both cases they have to reexamine their work according the paradigm-shift, which is a very unpleasant situation, for persons who afraid from deep changes.
2) They are already trained by another 'school of thought', therefore it is hard for them to see fundamental things from a different point of view.
hello3719
Aug7-04, 05:53 PM
Naturally well educated mathematicians are the first persons that have the biggest problems to understand a paradigm-shift because of more then a one reason, for example:
1) The foundation of their own work can be changed for better but also for worse, but in both cases they have to reexamine their work according the paradigm-shift, which is a very unpleasant situation, for persons who afraid from deep changes.
2) They are already trained by another 'school of thought', therefore it is hard for them to see fundamental things from a different point of view.
Mathematicians are the first people that would be happy for some deep changes. You just don't explain your definitions well. Every new term you introduce in your axioms should be explained using the most fundamental concepts (ex. < ,> = ...)
Dear hello3719,
I am glad to know that you have no problems with fundamental changes.
(ex. < ,> = ...) are simply and clealy explaind in:
http://www.geocities.com/complementarytheory/My-first-axioms.pdf (including its links).
Lama, it isn't fair for you to criticize anyone for not understanding what you're doing, nor is it fair to say that the fact that virtually everyone has panned what you've done.
You've failed to explain what the premise(s) and the conclusion(s) are in your tautology. You've invoked something called "number," which you've not defined, in order to explain your first two axioms. This isn't the way math, any math, is done. You can't just pull concepts out of the air.
If you want to be taken seriously, you need to present your definitions, then your axioms. Your axioms need to be understandable and explainable only by appealing to the definitions or previous axioms. When you have to pull out mathematical words that you haven't defined in order to explain your axioms, you're not being at all productive.
I'm trying my best to understand you and to give you the benifit of the doubt. Here's what we have so far:
Set: A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
1) Since your building up the numbers from scratch you need to give definitions for "finite" and "infinite."
2) Also, do you have commonplace definitions in mind for "order" and "multiplicity?" That is, is {x, x, y} the same set as {y, x}?
Tautology: x implies x.
1) Since you refernced wikipedia, then any mention of a tautology necessarily means you have a proposition in mind.
2) That is, "x implies" is the same as "if x, then x."
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
1) What is a singleton set? Since you're building the numbers up from scratch, you either have to admit to the standard way of constructing the counting numbers or explain "singleton set" w/o referencing numbers.
2) What are internal parts? Remember, you have to define this only with previously defined terms.
3) You say that a point can be defined only by tautology, which you equate with the operator "=". When one writes "x = x" is this the same as saying "if x, then x"?
See if you can take care of these questions, please?
Dear ex-xian,
There comes a time where you have no choice but to use your own finest internal intuition/reasoning interactions abilities to understand something.
From this moment you are in your own and nobody, included me, can help you to understand this thing.
In short, I did my best to represent my ideas in the simplest and clearest ways that I can.
All what I have to say about my work can be found in http://www.geocities.com/complementarytheory/My-first-axioms.pdf and its links.
From this point I can only answer to questions that shows that the person who asks them already made his paradigm-shift in his mind and leaped by using his own abilities to my new framework.
for example, you wrote:
1) Since your building up the numbers from scratch you need to give definitions for "finite" and "infinite."
All you need is to understand how the axiom of completeness
(The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.)
is related to the rest of my work, because one of the important things in my system is, that you have to understand the whole of it in order to understand a part of it, where what I call 'the whole of it' is the simplest level of understanding, which stands in the basis of my work, and cannot fully shared with others (by using written axioms) that are not already share within them this common source of the simplest state, which is beyond any definition.
And this is exactly the deep meaning of the words ‘paradigm-shift’.
There is always an unclosed gap that has to be closed by an intimate-private journey that a person does in the deep and fine silence of his both heart and mind.
Good Luck.
Lama
How do complex numbers fit into your new theory? I see only talk of real numbers here. I have read through all 28 pages of this thread and I have had time to look at all of your files on your website. Please explain i=sqrt(-1) in terms of your new notation. Please do not refer me to one of your pdf files, as I know that there is no reference to the complex numbers.
The very well known mathematision Michael Atiya, gave a very interesting lecture about a year ago at the conference about "The unity of mathematics". He said that we are waiting now to a new-Newton that will break the enigma of mathematics and Physics.
He gave the example of I ( root of -1) to how new concept bring high resistant when they are appearing.
Well, I hope that you can see the analogy to this complitly
new theory about mathematics as one whole - M(i).
Moshek :approve:
What in the world has this forum become?! Lama, what is the point of repeatedly posting links to your pdf files and saying the same thing over and over again to show the community your brilliant new way of doing math if you say:
"nobody, includ[ing] me, can help you to understand this thing".
Asking people to abandon their current way of thinking about math is not reasonable. If you want to share the wonderfulness of what you've found, you must prove that it works to others in a way that is comprehensible to people. Why else would you have tried to share these ideas, producing 28 pages of a single thread. This is what forums and discussion are for. That is why you are here. If you just want to advertise your new ideas without accepting criticism, then please just use your website.
Lama, please try to understand that the reason some people have demanded that you demonstrate some kind of knowledge of the current system of math is because there hundreds of years of theoretical and applied research to back up its use. Math is taught in schools and university the way it is because it works and has been put to use in so many real world applications; ie, computers, physics, economics, etc. That's not to say that math is a static thing, certainly not. It, as you have said, evolves through research and new ideas by bright minds, but the great thing is, each time a new breakthrough in mathematics is made, it ENCOMPASSES and is CONSISTENT WITH all the old ideas and methodologies that came before.
I have read through all of the previous messages in this thread and have gone through every single pdf file on your website. I am not convinced that you have discovered a new branch of mathematics in which the old mathematics is merely the "shadow" of yours.
Why not attempt to re-express your ideas in a new way that is compatible with the way it is done normally. I think you should start by defining every single unique term in your axioms in your own words and not simply use the definitions found on Wolfram's website. If your definitions and following axioms are inconsistent in some way as some people have pointed out so far, go back rework them further until they are consistent.
And don't be so defensive when someone points out a mistake in your reasoning! It's annoying. Please do not say, "I have clearly shown [such and such] on this file at my website." You sound like a broken record when do this. Everyone has probably looked at the link you're referring to already. We need a new and clearer explanation of what you're trying to say in order to understand you.
Lama, you can brand me as a "bodyguard of math". Yes, there are bodyguards of math, and for a good reason. Research in science and math is a vigorous process involving the constant judgment of one's submitted ideas by the other peers of the research community. A new idea cannot be accepted as truth without first being questioned and scrutinized. This is the power of the research process. Imagine how crazy and disorganized science and math would be if every single idea was immediately incorporated into the current body of knowledge. Your ideas should be able to predict and explain the behavior of a certain model or system, and, if this is successful, then those same ideas should be able to do the same on a different system or model. If the new idea fails in the respect just once, then it must be rejected in favor of the old ideas that can already correctly explain some phenomenon. At that point, it's time to go back to the drawing board. I can tell you from experience that less than 5% of any research that is done in any one research group or laboratory in an academic setting ever goes somewhere significant. Don't be surprised if someone shows that you are wrong. A good mathematician or scientist is able to admit when they are wrong.
And back to my original beef with what has been going on here in this thread. I’ll quote you again:
"nobody, includ[ing] me, can help you to understand this thing".
Let us suppose for the moment that you actually have come up with some new way of doing math that will lead to tremendous advancements in the field and in science. What a shame it would be if that idea were to die with you some day then. You must be able to convince people who study math that you are correct in order for that advancement to take place. If you cannot make us understand, then it is all for nothing and you should not be wasting our and your time on this forum. If you cannot do this, go and learn how to; go and share your ideas with a professor in mathematics that is near you. Maybe he/she can help you express your ideas in a way that most people can understand. Believe me, I think it’s good that people like you want to explore and discover new things in mathematics. Mathematics is a beautiful thing as you have said before and is a very noble pursuit; but it is not one to be taken alone. If you want us to embrace your ideas, you have to be willing to embrace our thoughts and ideas as well.
Well, I guess that settles it. You're not seriously trying to do math after all. Your trying to do mysticism and smear math words over it.
If you want to be a mystic, that's fine and good. But at least be honest about it.
All you need is to understand how the axiom of completeness
(The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.)
Then you need to give definitions for lowest bound, highest bound, finite, and scale. And you have to define them all w/o using numbers, since you're building everything from scratch. Either that or concede the standard way of defining finite sets--which would concede the standard way of building the natural numbers.
I agree, sometimes it sounds more like mysticism to me.
I agree, sometimes it sounds more like mysticism to me.
And I actually don't have a problem with mysticism per se, but I wish more of them subscribed to Wittgenstein's philosophy of mysticism.
"Of that which we cannot speak, we must pass over in silence."
I with you on that one. I would prefer mysticism to religion any day of the week. But look at how many times people in history have looked silly for trying to mix mysticism/religion with science; for example, Kepler tried to explain the orbits of the planets in terms of known 3D polyhedrons of the time because they were "perfect" or "divine". That's garbage. It was only after he let go of the idea of "perfect" shapes that considered using elliptical orbits and sucessfully formulated his laws.
