The Foundations of a Non-Naive Mathematics

[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 ! :tongue2:

 

[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 ! :tongue2:

simply over the year(s) doron has produced several garbage papers that have been refined to the current situation, he's cried wolf once too oftem for us to even bother trying to believe it is correct. do you know how long it took him to even offer some definition of what "uncertainty and redundancy" are? i don't mean in a metaphysical sense, purely in how to look at one of his diagrams and decide what corresponds to uncertainty, whatever metaphyiscal meaning we might have there. indeed he's not actually explained why it is that a diagram with such a property corresponds to being uncertain in anysense. how's that for an example?

 

[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.

 

[Lama]

If we use a structural point of view in this case, then 0.9999... is a single path of a base 10 fractal ( http://www.geocities.com/complementarytheory/9999.pdf pages 3,4 ), that exists upon infinitely many scale levels that cannot reach 1.

Also we can say that 0.999... = 0.9+0.09+0.009+0.0009+... and we can clearly see that this infinitely long addition cannot reach 1.

Therefore 0.999... < 1.

Another example:

Please look at this beautiful Koch Fractal members.cox.net/fractalenc/fr6g6s.577m2.html[/URL]

Now let us say the there is a 1-1 map between each fractal level of 0.9999... to each different blue level of Koch Fractal.

0.9999... = 1 if and only if we cannot find anymore a 1-1 map between some 0.000...9 to some Koch Fractal blue level.

Since Koch Fractal can be found in infinitely many blue levels and each blue level has a 1-1 map with some 0.000...9 fractal level, then we can conclude that 0.999... < 1.

Also we can say that 0.999... = 1 if and only if the outer contour of this multi-leveled Koch Fractal can be a smooth curve with no sharp edges.

It is clear that the outer contour line is not a smooth contour in any arbitrary examined scale level.

Therefore 0.999... < 1.

From this model you also can understand what is a "leap".

In short, any transition between a non smooth curve to a smooth curve, cannot be done but by a phase transition leap that also can be described by a smooth_XOR_no-smooth connection.

This model is better than any "abstract" mathematical definition, which leads us to "prove" that 0.9999... = 1.

Also by this "proof" we simply ignore infinitely many information forms that can be found in 0.9999... fractal.

Now think how many information forms are ignored by this trivial and sterile approach of standard Math.

 

[zeronem]

I only have one recommendation for Lama. I recommend this book for you Lama,

"Where Mathematics Comes From: How the embodied mind brings Mathematics into being", - by George Lakoff and Rafael E. Nunez

 

[hello3719]

If we use a structural point of view in this case, then 0.9999... is a one dimensional path of a base 10 fractal, that exists upon infinitely many scale levels that cannot reach 1.

Please define " one dimensional path of a base 10 fractal". There is a big problem here with definitions.

 

[Hurkyl]

Furthermore, since then most of the time you did your best to shut me off this forum at least 3 times, by using your power as the moderator of mathematics forum.

Not this forum, just the math forum. And I didn't "shut you off" until people started complaining.

I still recal one of our early disagreements; you liked to assert that the cardinal number (aleph_0 - 1) was smaller than the cardinal number aleph_0. Your posts conveyed that you thought you had a proof of this using your idea of cardinality. You refused to accept that your proof (whatever it was) was a proof about your idea of cardinality rather than the "normal" idea of cardinality. This pattern continued with just about every mathematical idea you considered, and ended when I started moderating your posts because you were hijacking threads to talk about your way of doing things when everyone else wanted to talk about the "normal" way.

 

[Lama]

To Hurkyl,

Instead of pluralism (which is the healthy essence of any non-trivial system) you and other "well-educated" colleagues of the academic system, prefer to protect the dogmatic core of your community, instead of let it be developed by an open dialog.

But form strategic point of view, each one of us doing his job in this evolution process, which means:

You play the current system and I play the mutation of it.

You will do your best to shut me down and I will do my best to survive and flourish.