The Foundations of a Non-Naive Mathematics

[matt grime]

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.

 

[Lama]

By the word 'is' I mean to '='

a = not_a is nothing but a false statmant in boolean logic.

 

[Lama]

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.

 

[PFanalog57]

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.

If the statement contradiction = not-contradiction is a contradiction is false, the statement contradiction = not-contradiction is not a contradiction is true?

 

[Lama]

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]

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?

 

[Lama]

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 statement 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.

 

[Lama]

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 referred 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.

 

[Lama]

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, which can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.

 

[matt grime]

"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.

 

[Lama]

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.

 

[Lama]

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]

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.

 

[Lama]

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]

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.

 

[Lama]

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.

 

[Lama]

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).

 

[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."

 

[Lama]

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 Langauge 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?

 

[Hurkyl]

Lama; what do you propose we use if we want to talk about quantity?

 

[Lama]

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 Langauge of Mathematics.

 

[Hurkyl]

Are you saying we should never want to talk about quantity?

 

[matt grime]

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)

 

[Lama]

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.

 

[Lama]

Are you saying we should never want to talk about quantity?

Do all what you want to do.

 

[Muddler]

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.

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 !

 

[Hurkyl]

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.

 

[Lama]

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 Langauge 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.

 

[Lama]

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: https://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 https://www.physicsforums.com/showpost.php?p=263204&postcount=69
and https://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 https://www.physicsforums.com/showpost.php?p=261949&postcount=59 thank you.

 

[Hurkyl]

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: https://www.physicsforums.com/showpo...49&postcount=59

It's very easy for one to quote things one does not understand.

 

[Lama]

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 !

 

[Hurkyl]

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.

 

[Lama]

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.

 

[Hurkyl]

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]

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.

 

[Lama]

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 behind this face.

 

[Lama]

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.

 

[Gza]

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.

 

[Lama]

Hi Gza,

Can you be clearer please?

 

[moshek]

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

 

[Lama]

Dear Moshek,

Thank you very much, I hope to see the beginning of a community of persons, which never afraid to re-examine any fundamental old or new concept of the Langauge of Mathematics, through an open dialog.

For me the language of Mathematics, is first of all an open and continuous dialog.

 

[matt grime]

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]

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.

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 !

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?

 

[Lama]

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:
https://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):
https://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]

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".

 

[Lama]

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]

"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]

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!

 

[Lama]

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]

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?
https://www.physicsforums.com/showthread.php?t=22866
There's a pretty lengthy discussion going on here that you might enjoy.