I with you on that one. I would prefer mysticism to religion any day of the week. But look at how many times people in history have looked silly for trying to mix mysticism/religion with science; for example, Kepler tried to explain the orbits of the planets in terms of known 3D polyhedrons of the time because they were "perfect" or "divine". That's garbage. It was only after he let go of the idea of "perfect" shapes that considered using elliptical orbits and sucessfully formulated his laws.
Perhaps the most blatant example of misplaced mysticism may be found in the philosopher Hegel.
He became so enthused by the idea that he'd found an overarching system(which "integrated" religion/morality/society/science/nature) that his "philosophy of nature" is just about the worst crank writing imaginable.
However, his earlier, less connected works (in which he didn't suffer from his unifying obsession to the same degree as later on) does contain bits an pieces of rather poignant and subtle social analysis.
If he had had less than the crank nature in him, he might well have developed a truly interesting social theory (placing him in the company of Max Weber, and others)
Perhaps the most blatant example of misplaced mysticism may be found in the philosopher Hegel.
He became so enthused by the idea that he'd found an overarching system(which "integrated" religion/morality/society/science/nature) that his "philosophy of nature" is just about the worst crank writing imaginable.
However, his earlier, less connected works (in which he didn't suffer from his unifying obsession to the same degree as later on) does contain bits an pieces of rather poignant and subtle social analysis.
If he had had less than the crank nature in him, he might well have developed a truly interesting social theory (placing him in the company of Max Weber, and others)
Yes, Hegel is a good example. My most favorite of his statements is when he claimed that the scientists were wasting their time searching for heavenly bodies--if they would just study philosophy they would realize that since 7 is the number of perfection, there can only be 7 and no more.
Incidentally, this brings me back to Lama/Dorian/Organic/etc. In E.T. Bells' Men of Mathematics, he is discussing the immortal Gauss, and what Gauss had to say about people like Hegel. He states that "those who wish to peck away at the foundations of mathematics would do well to sharpen their dull beaks on some hard mathematics first."
Or another of Gauss' immortal comments on Hegel:
"Der Mann ist verrückt"
LOL. I'm sure all matheticians have that to say about each other.
matt grime
Aug9-04, 04:02 AM
So instead of being able to give a tautology you've now had to remove it from your definition. Does this mean you're going to remove all your pointless posts where you refused to give a tautology and just reposted the same silly definition.
Now, if you could just remove direction from your new definition, which you've not defined then you're getting somewhere. Note, it appears that you're presuming the real numbers exist, but what are they in your system? You've just presumed they exist when they don't necessarily since they are just a construct in our system that you reject. Also note that you've admitted something we've been telling you for a while that you've ignored: that the organic numbers which you state are BASED on N, cannot therefore be more basic than them since you;ve used N in its definition.
So instead of being able to give a tautology you've now had to remove it from your definition
On the contrary, I added two statements, each one for each definition, see for yourself in: http://www.geocities.com/complementarytheory/My-first-axioms.pdf.
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Dear persons,
Thank you very much for your open-hearted replies.
1) I do not start every single concept from scratch, so words (for example) like, finite, and scale have their standard meaning.
2) Lowest bound or Highest bound are simply the edge values of a non-empty ordered set.
3) Scale factor is determined by the ratio of any R member to the entire R members.
As I wrote to ex-xian, I cannot do the whole work alone to convince someone about my point of view, if a person choose not to move from his spot, where things looks fundamentally different.
In short, you cannot stand in some place and say: “Please convince me to move to your place by describing what do you see from your place”.
I can do my best to describe (define) what I see, but the best advice that I can say is:
Please come to my place and see it by yourself, and only then we can argue about our different interpretation about what we see from this place.
Again, no one of you did this simple step, which is: To come to my place and see things by using his own eyes.
And this is the reason why I said that I have found that the most persons here acting like full_time_job bodyguards.
It is very important to check any new idea before we air our view about it, but it cannot be done if we are full_time_job bodyguards, which means, we do nothing to really see something by our own eyes, and instead we want that the mountain will come to us, instead of us to come and at least first see the mountain by using our own eyes.
And when we see the mountain by using our own eyes, then and only then we are in a position to decide if we want to clime on it, or not.
You have to understand that if you do not let yourself to see things by using your own eyes, we cannot move further in our dialog.
In short, I need your active participation on order to develop the dialog between us.
I hope that I explained my point of view about the dialog between you and me.
Please reply your remarks and insights.
Yours,
Lama
matt grime
Aug9-04, 04:44 AM
1) is patently false since you reject the proper definition of cardinality for example, so at best you occasionally use the proper definition of words, however, none of the words you've used in this item have a formal standard mathematical meaning that we can apply here with any certainty that doesn't require you to accept standard mathematics first. And even then "scale" is not well-understood.
2) Lowest and Highest imply an ordering, you've not proven any such exists in your system: what is the highest and lowest bound of the circle? See, you need to actually have the Real numbers as we know them already extant, and that is a contradictory position for you to adopt.
3) makes no sense (how do you define the ratio of an element of that set to the set? and that is a set yo'uve not even bothered to define)
and since you now say that you've not removed the requirement to be defined by tautology you muist still have some proposition in mind, so for the, what, 10th time what is that proposition?
what is the highest and lowest bound of the circle.
x1 and y1 are R members.
x1 and x2 are its lowest and highest bounds where x1 < x2, and its length is |x2-x1|*xs, where xs is any R member.
Matt why do you ignore my request for help that can be seen in my previous post?
matt grime
Aug9-04, 05:08 AM
you could also clarify what you mean by "indivisible" since I can "divide" the interval [0,2] into [0,1] and (1,2].
As it is your grasp of set theory is looking even shakier, and I bet I can make all your statements apply to Q or C, or P(3) if I felt like it, since they are so ill-defined. Of course, you are still presuming the existence in your model of something that you've not shown to exist. Your definitions are becoming increasingly circular: your set of reals is the set of points and intervals somehow, presumably, yet you can't define a point or interval without using the properties of your reals (order and equality, it is for instance not part of your axioms that this set even has an ordering, yet you use it).
matt grime
Aug9-04, 05:10 AM
Where did you set of reals obtain its ordering? Why is length of an interval not well defined? What is R in your system? What is the tautology that defines x1 and y1? were did the abs value operator come from?
you could also clarify what you mean by "indivisible" since I can "divide" the interval [0,2] into [0,1] and (1,2].
I clearly and simply clarify what I mean by "indivisible".
In my system [0,1] and [1,2] are two {.}_AND_{._.} indivisible elements where [0,2] is also {.}_AND_{._.} indivisible element.
Also I have a different interpretation to (r1,r2].
Again you cannot see (understand) my framework in terms of your framework,
and if you can't grasp this, then we cannot communicate.
matt grime
Aug9-04, 06:24 AM
It is not that I cannot understand your terms, but that you haven't defined them. incidentally the correct mathematical term is connected, not indivisible.
You have still not given this tautology that you say exists.
A note for everyone: shall we refuse to comment or in anyway communicate in this thread until that simple request has been met?
Let us absolutely make clear what is required:
something is alleged to be defined (only) by a tautology. That thing is a set. A tautology is a proposition. Please give the (an) example of this (or any other) tautology that defines this (or any other) set.
Until you can do that you have no need to post anything else, Doron.
Matt, please read this again and pay attention to * and * propositions.
Tautology:
x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
(tautology is also known as the opposite of a contradiction).
(EDIT: instead of the above definition, I change Tautology to: The identity of a thing to itself.
It means that in my framework we do not need 'if, then' to define a Tautology)
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
Urelement:(no internal parts)
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.
A definition for a point:
A singleton set p that can be defined only by tautology* ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.
(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)
By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}.
Now we have the minimal building-blocks that allow us to define the standard R members.
(edit:
*A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
*A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.)
The axiom of independency:
p and s cannot be defined by each other.
By the above axiom {.} and {._.} are independed building blocks.
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Now please tell us dear Matt why * and * are not tautological propositions?
(Logical statements in which the conclusions are equivalent to the premises)
incidentally the correct mathematical term is connected, not indivisible.
No, {._.} or {.} are indivisible exactly as I say.
If you want to force your point of view on my framework then do not west your time, beacuse it will nor work and you will stay in your spot.
That thing is a set
No Matt, that thing is a content of a set, which its existence cleary and simply defined by an axiom and astatment for each case of '.' or '_' .
matt grime
Aug9-04, 07:31 AM
They are not even propositions, Doron.
Ok Matt since you say that you understand my first two axioms, then please help me and write their logical propositions (statmants), because I do not know what do you mean when you say 'propositions' in this case.
Thank you.
matt grime
Aug9-04, 07:55 AM
A proposition is a statement of the form If X then Y where X and Y are some mathematical statements to which we can assign truth values.
Example:
If an integer, n, is divisible by 4, then n is even.
We start from the antecedent (n is divisible by 4) and come to the conclusion (n is even).
This proposition is true.
A proposition here would be:
If P is a point then p is defined only by a tautology.
