The Foundations of a Non-Naive Mathematics

[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.

 

[kaiser soze]

Mathematics was developed and taught long before academic institutions existed, yet its fundamental concepts and foundationds are still valid.

Kaiser.

 

[Lama]

Mathematics was developed and taught long before academic institutions existed.

Yes, it developed by people who where not forced by external methods or exteral reasons or lack of time.

yet its fundamental concepts and foundationds are still valid.

Not all of them.

 

[kaiser soze]

The least I can offer you, is to accept and learn from others, who may have more knowledge from you in a given area - that is if you wish to interact with them.

Kaiser.

 

[Lama]

Dear kaiser soze,

Please show some detailed example, which cearly show why my work failes because I am not listening to persons that have more knowledge in this given area.

 

[kaiser soze]

you refuse to accept the simple fact that the LIMIT of the sequence 0.9, 9.99, 0.999, ... is 1.

Kaiser.

 

[Lama]

Don't you see the simple fact, which can clearly understood by https://www.physicsforums.com/showpost.php?p=267089&postcount=101 that 1 is not the limit of your sequence?

 

[kaiser soze]

I rest my case.

 

[terrabyte]

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.

you're essentially limiting the structure of your numerical system by this "entity" we call infinity. now, infinity is a useful term, we use it frequently to extend expressions out to incredible precision, but since the term itself is not closed, there lacks a decided "conclusion" for formulas utilizing infinity. thus any formulas that use infinity are relegated to being approximations, albeit really damn good ones.

the limit of a sequence is something that is NEVER reached. hence defined as such 1 IS the limit of .9+.09+.009...

but from that statement above it's logically sound that since it is NEVER reached, the sequence can NEVER be equal to 1.

the flaw is in the system, whether we choose to fix it or ignore it is the question...

 

[sparkster]

Wow...I came over here hoping to find some info on the recent experiment that supposedly disproves the Copenhagen interpretation of QM. Imagine my surprise when I see my friend Dorian (known as Lama here) posting this stuff here. And I was even quoted! I am indeed honored.

If anyone is interested, my dialogues with him are here, http://www.iidb.org/vbb/showthread.php?t=89321

 

[Lama]

Welcome my dear ex-xian,

I invite you to continue our dialog here or in your forum.

An information of this recent experiment you can find here:

http://drauh.typepad.com/blog/2004/04/

http://www.kathryncramer.com/wblog/archives/000530.html

Since Quantum elements are both wave and particle, no one of these properties can completely disappear, so I do not see any new point in this recent experiment.

 

[Hurkyl]

the academic system does its best to force its methods on the minds of the students

Do you see the irony in that you are trying to force your ideas on others?

 

[Lama]

Do you see the irony in that you are trying to force your ideas on others?

Am I an academic institute that gets money for my knowledge and forces people to show that they got it, by using an industry of examinations that if they do not pass them they will not get their diploma?

Can you please show how I force you to agree with me?

By the way, I am still waiting to you at https://www.physicsforums.com/showpost.php?p=267696&postcount=116

 

[sparkster]

Welcome my dear ex-xian,

I invite you to continue our dialog here or in your forum.

An information of this recent experiment you can find here:

http://drauh.typepad.com/blog/2004/04/

http://www.kathryncramer.com/wblog/archives/000530.html

Since Quantum elements are both wave and particle, no one of these properties can completely disappear, so I do not see any new point in this recent experiment.

Then you don't follow what's happening. Apparantly, this is the first time that light has been both a wave and a particle and not either/or.

Anyway, it hasn't been published, it hasn't been repeated. It's very preliminary, yet you insist on making dogmatic, authoritative statements. This is indicitive of your way of rejecting established science out of hand in order to favor your own bizarre theories.

With all due respect, I'll wait for real scientists who have spent decades doing this to decide rather than taking your opinion.

About your theories. I'm getting ready for graduate school next month, which I'm spending all my free time doing actual mathematics. I've responded to your posts again and again. You ignore what I write (the construction of the reals w/o using points--which invalidates your theory), take my words and use your own defintion (the "well-defined" bit), or you totally misunderstand what I write and put your own odd spin on it (my example of a delta-epsilon proof).

 

[sparkster]

Yes, it developed by people who where not forced by external methods or exteral reasons or lack of time.

Not all of them.

You've never even come close to showing this.

 

[sparkster]

you're essentially limiting the structure of your numerical system by this "entity" we call infinity. now, infinity is a useful term, we use it frequently to extend expressions out to incredible precision, but since the term itself is not closed, there lacks a decided "conclusion" for formulas utilizing infinity. thus any formulas that use infinity are relegated to being approximations, albeit really damn good ones.

the limit of a sequence is something that is NEVER reached. hence defined as such 1 IS the limit of .9+.09+.009...

but from that statement above it's logically sound that since it is NEVER reached, the sequence can NEVER be equal to 1.

the flaw is in the system, whether we choose to fix it or ignore it is the question...

