The Foundations of a Non-Naive Mathematics

[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