Note, that that in itself is NOT the tautology that defines p, if it were you'd have a circular argument, which is not the tautology you require since it does not tell us what p is (ie offer any indication p even exists). You cannot rely on the definition of p to define p like that, since it does not prove that any such "p" exists. You must therefore offer another model of some such p, and by doing so you will contradict the minimality that it is only defined by (this alleged) tautology as you will be defining it in some other way. Sorry, but as we've been telling you your definition is not consistent.
In short, you cannot define things by tautologies like this, and you cannot prove that any such p exists since it contradicts its own definition. That is not to say it cannot exist, but that we cannot know it exists.
It makes someone wonder what you thought you were doing? An ungenerous person might think you'd been displaying ignorance of things you claim to understand. After all, it's you who's been posting the wikipedia definitions, did you not understand them?
And I didn't say I understood your two axioms, I said I was happy to accept them, and asked you to provide examples and models of things satisfying those axioms. You didn't, and, it appears, can't.
An example of a tautology (in mathematics)
If A then (if A then A)
which is equivalen to (not(A)) or ((not(A)) or A), which is equivalent to (not(A))orA, which is always true irrespective of the truth of A.
note that this of course requires you to have boolean logic and excluded middle and so, some things you reject, so actually you are once again possibly misusing a mathematical term, since your tautologies, may not be our tautologies, even after you've made a propositional form out of it. More contradictions again.
The best I can do is:
If a content of a set is a singleton and a urelement and has no directions, then it is a point.
If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
I understans tautology in a different way than you.
For me tautology is not if,then proposition but a self avident existence of a thing to itself for example:
{}={}, {.}={.}, {._.}={._.}, {__}={__}
This is the reason why I call '=' 'only by tautology'.
matt grime
Aug9-04, 08:11 AM
And you have removed the requirement of some tautology then. Which is good, because that was wrong.
Now, you just need to explain what a direction is.
It is still not clear of you've "defined R" and now define points and intervals in terms of R, or if you are defining points and intervals and then will define what R is.
Which of those two orders is it?
It appears, since you are talking about direction, and equality and ordering (<,> etc) that R must exist in your alleged system, so what is R?
matt grime
Aug9-04, 08:25 AM
So, your next requirement is to tell us all what YOUR reals are. Until you do that we can do nothing.
Ok, let us write is again:
If a content of a set is a singleton and a urelement and has no directions, then it is a point.
If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
(more detailed explanation of the first two definitions:
Let us examine these first two definitions by using the symmetry concept:
1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.
It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.
2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'
In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .
In short, in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise (mathworld.wolfram.com/Tautology.html)
As you see, in my framework '<','>' symbols have a deeper meaning then 'order'.
Actually, in order to talk about 'order' we first need a 'direction'.
matt grime
Aug9-04, 08:31 AM
For me tautology is not if,then proposition but a self avident existence of a thing to itself for example:
a tautology is not required to be a proposition, merely a statement that is true irrespective of the truth values of the elements in it.
A=>A is a tautology
A=>B is not
A<=>A is not
(A=>(B=>C) => (A=>C) is i think a tautology.
it is not a self evident truth.
so stop saying tautology and pointing to the mathematical definition as if it is the one you are using. everything is in some sense equivalent to itself (tautologous in your new fangled interpretation) so it is impossible to say what is only defined by a tautology, since we cannot define it in any other way. that attempt at a definition is illogical.
Now, what is R in your system?
matt grime
Aug9-04, 08:34 AM
I almost give up: what you've shown is that the definition of something implies itself. that is always true and not that useful directly. of course, your use of the word definition is different from everyone elses.
Now, what is R? in you system of course.
Since you missed it then here it is again:
If a content of a set is a singleton and a urelement and has no directions, then it is a point.
If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
(more detailed explanation of the first two definitions:
Let us examine these first two definitions by using the symmetry concept:
1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.
It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.
2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'
In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .
In short, in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise (mathworld.wolfram.com/Tautology.html)
As you see, in my framework '<','>' symbols have a deeper meaning then 'order'.
Actually, in order to talk about 'order' we first need a 'direction'.
Also by my system a=a is a tautology, and we do not need 'if,then' to define it.
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Do you have any remarks before we visit R?
(I have changed Tautology to: The identity of a thing to itself)
Matt, there are some points that we have to check before we examine R by my framework:
1) In an included-middle reasoning contradiction is not used because any two opposites are simultaneously preventing/defining their middle domain.
2) A thing is defined by its identity to itself, and we do not need the 'if, then' proposition (as we do in an excluded-middle reasoning) in order to define the existence of something.
3) By this reasoning we distinguish between a true statement and a tautology, which by included-middle reasoning is simpler and stronger then any existence that must an 'if, then' to exist.
4) The included-middle interpretation of a Tautology is circular only if we look at it from an excluded-middle reasoning. But then we must realize that we see and understand things which are not from an included-middle point of view.
If you stay in standard point of view, then you cannot understand my new framework (and in this sentence I used the standard 'if, then' reasoning)
Included-middle reasoning is the logic of mutual communication between opposites (and in this sentence I used the self identity of a thing to itself)
In short, I gave just now a simple demonstration that I understand very well the standard point of view.
Can you do something which demonstrates that you are able to write something from an included-middle logical reasoning?
If you cannot do it at this stage, then your mathematical skills will not help you to understand my system (and here I used again the 'if, then' reasoning).
An included-middle reasoning can be understood only by an included-middle reasoning (and here I used an included-middle reasoning).
Some examples:
By my system 4 not= 2+2.
By my system 4=4 and 2+2=2+2.
Now can say: But you can do nothing with these two trivial and circular equations, right?
My answer is: In included-middle reasoning we can do very interesting things because 4 not= 2+2, for example:
For example, let us represent the variations of cardinals(*) 2,3,4:
Let Redundancy be more then one copy of the same value can be found.
Let Uncertainty be more than one unique value can be found.
Let XOR be #
Let a=0,b=1,c=2,d=3 then we get:
b b
# #
{a, a, {a, b}
. . . .
| | | |
|__|_ |__|
| |
{x,x} {{x},x}
c c c
# # #
b b b b b
# # # # #
{a, a, a,} {a, a, c} {a, b, b}
. . . . . . . . .
| | | | | | | | |
| | | |__|_ | |__|_ |
| | | | | | |
|__|__|_ |_____| |_____|
| | |
| | |
{{x,x,x} {{x,x},x} {{x},x},x}
Uncertainty
<-Redundancy->^
d d d d |
# # # # |
c c c c |
# # # # |
b b b b |
# # # # |
{a, a, a, a} V {a, b, c, d}
. . . . . . . .
| | | | | | | |
| | | | |__| | |
| | | | | | | <--(Standard Math language uses only
| | | | |_____| | this no-redundancy_
| | | | | | no-uncertainty_symmetry)
|__|__|__|_ |________|
| |
={x,x,x,x} ={{{{x},x},x},x}
============>>>
Uncertainty
<-Redundancy->^
d d d d | d d d d
# # # # | # # # #
c c c c | c c c c
# # # # | # # # #
b b b b | b b b b b b b b b b
# # # # | # # # # # # # # # #
{a, a, a, a} V {a, a, a, a} {a, b, a, a} {a, a, a, a}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
{x,x,x,x} {{x,x},x,x} {{{x},x},x,x} {{x,x},{x,x}}
c c c
# # #
b b b b b b b
# # # # # # #
{a, b, a, a} {a, b, a, b} {a, a, a, d} {a, a, c, d}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
{{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x} {{{x,x},x},x}
a, b, c, d}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this
|_____| | no-redundancy_no-uncertainty_symmetry)
| |
|________|
|
{{{{x},x},x},x}
Also please pay attantion that the last form is the standard R members 0,1,2,3:
0 = .
1 = 0______1
2 = 0____________2
3 = 0___________________3
And the standrard R is nothing but the above 2-D representation
in a 1-D representation:
0______1______2______3
And because no R member is both Multiset_AND_Set, I call it: The "shadow" of my new number system.
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(*) Please pay attention that we are not talking about the natural numbers 2,3,4 but the cardinals 2,3,4.
It means that our Organic Natural Numbers are actually a general representation of information-trees, where any finite quantity of names of R members can be described by them, for example:
Instead of a=0,b=1,c=2,d=3 we can use a=0,b=.5,c=3,d=pi and then we use the same information-trees above.
I called these general information-trees 'Organic Natural Numbers' because:
1) These information-trees of cardinals are always having a structure, which is based on N members.
2) They can be used as natural (not forced) and general representation for any interaction between complementary states, which simultaneously preventing/defining their middle domain.
3) Because no R member is divisible by my system, it has its own organic (complete) unique and independent self existence.
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Matt, please reply your comments to this post before we continue, thank you.
Since you're trying to redefine another word, in this case "tautology," it's better to just take it out of your definition.
1) I do not start every single concept from scratch, so words (for example) like, finite, and scale have their standard meaning.
If "finite" has it's standard mathematical meaning, then you are assuming the existence of the natural numbers. Here's Rudin's definition of a finite set
For any positive integer (natural number) n, let J_n be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers (or natural numbers)....A is finite if A has the same cardinality of J_n for some n
Again, if you're using finite as a concept in your definitions, you must accept the standard definitions for the natural numbers (and also cardinality). If you do not concede this, then you cannot use the concept of finite, or else define it yourself.
