The Foundations of a Non-Naive Mathematics

[Lama]

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:

---------------------
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.
---------------------

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*.)

--------------------------------
*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.
--------------------------------



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.

 

[Lama]

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]

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?

 

[Lama]

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 Langauge of Mathematics.

The Langauge of Mathematics become meaningful only if it uses the products of the interactions between {.} and {._.}.

It means that any meaningful thing in the Langauge of Mathematics it at least {.} and {._.}.

Strictly speaking, The Langauge of Mathematics is at least an absolute/relative system.

 

[matt grime]

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.

 

[NateTG]

It's Organic.

So, continue posting at your own risk.

 

[Lama]

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]

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?

 

[Lama]

Matt please show us that you understand my answer.

 

[matt grime]

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.

 

[Lama]

What you wrote is utter utter rubbish.

So, you demostrated that you do not understand my answer.

 

[matt grime]

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.

 

[Lama]

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]

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.

 

[Lama]

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]

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.

 

[Lama]

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]

*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.

 

[Lama]

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]

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]

wow, people on this board sure are cranky... I'm going to go read a book.

 

[chroot]

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

 

[Lama]

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]

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.

 

[Lama]

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?" ??

 

[matt grime]

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

 

[Lama]

Matt please give a detailed answer to post #23, thank you.

 

[Lama]

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]

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 language.

 

[Lama]

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 language.

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]

Wow ! What fun !

 

[matt grime]

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]

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.

 

[Lama]

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 Langauge 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 Langauge of Mathematics" cen be found in the front page of my website here: http://www.geocities.com/complementarytheory/CATpage.html

----------------------------------------------------------------------------------------------

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]

"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.

 

[Lama]

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 Langauge 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 Langauge 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]

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.

 

[Lama]

ahrkron, my heart with you, because my work has a strange property.

When you look at it, you see your own real face.

 

[Lama]

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 help here, so here it is:

Thanks for the kind words...(this thread is becoming a mush fest ).

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]

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]

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.

 

[Lama]

Well Matt you are well-known by your profound and quick conclusions.
-----------------------------------------------------------------------------------

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?

-------------------------------------------------------------------------------------

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 Langauge 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.

-------------------------------------------------------------------------------------

Did you mean "let us say that there exists delta such that delta >0" ?

Yes.

 

[kaiser soze]

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.

 

[Lama]

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 Langauge of Mathematics, can have different interpretations in different frameworks.

 

[kaiser soze]

Your statements make very little sense. I think you do not really understand what epsilon/delta proofs are used for.

 

[Lama]

Dear kaiser soze,

Math has many faces, and if you choose to ingore it, then we have no dialog between us.

 

[PFanalog57]

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.

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" .

 

[Lama]

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 statement.

 

[PFanalog57]

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 statement.

contradiction = not-contradiction is a contradiction

true?

or

false?

 

[Lama]

There is no question here.

a is not_a is nothing but a false statement in boolean logic, because no identity can be in more than one unique state in boolean logic.