The Foundations of a Non-Naive Mathematics

[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.

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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.

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(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.

 

[Lama]

Dear zeronem,

Thank you for this information.

 

[Lama]

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

There is no big problem here if you understand this model:

http://www.geocities.com/complementarytheory/9999.pdf

I'll be glad to get your opinion, thank you.

 

[zeronem]

By the way, what is your Mathematical Background Lama? How far have you gotten in your math classes at school?

The answer to these questions will let us know exactly who we are talking to.

 

[moshek]

Zeronem:

If you talk like this,
it is better for you
not to know
who you are talking with.

Moshek

 

[matt grime]

"I am a mutation"

well, you said it, Doron.

Muddler, as to the proof that 0.99.. =1 in the reals (base ten expansion), it would appear that Doron is once more using his own definitions to conclude something about ours.

Something that should warn you he's off on one is that he doesn't ever use any of the parts of the definition of the real numbers to prove anything.

Here is a simple proof:

consider the sequences 1,1,1,1,1... and 0.9, 0.99, 0.999, now the "symbol" 0.99999... means the limit of the second. Now, it follows that these are clearly both cauchy sequences, and the difference between them is a cauchy sequence tending to zero, hence, that the sequences represent the same real number. (Which follows from the characterization of the reals as equivalence classes of cauchy sequences of rational numbers).

In fact we have all seen his characterzation of duality and scale factor and such. Now, all of the things he says about the reals are valid about the rationals and hence do not in anyway characterize reals uniquely.

 

[terrabyte]

.999... ≠ 1 is one of my favorite topics

so many exceedingly smart people get it wrong. it is quite clear that when functioning under limits you cannot expect your tried and true methods to yield correct results.

.999... is NOT 1. it's simple and logical

 

[matt grime]

In the reals, in base 10 representations, 0.999... and 1 are equivalent, and equal in this sense. they may be taken to represent different cauchy sequences, but they are unequivocally the same real number, Terrabyte. Please, offer a reason, mathematically sound, as to why they are not equal.

 

[Hurkyl]

you and other "well-educated" colleagues of the academic system, prefer to protect the dogmatic core of your community, instead of let it be developed by an open dialog.

As long as you continue to believe this, you will have shut out a great store of knowledge and innovation.

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

As long as you reject criticism, you will not flourish. It is just as important, if not more so, to understand one's mistakes than to understand one's successes.



Do you remember long ago when I was working with you? I stated that I felt it important to understand how your ideas related to mathematics, and eventually decided that, among anything I could try to do, helping you understand that relation would be the most beneficial to you. I still think that is true.

 

[Lama]

Now, all of the things he says about the reals are valid about the rationals and hence do not in anyway characterize reals uniquely.

Please read again http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

You can find there:

1) A point:
A singleton p that can be defined only by tautology ('='), where p has no internal parts.

2) An interval (segment):
A singleton s that can be defined by tautology ('=') or '<' or '>' , where s has no internal parts.

3) A number is both p_AND_s

4) A general representation of an interval (segment) is {._.}.

5) A general representation of a point is {.}.

6) Only {._.} can be affected by a scale factor.


Both Q and R members are effected by using one of them as a scale factor over the entire Real-Line.

The difference between Q and R members is still the inability to define an R member (irrational) by a ratio between at least two integers.

But this is not the new case here.

The new thing is that any given number is now both absolute and relative element, because it is defined by ({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}.

 

[Lama]

By the way, what is your Mathematical Background Lama? How far have you gotten in your math classes at school?

The answer to these questions will let us know exactly who we are talking to.

I am an autodidact that dealing (with a lot of love and patience) with the Langauge of Mathematics for more then 20.

 

[Lama]

As long as you reject criticism...

On the contrary, I seek for criticism (otherwise I would not be hear) and you are maybe one of the best.

But the sad thing is that I used your criticism to develop my system, and you react as if nothing has been changed during the years.

To be more specific: please criticize https://www.physicsforums.com/showpost.php?p=267089&postcount=101

Thank you.

 

[kaiser soze]

You receive criticism only when you agree with it. If you really want to learn, you need to be able to accept criticism even if you do not initially agree; this because you acknowlegde that whom ever criticizes you knows more than you do in the acclaimed subject...

Kaiser.

 

[Lama]

You receive criticism only when you agree with it...

Please show me an example of how a person can receive criticism, and on the same time he does not agree with it.

In short, 'receive' and 'agree' are synonyms in this case, isn't it?

 

[kaiser soze]

I'll try an analogy: when a parent says to his little child "do not speak to strangers", the child should accept this "criticism", even if the reason is unclear at that moment. When this child grows up, it will become clear to him why this "criticism" was given.

Kaiser.

 

[Lama]

I'll try an analogy: when a parent says to his little child "do not speak to strangers", the child should accept this "criticism", even if the reason is unclear at that moment. When this child grows up, it will become clear to him why this "criticism" was given.

Your analogy does not hold in this case, because we are talking about fundamental things which are the heart bits that give life to the language of Mathematics.

In this most simple level we do not need sophisticated methods but very sensitive intuition/reason gentle interactions, which become our self-evident building-blocks that are used as the milestones of our research.

What I have found is, that instead of developing this gift of self research that exists in most of the young students, the academic system does its best to force its methods on the minds of the students, because there is no time and no money to develop the unique internal skills of each student.

As a result the student is forced to be under the doctrine of some external method which he must agree with it in every step along the "learning" process.

After couple of years under this forcing external race-like method, most of the students are not able to use anymore their natural skills of self learning, and most of them become no more then good technicians of the academic system.

I have found that a lot of doctors and professors of Mathematics simply lost any ability to understand simple things.

My work is on these simple things, and I developed my skills to work and create in this simple level, for more then 20 years.

No external reasons like money or academic title where my motivations and no forcing external race-like methods washed my natural ability to understand simple things.