By accepting the natural numbers, you must either construct them yourself, or accept the standard construction. This in turn forces you to accept standard ZF set theory. Here are the axioms:
1. The empty set is a set as is every member of a set.
2. If X is a set and, for each x in X, P(x) is a proposition, then
{x in X: P(x)} is a set.
3. If X is a set and Y is a set, so is {X, Y}.
4. If X is a set, {z: z in x for some x in X} is a set.
This set the “union of X”.
5. If X is a nonempty set, {z: zin x for each x in X} is a set.
This set is the “intersection of X”
6. If X is a set, {z: z is a subset of X} is a set.
This set is called the power set of X.
7. The set N of all natural numbers is a set.
8. No set is a member of itself.
9. If X is a set and Y is a set, so is X cross Y.
I left off the axiom of choice, which is often included.
Are you prepared to concede all this to have your real numbers? If not, you must provide your own construction of the naturals.
matt grime
Aug10-04, 04:17 AM
My simple comment is that you evidently do not understand the distinction between equivalent and equal, but heck, you're not very well educated mathematically, so whose fault is that?
99% of your last post is nonsensical unless you actually get round to defining all of the terms you use so that the rest of the world might know what you're doing. You should also put a huge disclaimer saying:
contrary to what I said earlier, when I use words I do not in fact use them with their proper well understood meanings.
You're talking about something you've labelled R butnot actually saying what it is. AGAIN.
You also don't seem to understand that, whilst you say your interpretation of "tautology" is different from that in formal mathematics, you cannot then cite its definition! And there is a difference between statement being tautologous, and it being true, since everything satisfies its own definition, then including this "tautology" is unnecessary.
It reminds me very much of your confusion over the Collatz conjecture, that you thought you'd figured out.
Matt,
My simple comment is that you evidently do not understand the distinction between equivalent and equal, but heck, you're not very well educated mathematically, so whose fault is that?
I know exactly the difference between:
_____
_____ AND ______ NOTATIONS
_____ ______
The first one is stronger then the other.
By standard Math, if internal structural properties are omitted and only quantity remains, then and only then (and because of this trivial, again not abstract but trivial approach) there is a difference between 'identical' and 'equal'.
But you see, in my framework a '=' notation has one and only one meaning, which is the identity of a thing to itself, and the reason is:
Any given element in my system is unique and cannot be represented by any other element but itself.
And by this fundamental approach, my system is sensitive to any information change between elements, and can use these differences to make richer and deeper Math.
You are still standing in your spot and observe my system from your standard point of view, and I am talling you again, it will not work.
You're talking about something you've labelled R butnot actually saying what it is. AGAIN.
I alreadry crearly and simply defined my number system, and it is very easy to see how R members are the "shadows" of my number system.
Here it is again, and if you still standing in your spot, you will not understand it:
Tautology:
The identity of a thing to itself.
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
Urelement:(no internal parts)
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.
A definition for a point:
A singleton set p that can be defined only by tautology*('='), where p has no internal parts.
-----------------------------------------------------------------------------------------------
Now let us move to the next step in order to define what is a number in my system.
First let us examine a well-known relation between mathematical objects and their representations.
=>> is ‘represented by’
|{}|=>>0 ; |{{}}|=>>|{0}|=>>1 ; |{{},{{}}}|=>>|{0,{0}}|=>>|{0,1}|=>>2 ;
|{{},{{},{{}}}}|=>>|{0,{0,{0}}}|=>>|{0,1,2}|=>>3 ; …
A definition for an interval (segment):
A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.
(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)
By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}.
Now we have the minimal building-blocks that allows us to define the standard R members.
(more detailed explanation of the first two definitions:
Let us examine these first two definitions by using the symmetry concept:
1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.
It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.
2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'
In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .
*A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
*A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
(In standard Math we had to write:
A point: If a content of a set is a singleton and a urelement and has no directions, then it is a point.
A segment: If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
But in this framework A=A is a tautology, and we do not need an ‘if, then’ proposition in order to define it) )
The axiom of independency:
p and s cannot be defined by each other.
By the above axiom {.} and {._.} are independed building blocks.
The axiom of complementarity:
p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).
By the above axiom we define the basic property of the middle domain between {.} and {._.}
The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
The above axiom allows us to:
1) To define the internal structure of standard R members.
2) To define the internal structures of my new number system.
The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.
The above axiom allows us to construct a collection of R members and also a collection of my new number system.
First, let us see how we use my method to construct a collection of R members.
R members are constructed like this:
1) First let us examine how we represent a number by my system:
=>> is ‘represented by’
a) |{}|=>>0
b) There is 1-1 and onto between ‘0’ and the left point of {._.} and we get {‘0’_.}
c) |{{}}|=>>|{0}|=>>1
e) There is 1-1 and onto between ‘1’ and the right point of {._.} and we get {‘0’_’1’}
In short, {.} is the initial place of R collection, which is represented by ‘0’, where {‘0’_.} is the initial place of the second place of R collection, which is represented by ‘1’, and we get our first two must-have building-blocks of R collection.
2) When we get {‘0’_’1’} we have our two must-have numbers, which are ‘0’ and _’1’.
Be aware that ‘0’ is the representation of {.} where ‘1’ is the representation of {._.}.
3) If we get {.}_AND_{._.}, then and only then we have the minimal must-have information to construct the entire R collection because:
a) We have ‘0’ AND _’1’ that give us the to basic scale factors 0 and _1.
b) We also have our initial domain _1, which standing in the basis of any arbitrary scale factor that is determined by the ratio between the initial domain _1 and another segment that is smaller or bigger than the initial domain _1 , for example:
0 = .
1 = 0______1
2 = 0____________2
3 = 0___________________3
.5 = 0__.5
pi = 0______________________pi
The negative numbers are the left mirror image of the above numbers.
There is no division in my number system because both {.} and {._.} are indivisible by definition.
In short, any segment is an independent element, that clearly can be shown in the above 2-D representation.
If we use a 1-D representation, we get the standard Real-line representation, but then we can understand that division is only an illusion of an overlap of independent elements when they are put on top of each other in a 1-D representation, for example:
0__.5 __1_____2_____3__pi
(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.
Since in my system nothing is divisible, then '/' stands for a ratio between at least any given two (indivisible) numbers.
-----------------------------------------------------------------------------------
If "finite" has it's standard mathematical meaning, then you are assuming the existence of the natural numbers. Here's Rudin's definition of a finite set
Let us stop here (before we continue to my new number system) to get your remarks.
I already gave you my answer to 'finite', which is:
The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.
Then you asked me what is lowest and highest bounds, and my answer is:
The ends of some given element, where beyond them it cannot be found.
This last definition (and its opposite) is for {},{.},{._.} and {__} sets.
CrankFan
Aug10-04, 10:19 AM
No matter how many times you repost your incomprehensible crap it's still incomprehensible crap.
arildno
Aug10-04, 10:23 AM
No matter how many times you repost your incomprehensible crap it's still incomprehensible crap.
What's really sad is that he's been into this crap for over 20 years, according to some of his more lucid statements..
matt grime
Aug10-04, 10:50 AM
So, you've got something that is your version of R, though you've not shown that it is a field, that it has a metric (though you've used it, preversely), that it is complete, etc... of course I understand that this is not necessary because your R is not the R as understood by the rest of humanity, just as those aren't naturals as we know them, tautology has a new meaning, as does =, symmetry you use without explanation, and i'm very impressed that you've introduced another spurious and unexplained property (tight). also, you're presuming that your set R has a direction given by an order, something else you've not shown to be true, nor that there is anything in those intervals, or that pi even exists in that system as a deducible fact from the axioms.
your axioms are rather clever in that part of them is that anything you choose to exist exists without proof.
arildno and CrankFan,
No matter how many times you will look at my work from your spot of reasoning, you will not be able to understand it, unless you come to my spot.
If you do not want to do it or cannot do it, then do not west your time in this thread, because you will see nothing, even after 20 years.
arildno
Aug10-04, 12:43 PM
arildno and CrankFan,
No matter how many times you will look at my work from your spot of reasoning, you will not be able to understand it, unless you come to my spot.
If you do not want to do it or cannot do it, then do not west your time in this thread, because you will see nothing, even after 20 years.
Please explain what a "spot of reasoning" is.
And, mind you, no silly analogies this time.
your axioms are rather clever in that part of them is that anything you choose to exist exists without proof.
Yes Matt isn't it beautiful and simple?
1) A thing is always consistent with itself, otherwise we cannot find it.
2) Since we can find it we do not need to prove its existence.
3) Now all we need is to find interesting and useful ways of interactions between our tautologies.
4) The definitions of our used concepts help us to construct some agreement that can help us to work together, but we have to be aware that this common agreement is only a tool for the mathematical creation itself, which is always a unique creation of the person which creates it, and it is never a technical tuning of the agreement.
The mathematical unique creation changes the agreement, because it is an open thing by nature, where any agreement is a closed thing by nature that has to fit itself to Math development when time comes.
During the last 2500 years we can see that meaningful developments in Math always changed the common agreement, where persons that had no creative minds where forced to change their common agreements in order to fit themselves to the new development.
And why they where forced to do it?
Because this development was not their creation, and this creation broke their agreement, which is the only playground for non-creative persons.
-----------------------------------------------------------------------------------
Edit:
I want to add some notes about '=' notation which is in my system is first of all used for tautology, where tautology in my system is the identity of a thing to itself.
But I also continue to use this symbol in the standard way for example:
4=2+2 but this equation is different then 4=4 and 2+2=2+2 which are tautologies
(for eample:
4=0_______4 ; 2=0___2 ; 2+2=0___2 + 0___2 ).
By 4=2+2 we clearly mean that we care only about the quantity,
because it is clearly understood that (0_______4) is not (0___2 + 0___2) by tautology.
Actually we use quantity as a "glue" to connect different tautologies to each other, for example:
For example, let us represent the variations of cardinals(*) 2,3,4:
Let Redundancy be more then one copy of the same value can be found.
Let Uncertainty be more than one unique value can be found.
Let XOR be #
Let a=0,b=1,c=2,d=3 then we get:
b b
# #
{a, a, {a, b}
. . . .
| | | |
|__|_ |__|
| |
{x,x} {{x},x}
c c c
# # #
b b b b b
# # # # #
{a, a, a,} {a, a, c} {a, b, b}
. . . . . . . . .
| | | | | | | | |
| | | |__|_ | |__|_ |
| | | | | | |
|__|__|_ |_____| |_____|
| | |
| | |
{{x,x,x} {{x,x},x} {{x},x},x}
Uncertainty
<-Redundancy->^
d d d d |
# # # # |
c c c c |
# # # # |
b b b b |
# # # # |
{a, a, a, a} V {a, b, c, d}
. . . . . . . .
| | | | | | | |
| | | | |__| | |
| | | | | | | <--(Standard Math language uses only
| | | | |_____| | this no-redundancy_
| | | | | | no-uncertainty_symmetry)
|__|__|__|_ |________|
| |
={x,x,x,x} ={{{{x},x},x},x}
============>>>
Uncertainty
<-Redundancy->^
d d d d | d d d d
# # # # | # # # #
c c c c | c c c c
# # # # | # # # #
b b b b | b b b b b b b b b b
# # # # | # # # # # # # # # #
{a, a, a, a} V {a, a, a, a} {a, b, a, a} {a, a, a, a}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
{x,x,x,x} {{x,x},x,x} {{{x},x},x,x} {{x,x},{x,x}}
c c c
# # #
b b b b b b b
# # # # # # #
{a, b, a, a} {a, b, a, b} {a, a, a, d} {a, a, c, d}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
{{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x} {{{x,x},x},x}
a, b, c, d}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this
|_____| | no-redundancy_no-uncertainty_symmetry)
| |
|________|
|
{{{{x},x},x},x}
CrankFan
Aug10-04, 01:26 PM
Lama,
What makes you so certain that you aren't the one who is misunderstanding?
Why are you so insistent that all of the people who have criticized your presentation for being ambiguous aren't fully grasping it, and need to see a repost of one of your papers? Maybe it's the other way around, maybe you aren't entirely grasping the nature of their criticism. Maybe you need to work harder to understand criticism of your work and stop automatically dismissing it.
If you are sincerely interested in communicating your ideas to people in this forum then I think the best approach for you is to find some rigorous mathematics texts on a subject you think you would enjoy and then study those texts in earnest. Follow all of the proofs, do all of the exercises, suppress your ego and seek help from people when you are genuinely stuck. Be sincere in your approach to learn and understand it.
If you do this, you will learn by example (of the author) and through practice (completing the exercises) what is roughly considered to be an acceptable definition and proof. Then, you can utilize this knowledge to describe your ideas so that they can be understood by your target audience on this forum.
What makes you so certain that you aren't the one who is misunderstanding?
Please understand that you did not show any understading of my work.
On the contrary I showed that understand the standard framework.
Deeviant
Aug10-04, 01:58 PM
Lama, I think you would have more success creating a new religious belief than a new system of math. You skillsets would serve you better in that endeavor.
arildno
Aug10-04, 02:05 PM
On the contrary I showed that understand the standard framework.
Eeh, whenever you've been pinned down to specifics (for example, in your usage of "tautology" or "="), you have consistently shown a lack of understanding of these terms; you have been using them with completely different meanings than standard usage
you have been using them with completely different meanings than standard usage.
Yes, you are right because my framework is based on an included-middle reasoning.
Lama, I think you would have more success creating a new religious belief than a new system of math.
Do you say it after you showed us that you understand what is an included-middle reasoning?
arildno
Aug10-04, 02:30 PM
Yes, you are right because my framework is based on an included-middle reasoning.
Please explain what included-middle reasoning is, and why it is valid to term it a "reasoning"
Please explain what a "spot of reasoning" is.
And, mind you, no silly analogies this time.
Each one of us can understand the other by using its logical reasoning method.
You use the excluded-middle reasoning and I use an included-middle reasoning, which is deeper, simpler and richer then the excluded-middle reasoning.
In short, excluded-middle reasoning is nothing but a trivial sub-system of included-middle reasoning, and included-middle reasoning cannot be understood from this "spot of reasoning".
arildno
Aug10-04, 02:37 PM
You are an arrogant idiot. Goodbye
Please explain what included-middle reasoning is, and why it is valid to term it a "reasoning"
Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf and see for yourself not just what is an included-middle reasoning, but how I use it to develop my framework.
You are an arrogant idiot. Goodbye
Why, is it because of something that you have found in my article?
anti_crank
Aug10-04, 02:56 PM
Each one of us can understand the other by using its logical reasoning method.
You use the excluded-middle reasoning and I use an included-middle reasoning, which is deeper, simpler and richer then the excluded-middle reasoning.
In short, excluded-middle reasoning is nothing but a trivial sub-system of included-middle reasoning, and included-middle reasoning cannot be understood from this "spot of reasoning".
Lama, please try to understand that this is not a valid response to the question you were asked. You have simply made grandiose statements that your reasoning is so much better than the one we've been using for 200 years, but failed to offer a single statement in support of that. Show how it can be "simpler, richer and deeper" than what we have. Show how the excluded middle reasoning is a trivial subsystem of it (which I think is not true as the two systems are contradictory. Feel free to correct me if you can convincingly show otherwise). I don't know whether you realize that absolutely everyone here says simiar things about their pet theories, and such claims always have the opposite effect of what they intend. You may also not realize that people don't like being constantly given reading assignments, since it is your job to make your case through reasonably self-contained posts. You may not be aware that in some forums, this practice is considered spam and such posts are summarily erased.
Call me a bodyguard too if you like, as you've expressed your dislike of my methods before. It is ironic that for one who is having such a hard time making himself understood (which you seem to genuinely want), you have not even tried to understand the method to my madness. As mathematics is not subject to the constraint of having to describe an experimentally accessible world, your system can coexist with any other system, assuming of course it is consistent, satisfies the standards of rigor, and it is useful in that it allows desirable results to be drawn. I am not a mathematician so I am not qualified to judge that, hence I do not want you to waste your time on me. Feel free to put my advice to whatever use you wish.
ex-xian
Aug10-04, 03:01 PM
Yes, you are right because my framework is based on an included-middle reasoning.
What does included middle reasoning have to do with you changing the defintion of tautology?
If "finite" has it's standard mathematical meaning, then you are assuming the existence of the natural numbers. Here's Rudin's definition of a finite set
I already gave you my answer to 'finite', which is:
The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.
This is circular reasoning! I asked to define finite w/o "standard" mathematics or else concede everything I specified above. You respond by "answering" finite with an axiom that includes the disputed definition.
This is really getting silly.
Dear anti-crank,
What I was doing in the last 2 years in this forum is to develop my framework by trying the best I can to explain my non-standard ideas.
Instead of look around you and see what other people are saying, I suggesting you to read by yourself the fruits of my efforts.
You do not need complecated mathematical teachings to understand my work because my work is on the most fundamental concepts of the Language of mathematics, and because it is new in cannot be compared with the goals and achievements of the standard framework.
Please read:
http://www.geocities.com/complementarytheory/No-Naive-Math.pdf and all of it links,
and also http://www.geocities.com/complementarytheory/My-first-axioms.pdf.
More general view of my work can be found in:
http://www.geocities.com/complementarytheory/CATpage.html
After you have your personal opinion on my work, I'll be glad to discuss with you about it.
Yours,
Lama
This is circular reasoning! I asked to define finite w/o "standard" mathematics or else concede everything I specified above. You respond by "answering" finite with an axiom that includes the disputed definition.
This is really getting silly.
Then you missed the good part of this post where I wrote:
Then you asked me what is lowest and highest bounds, and my answer is:
The ends of some given element, where beyond them it cannot be found.
This last definition (and its opposite) is for {},{.},{._.} and {__} sets.
As you see I have two basic states for finite things:
One is for singletons like {},{.},{._.} and {__} sets and the other is for collections of {.}_AND_{._.} elements.
If each {.}_AND_{._.} has a unique notation (for example: '1', 'pi' , (1/3=@) '@', (.99999...=&) '&', ...), then a collection with more the one notation can have finite or infinitely many notations, and we do not need ZF axioms for this.
Also please pay attention that Cantor's second diagonal proof does not hold here because we are using a single and unique notation for any given number.
ex-xian
Aug10-04, 03:58 PM
Then you missed the good part of this post where I wrote:
As you see I have two basic states for finite things:
But you still have never given a defintion for what finite is.
But you still have never given a defintion for what finite is.
Dear ex-xian, please show why do we need a special definition for 'finite'
by demonstrate how we can understand 'finite' in more than one way, thank you.
Anyway I can use this:
a) A colloction of more then one element where its lowest bound and its highest bound are included in it.
b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.
ex-xian
Aug10-04, 06:35 PM
Dear ex-xian, please show why do we need a special definition for 'finite'
by demonstrate how we can understand 'finite' in more than one way, thank you.
Anyway I can use this:
a) A colloction of more then one element where its lowest bound and its highest bound are included in it.
b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.
Ok, but what are the "ends of some given element?" What does "beyond them" mean? As Matt pointed out, you're assigned order to sets without specifying what dicates that order.
you're assigned order to sets without specifying what dicates that order
Instead of repeating Matt's point of view, all you have to do is to read my papers, and then you will discover that my elements ({} and {__} are excluded) are {.}_AND_{._.}, where {.}_AND_{._.} are at least Multiset_AND_Set which is ordered by its own internal symmetry degrees, when a given quantity remains unchanged.
But if we return to standard Math where we have Set_XOR_Multiset (and any fundamental concept is based only on quantity), then in the case of a multiset {1,1,1,1,1} for example, we cannot order its content, but even in this case, it has ends that beyond them it cannot be found.
For example: its lowest bound is '{1,' and its highest bound is ',1}’.
In the case of a "normal" set, where any member is a unique member, the lowest bound is '{x1,' and the highest bound is ',xn}' and we don't care about the internal order of the set, so as you see in both cases the standard point of view remains the same, which means, we care only about the cardinality of some set or multiset contents, and we do not care about their order.
So as you see we can use:
1) Peano axioms of the natural numbers (where a definition of a set does not exist).
2) then we can use the definition of a set as:
Set:
A set is a collection of objects in which order has no significance, and multiplicity is also ignored.
3) And then we can use:
“For any positive integer (natural number) n, let J_n be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers (or natural numbers)....A is finite if A has the same cardinality of J_n for some n” (and this is an example that clearly shows that 'finite' in standard Math is based only on 'quantity').
Let us check again my definition for 'finite':
a) A singleton or a collection of more then one element where its lowest bound and its highest bound are included in it.
b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.
As you see my definition is stronger, simpler and richer then the quantity-only definition, because it works for both quantity and structure.
Ok, but what are the "ends of some given element?" What does "beyond them" mean? As Matt pointed out, you're assigned order to sets without specifying what dicates that order.
We can do this endless game of questions also to:
1. The empty set is a set as is every member of a set.
(What is: empty, set, every, member?)
2. If X is a set and, for each x in X, P(x) is a proposition, then
{x in X: P(x)} is a set.
(What is: set, proposition?)
3. If X is a set and Y is a set, so is {X, Y}.
(What is: set?)
4. If X is a set, {z: z in x for some x in X} is a set.
This set the “union of X”.
(What is: set, union?)
5. If X is a nonempty set, {z: zin x for each x in X} is a set.
This set is the “intersection of X”
(What is: set, nonempty, intersection?)
6. If X is a set, {z: z is a subset of X} is a set.
This set is called the power set of X.
(What is: subset?)
7. The set N of all natural numbers is a set.
(What is: set?)
8. No set is a member of itself.
(What is: set, member, itself?)
And in a very short time we get questions like:
What is a definition?
What is What is?
And so on ...
So, as you see ex-xian, any agreement is always based on some arbitrary decision between some community of people that stop the game of endless questions end start to work with each other on the basis of this agreement.
And the Community of Mathematician is not a special community.
matt grime
Aug11-04, 04:55 AM
Dear ex-xian, please show why do we need a special definition for 'finite'
by demonstrate how we can understand 'finite' in more than one way, thank you.
Anyway I can use this:
a) A colloction of more then one element where its lowest bound and its highest bound are included in it.
b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.
But this is rubbish. [0,1] is a set with an infinite cardinality (using the words properly, and we cannot use your definitions becuase they make no sense), that "contains its end points". And this would only be something that applies to sets that have "ends" which is stupid. (the collection of all isomorphism classes of groups of order 6 is finite, but doesn't have any bounds)
(the collection of all isomorphism classes of groups of order 6 is finite, but doesn't have any bounds)
Ho, yes they have, you cannot find the 7 of them and you cannot find 0 of them, but you can find 1 to 6 of them.
[0,1] is a set with an infinite cardinality
Only if there are also infinitely many scales.
Please read post #471, thank you.
Also you missed this so I copeid it for you:
I want to add some notes about '=' notation which is in my system is first of all used for tautology, where tautology in my system is the identity of a thing to itself.
But I also continue to use this symbol in the standard way for example:
4=2+2 but this equation is different then 4=4 and 2+2=2+2 which are tautologies
(for eample:
4=0_______4 ; 2=0___2 ; 2+2=0___2 + 0___2 ).
By 4=2+2 we clearly mean that we care only about the quantity,
because it is clearly understood that (0_______4) is not (0___2 + 0___2) by tautology.
Actually we use quantity as a "glue" to connect different tautologies to each other.
For example, let us represent the variations of cardinals(*) 2,3,4:
Let Redundancy be more then one copy of the same value can be found.
Let Uncertainty be more than one unique value can be found.
Let XOR be #
Let a=0,b=1,c=2,d=3 then we get:
b b
# #
{a, a, {a, b}
. . . .
| | | |
|__|_ |__|
| |
{x,x} {{x},x}
c c c
# # #
b b b b b
# # # # #
{a, a, a,} {a, a, c} {a, b, b}
. . . . . . . . .
| | | | | | | | |
| | | |__|_ | |__|_ |
| | | | | | |
|__|__|_ |_____| |_____|
| | |
| | |
{{x,x,x} {{x,x},x} {{x},x},x}
Uncertainty
<-Redundancy->^
d d d d |
# # # # |
c c c c |
# # # # |
b b b b |
# # # # |
{a, a, a, a} V {a, b, c, d}
. . . . . . . .
| | | | | | | |
| | | | |__| | |
| | | | | | | <--(Standard Math language uses only
| | | | |_____| | this no-redundancy_
| | | | | | no-uncertainty_symmetry)
|__|__|__|_ |________|
| |
={x,x,x,x} ={{{{x},x},x},x}
============>>>
Uncertainty
<-Redundancy->^
d d d d | d d d d
# # # # | # # # #
c c c c | c c c c
# # # # | # # # #
b b b b | b b b b b b b b b b
# # # # | # # # # # # # # # #
{a, a, a, a} V {a, a, a, a} {a, b, a, a} {a, a, a, a}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
{x,x,x,x} {{x,x},x,x} {{{x},x},x,x} {{x,x},{x,x}}
c c c
# # #
b b b b b b b
# # # # # # #
{a, b, a, a} {a, b, a, b} {a, a, a, d} {a, a, c, d}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
{{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x} {{{x,x},x},x}
a, b, c, d}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this
|_____| | no-redundancy_no-uncertainty_symmetry)
| |
|________|
|
{{{{x},x},x},x}
--------------------------------------------------------------------------------
(*) Please pay attention that we are not talking about the natural numbers 2,3,4 but the cardinals 2,3,4.
It means that our Organic Natural Numbers are actually a general representation of information-trees, where any finite quantity of names of R members can be described by them, for example:
Instead of a=0,b=1,c=2,d=3 we can use a=0,b=.5,c=3,d=pi and then we use the same information-trees above.
matt grime
Aug11-04, 07:35 AM
More demonstrations of ignorance? wow you're good.
there are exactly TWO isomorphim classes of groups of order 6, now which of them is the lowest bound and which is the highest bound? it is a finite set after all, so it must have them in your logic.
the rest as i think ex-xian said is spamming. you've no need to repost it, it is "vandalism" to do so in some people's opinions.
Matt,
Pease read my posts, because I do not see any detailed comments of you on posts #471 and #473.
there are exactly TWO...
So, it is 2 instead of 6, but it does not matter because both of these cardinals are finite cardinals, and in this case you say bye,bye in 0 and 3.
HallsofIvy
Aug11-04, 08:08 AM
Originally Posted by Matt Grime
(the collection of all isomorphism classes of groups of order 6 is finite, but doesn't have any bounds)
Originally Posted by Lama
Ho, yes they have, you cannot find the 7 of them and you cannot find 0 of them, but you can find 1 to 6 of them.
You have absolutely no idea what Matt Grime is talking about, do you?
You asserted that any finite set must contain upper and lower bounds as part of your definition of "finite". Matt gave an example in which the members of the set are not linearly ordered and so your statement makes no sense for that set.
And please stop responding to every criticism with "read all of ...". The fact that someone points out that your second sentence is nonsense doesn't mean that he hasn't already waded through the whole thing.
You are correct that "You do not need complecated mathematical teachings to understand my work". What we need are actual definitions not hand waving and vague examples of things that seem to you to illustrate an idea.
matt grime
Aug11-04, 08:09 AM
But the set of iso classes isn't a set of numbers, so it has no ordering in it, it has no way of describing an upper or lower limit of the elements in it. The set is a collection of things, is it finite? You're confunsing a set and its cardinality. The bounds you gave are the least integer strictly greater than the cardinality of the set in question, and the greatest integer strictly less than ... They bound the cardinality, but in no way form a bound for the set which is what your definition requires.
Matt and HallsofIvy,
In my system there is no such thing like lower and upper bounds.
In my system I use the terms lowest and highest bounds, which are totally different things.
Since you do not really read my posts in order to understan them, I don't no how we can communicate with each other.
matt grime
Aug11-04, 08:22 AM
ok, rewrite my post in your head with lowest and highest then. the question still makes sense and the answer doesn't.
the point is quite simple, the only sets you can talk about being finite are those with highest and lowest bounds by your own admission, so this cannot be applied to any set other than something with an ordering and even that makes little sense since i an produce infinite (in the proper sense) sets with highest and lowest memebers
Let us check again my definition for 'finite':
a) A singleton or a collection of more then one element where its lowest bound and its highest bound are included in it.
b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.
And there is nothing here, which is connected to order!!!
As you see my definition is stronger, simpler and richer then the quantity-only definition, because it works for both quantity and structure.
This is a part of post #471, and if you don't read it, as I can clearly understand from what is written in your last post, then you keep talking to yourself.
matt grime
Aug11-04, 08:28 AM
it doesn't even work for quantity. consider the set underlying the ordinal w+1, then it more than one element in it, and contains a least element and a greatest element, hence ordinals are not allowable in your system, unless you're going to claim w+1 is a finite ordinal?
matt grime
Aug11-04, 08:40 AM
"Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found"
what on earth is "it" in that sentence referring to? there must be some order on something otherwise you cannot talk of highest or lowest, so what is ordered?
There are two kinds of ordinals in my system, internal and external.
The external ordinals are any change in the quantity of {.}_AND_{._.}.
The internal ordinals are any change in the symmetrical degrees of {.}_AND_{.}, where the cardinality of {.}_AND_{._.} is unchanged.
there must be some order on something otherwise you cannot talk of highest or lowest, so what is ordered?
By 'lowest' and 'highest' I mean lowest existence and highest existence of some element, and there is no connection to order here.
matt grime
Aug11-04, 08:55 AM
you#ve just redefined ordinal then.
"lowest existence" what does that mean? please give me some example where you may talk of things being lowest and highest which is not from some order?
Take for example the multiset {1,1,1,1,1}.
It cannot be ordered but cardinals 0 and 6 are the lowest and highest bounds of it (I already gave this example in post #471).
Also the notations '{' and '}' are also the bounds of existence of any concept that we research, isn't it?
matt grime
Aug11-04, 09:11 AM
Take for example the multiset {1,1,1,1,1}.
It cannot be ordered but cardinals 0 and 6 are the lowest and highest bounds of it (I already gave this example in post #471).
Also the notations '{' and '}' are also the bounds of existence of any concept that we research, isn't it?
what has this post to do with anything? why is 6 a bound of the set? it is a bound for that set which allows repeated elements, but? that is not what your axiom for "finiteness"states, since it requires that the "end" elements are in the set for it tobe finite. what is the first and last element of a set without an order on the elements in it?
psot 482, the question about what the"it" refers to in your axiom is still unanswered
Beyond your end you cannot be found anymore, isn't it?
To what "it" do you mean exactly?
what is the first and last element of a set without an order on the elements in it?
I do not care about the internal order of the given set, its end is always determinated by its cardinality.
matt grime
Aug11-04, 09:42 AM
"Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found"
this is your axiom, so what does the "it" refer to? it's your sentence, presumably you know what the pronoun references.
"I do not care about the internal order of the given set, its end is always determinated by its cardinality"
that is also wrong trivially. {2} and {3} both have cardinality 1, the "end" is not determined by the cardinality. you're confusing a set and its cardinality again.
{2} and {3} both have cardinality 1
Yes you are right, therefore the values beyond their ends are cardinals 0 and 2.
"Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found"
It is an element or a collection .
I think that I made a mistake in my previous posts:
The cardinals are not just the ends of a given finite set, but they are also the values that beyond its ends.
I am really Sorry about the confusion.
Here it is again my definition of 'finite' to clarify this mistake:
a) A singleton or a collection of more then one element where its lowest bound and its highest bound are included in it.
b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.
ex-xian
Aug11-04, 11:39 AM
You're a really sad person. You've spent 20 years at this?! I've tried being patient, asking specific questions, but you are clearly too ignorant of what mathematics really is to be able to answer them.
Now, of course, you're going to rationalize this post by called me a bodyguard of mathematics, telling me that I won't come to where you're standing to look at your point of view, or make ludicrous mystical comments. Of course you're free to do this, but none of that will change the fact that you don't know math and that you're theories are just the ramblings of another (unoriginal) crank.
Have fun trying to be original and deluding yourself. Have fun not answering questions and chaning the subject when you can no longer avoid the major and damning problems that are pointed out in your "theories" (and I use that term very loosely).
My last suggestion to you: actually learn some mathematics. Even if you don't believe it when you're learning it, you'll see how it works and be in a better place to critique it. When people see that you actually understand what you are trying to change, you a lot less likely to be laughed at. As things stand now, you theories are just that...laughable at best. And tragic (that you actually think that qualify as intellectual, scholarly work) at worst.
Let us clarify the 'finite' concept in my framework:
In my system I have 4 building-blocks, which are:
{}, {.}, {._.}, {__}
The cardinal of {} is 0.
The cardinal of {.} is one of many.
The cardinal of {._.} is one of many.
The cardinal of {__} is The one.
The bounds of lowest and highest existence (the ends) of these building-blocks
are determined by their cardinality, for example:
(in this example I omitted {.}_AND_{._.} and used only their building-blocks)
The lowest and highest bounds of {.} are cardinals 1 to 1.
The lowest and highest bounds of {._.} are cardinals 1 to 1.
The lowest and highest bounds of {} are cardinals 0 to 0.
The lowest and highest bounds of {__} are cardinals The 1 to The 1.
The lowest and highest bounds of {{.},{._.},{.}} are cardinals 1 to 3.
The cardinals beyond {.} are 0, n>1 and the 1.
The cardinals beyond {._.} are 0, n>1 and the 1.
The cardinals beyond {} are n>0 and The 1.
The cardinals beyond {___} are any cardinal which is not The 1.
The cardinals beyond some n are 0 and any j where j>n.
I give up ...
Dear ex-xian,
Do you say that after you read
http://www.physicsforums.com/showpost.php?p=282481&postcount=471 ?
Also please read my previous post (#492), thank you.
ex-xian
Aug11-04, 01:01 PM
Dear ex-xian,
Do you say that after you read
http://www.physicsforums.com/showpost.php?p=282481&postcount=471 ?
Also please read my previous post (#492), thank you.
Oh yes, I read it. I just reenforced my belief that you aren't interested in learning actual math or listening to critiques of your lame, pointless, and ignorant theories. Naturally, you're going to see this as a victory for yourself and the profoundly stupid ideas you write about. Please be assured that this is not the case--I'm not giving up because you've put me in my place. Rather because talking to you is like talking to a child. You can't/won't comprehend what other people write when they don't agree with you. Instead of appempting to recieve criticism (which you asked for), you accuse others of being bodyguards of math, refusing to come to where you are, or other ridiculous and childish excuses.
You're either a troll or a pitiful, ignorant individual.
...because talking to you is like talking to a child.
Ho dear ex-xian, this is the best complement that I can get, and I which to be opened like a child for the rest of my life.
Now, did read also post #492 ?
yesicanread
Aug11-04, 01:14 PM
contradiction = not-contradiction
How's this for a contradiction. Do you understand my reasoning Lama ?
So ( A = (Set = (2*A))) = Triangle inequality theorem = Triangle inequality theorem = contradiction
yesicanread
Aug11-04, 01:43 PM
How's this for a contradiction. Do you understand my reasoning Lama ?
So ( A = (Set = (2*A))) = Triangle inequality theorem = Triangle inequality theorem = contradiction
The full theory that uses this iiis...
(A = ( Set = ( 2 * A ))) = ( Set = ( 2 * A)) = Three right angles I'm thinking.
If I hadn't posted here before, I wouldn't have seen this.
ex-xian
Aug11-04, 01:54 PM
Ho dear ex-xian, this is the best complement that I can get, and I which to be opened like a child for the rest of my life.
In that case, let me explain. I did not me "like a child" in the sense of a child's sense of wonder, amazement, and openness. On the contrary, I meant a bratty, stubborn child who clings to what he wants in spite of the adults' explanations. I hope that's more clear.
Now, did read also post #492 ?
I've read all your gibberish, yes.
arildno
Aug11-04, 01:59 PM
I which to be opened like a child for the rest of my life.
So that's why you lobotomized yourself 20 years ago?
I've read all your gibberish, yes
You read but did not understand a single note of it, because you are not able to grasp that mathematics is not just an external method that can be learned in school, but first of all the creative abilities of parsons who can see beyond any given external method, which is based on some agreement between a group of people.
I do not belong to your school of thought, but I clearly showed that I understand the fundamental concepts of your school of thought.
On the contrary I did not show any ability to see things beyond your school of thought, because in your school of thought, you do not learn how to use your creative abilities, but only how to be a good technician of the academic standard system.
But I believe that you can do much more then that, if you reduce your aggressive attitude to unfamiliar ways of thinking, which are not belong to your school of thought.
arildno
Aug11-04, 02:35 PM
I do not belong to your school of thought..
You do not belong to any school of thought, because you lack the ability to think (i.e, what is usually considered the defining feature of a human being)
You do not belong to any school of thought, because you lack the ability to think (i.e, what is usually considered the defining feature of a human being)
Dear arildno, so why do you waste your time in a thread of someone who cannot think?
On the contrary I learn How to develop my system because of your questions,
and I really want to thank you for that.
ex-xian
Aug11-04, 03:17 PM
You read but did not understand a single note of it, because you are not able to grasp that mathematics is not just an external method that can be learned in school, but first of all the creative abilities of parsons who can see beyond any given external method, which is based on some agreement between a group of people.
What are you smoking and where can I get some?
I do not belong to your school of thought, but I clearly showed that I understand the fundamental concepts of your school of thought.
No, you didn't. Amazing how you post such blatant lies. You displayed your ignorance of even the most simple definitions, of the most basic concepts. When challenged, you asked that we no longer discuss standard math and instead focus on your theories. Here's a hint about mathematics--you can't learn it just from widipedia and mathworld.
Tell you what. See if you can answer these questions about "standard" mathematics.
With a delta-epsilon proof, show that \lim_{x\rightarrow 2} (x^2+x-6) = 0.
Find \frac{d}{dx} (x^2+x-6) by first principles.
Prove that \mathbb{Z}_6 is an abelian group with respect addition.
Solve this integral, \int xe^{-x} \,dx
If you can do these, then you will have demonstrated a very basic understanding of mathematics. Also, provide detailed explanations of your solutions.
On the contrary I did not show any ability to see things beyond your school of thought, because in your school of thought, you do not learn how to use your creative abilities, but only how to be a good technician of the academic standard system.
Well, you're typo is actually correct. Other than that, more self-rationalizing. Sad really. Also, the fact that you consider mathematics nothing more than techniques that you think to be sucessful at mathematics one does not need creativity shows that really haven't studied math at all. Spend some time proving things about and studying the hilbert space \l^2 and tell me no creativity is involved.
But I believe that you can do much more then that, if you reduce your aggressive attitude to unfamiliar ways of thinking, which are not belong to your school of thought.
If I had "aggressive" attitudes toward unfamiliar areas of thought, I wouldn't study mathematics, physics, philosophy, or anything interesting. The point is to explore unfamiliar things that are coherent, not gibberish. Yours is the latter, not the former.
With a delta-epsilon proof, show that...
I see that you forgot about:
----------------------------------------------------------------------------------
kaiser:
off course I agree with this definition. I meant for you to provide the definition for the limit of S(n), no need delta epsilon at this point. A limit can be defined using epsilon and S(n). At any case, I am not interested in your definitions at the moment. I need to be convinced that you understand and know how to use the fundamental "conventional" mathematical definitions before we can move on to your definitions.
----------------------------------------------------------------------------------
Lama:
Ok, the main persons in modern Math that are related to the so called rigorous definition of the limit concept are Cauchy and Weierstrass.
Cauchy said:" When some sequence of values that are related one after the other to the same variable, are approaching to some constant, in such a way that they will be distinguished from this constant in any arbitrary smaller sizes that are chosen by us, then we can say that this constant is the limit of these infinitely many values that approaching to it."
Weierstrass took this informal definition and gave this rigorous arithmetical definition:
The sequence S1,S2,S3, … ,Sn, ... is approaching to (limit) S if for any given positive and arbitrary small number (e > 0) we can find a matched place (N) in the sequence, in such a way that the absolute value S-Sn (|S-Sn|) become smaller then any given epsilon, starting from this particular place in the sequence
(|S-Sn| < e for any N < n).
----------------------------------------------------------------------------------
And then I clearly and simply explain in http://www.physicsforums.com/showpost.php?p=275250&postcount=318 why I do not share these reasoning with the rest of you.
Again, my work is in the most fundamental levels of the basic concepts of the language of Mathematics, and all I need is to show that I understand the reasoning behind these basic concepts, and I clearly showed that I understand the reasoning behind it.
Now after I showed it, I want you to show that you can understand my different point of view about these fundamental concepts.
And you have to understand that in this gentle level your technical skills cannot help you, but only your most simple ability to understand deep things, that cannot be thought by any external method.
It is like a swing in jazz, you have it or you don't have it, and no school can give you the swing.
So, I have a very simple question to you.
Can you take off your boxing gloves and try to understand my reasoning with your bear hands?
ex-xian
Aug11-04, 04:15 PM
<List of excuses>
I didn't ask you if you agreed with it, I challenged you to prove that you actually understand math. Will you show that you do, admit that you don't, or continue to make excuses? I think you'll chose the last.
kaiser soze
Aug11-04, 04:18 PM
Lama, quoting definition you downloaded from the web does not prove you understand them. I agree with ex-xian, prove him and us that you know basic math by accepting his challenge.
Kaiser.
didn't ask you if you agreed with it, I challenged you to prove that you actually understand math. Will you show that you do, admit that you don't, or continue to make excuses? I think you'll chose the last.
So you cannot do it, because you cannot understand anything of what I wrote to you in post #504.
Again I do not need to show that I know how to use your reasoning to prove things, all I need to show is that I understand the most fundamental reasoning behind it, and I clearly showed it in my previous post.
If you do not understand it then you cannot understand what is the meaning of fundamental work on the most basic concepts of the Language of Mathematics.
prove him and us that you know basic math by accepting his challenge
By these words you show that you do not understand the meaning of the words 'basic Math', and so is ex-xian.
Basic Math is exactly what I am doing in my work, when you understand 'basic math' as basic techniques of your reasoning.
arildno
Aug11-04, 04:35 PM
Your a ****ing retard, Lama!
Stick your head up your *** and eat your own **** if you like, but you won't get any here to enjoy the taste of it
Well arildno I like to eat @@@@@ more then **** , so you see we have a different taste.
ex-xian
Aug11-04, 04:50 PM
Again I do not need to show that I know how to use your reasoning to prove things, all I need to show is that I understand the most fundamental reasoning behind it, and I clearly showed it in my previous post.
Oh really. Then why did you write this?
Please demonstrate some fundamental mathematical idea, which can clearly show that I do not understand (again, not disagree with, but do not understand) its standard interpretation.
If you do not understand it then you cannot understand what is the meaning of fundamental work on the most basic concepts of the Language of Mathematics.
I freely admit that I don't understand much of what you write. It's total gibberish. Will you also admit that you don't understand math?
prove him and us that you know basic math by accepting his challenge
By these words you show that you do not understand the meaning of the words 'basic Math', and so is ex-xian.
Basic Math is exactly what I am doing in my work, when you understand 'basic math' as basic techniques of your reasoning.
Once again, you change the meaning of words to suit yourself. You claimed that standard math is a shadow your system and that you have a strong, solid grasp on standard math. Personally, I think you were lying and I think it's been demonstrated by many people that you don't understand mathematics. Prove us wrong.
So ex-xian and kaiser soze,
If you still cannot understand by reading my work that I cannot write what I write without a deep understanding of standard reasoning, then you have no ability to see beyond what you learned in your school of thought.
ex-xian
Aug11-04, 05:05 PM
So ex-xian and kaiser soze,
If you still cannot understand by reading my work that I cannot write what I write without a deep understanding of standard reasoning, then you have no ability to see beyond what you learned in your school of thought.
Prove that you have this understanding. You offered to before! Here, I'll quote you again.
Please demonstrate some fundamental mathematical idea, which can clearly show that I do not understand (again, not disagree with, but do not understand) its standard interpretation.
If you don't, you've just lost what little credibility you might have retained.
Ex-xian,
Do you know what is a fundamental Mathematical Idea (and I do not mean to some basic techniques to prove things)?
ex-xian
Aug11-04, 05:26 PM
Do the problems, which you asked for, or admit that you can't. It's as simple as that.
A correction: the operation for the abelian group problem should have been circle addition, not addition. Addition isn't an operation for Zmodn.
Tom Mattson
Aug11-04, 07:29 PM
It seems that for every single question that is asked of Lama, he either cuts and pastes old posts or posts a link to his book. This has gone on for 26 pages, and there is no indication that it will not go on for another 26, if left on its own. Since there is no need to waste bandwidth at PF on that sort of back-and-forthing, I'm putting a stop to it now.
Here's that link again: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
Apart from that, say goodnight to this thread.
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