There's no flaw in the system. When I was taking my calculus classes, my teachers always made a point to emphasize that when we say a sequence or series equals a number, we really mean that the sequence has limit of that number. Saying "equals" is just shorthand. That's not to say, however, that 0.99... is not equal to 1.

Here's a really simple way of showing that 0.99... = 1 that doesn't involve limits.

Clearly, 1/3 = 0.333...
So 3(1/3) = 3(0.333...)
Thus 1 = 0.999...

 

[Lama]

This is not the first time that both wave and particle properties of a quantum element, simultaneously appearing in a physical experiment.

When both stils are opened and we check the photons after they passé both of them, we can change gradually the wave picture to a particle picture, and vice versa.

 

[sparkster]

This is not the first time that both wave and particle properties of a quantum element, simultaneously appearing in a physical experiment.

When both stils are opened and we check the photons after they passé both of them, we can change gradually the wave picture to a particle picture, and vice versa.

Thanks, I think everyone knows this. And this is totally not what I'm talking about. You should actually read up on this topic before you try to discuss it anymore.

 

[Lama]

ex-xian,

I'll be glad to get your opinion on https://www.physicsforums.com/showpost.php?p=267089&postcount=101

 

[sparkster]

ex-xian,

I'll be glad to get your opinion on https://www.physicsforums.com/showpost.php?p=267089&postcount=101

My opinion is that it's juvevile and wrong.

Assume that .99.. < 1. Then 1/3(.999) < 1/3 (1), and .33... < 1/3. A contradiction, therefore .999 >/= 1. By a similar argument, .99... is not > 1. Therefore .99.. = 1.

Please, instead of citing old posts or referncing your own papers, actually address what I wrote if you want to prove me wrong.

 

[Lama]

Please give your detailed explanation why do you think it is wrong?

 

[sparkster]

Please give your detailed explanation why do you think it is wrong?

See my edit. Also, I find that you saying that your fractal analogy is better or less abstract than a rigorous mathematical proof to be absurd in the highest degree.

 

[Hurkyl]

Are you relied upon by employers and applicants alike to certify you know what you claim to know? And you do realize there's a difference between being forced to pass examination to receive a diploma and having ideas forced on you?

Can you please show how I force you to agree with me?

You refuse to participate in a discussion that don't involve your ideas, and when you were allowed to post in the math forum, you attempted several times to take over threads where mathematics was being discussed.

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.

0.999... is not a "one dimensional path". You've provided no definition of terms "base 10 fractal", "infinitely many scale levels". And obviously, since there is no definition available for these terms, you obviously cannot know that it "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 sequence cannot reach 1.

What sequence? 0.999... is a decimal number. 0.9+0.09+0.009+... is an infinite sum (which equals 1, which can easily be computed from the fact that this is a geometric series) In any case, it is not clear that "this sequence cannot reach 1".

Therefore 0.999... < 1.

As you have not made any valid statements yet, you cannot conclude that 0.999... < 1.

Another example:

Please look at this beautiful Koch Fractal http://members.cox.net/fractalenc/fr6g6s.577m2.html

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.

0.999... is not a fractal, and I'm fairly sure there is no such term as "fractal level". Furthermore, you are merely saying (aka assuming) there is a 1-1 map between two things; unless you can show this assumption is true, you cannot be sure that any of your conclusions are true.

0.9999... = 1 if and only if we cannot find anymore a 1-1 map between some 0. ...9 to some Koch Fractal blue level

What is 0. ...9? And you've given no reason why this statement should be true.

Since Koch Fractal can be found in infinitely many blue levels and each blue level has a 1-1 map with some 0. ...9 fractal level, then we can conclude that 0.999... < 1.

Ignoring what comes before "each blue level", I think this actually logically follows from your previous statements. However, you've suggested no reason why your previous statements might be true.

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.

You've given no reason why we can say that.

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

It is true that the boundary of the Koch snowflake is not smooth. You've not given any definition of "scale level".

Therefore 0.999... < 1.

Again, this follows from previous statements, but you've given no reason to think those previous statements are true.

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

No, I cannot.

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.

You've not provided definitions for "Transition", "phase transition leap", or "smooth_XOR_no-smooth connection".

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

Better in what way?

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

Since there is nothing about the real numbers called "infinitely many information forms" nor is 0.999... a fractal, ignoring these is a good thing.

Now think how many information forms are ignored by this trivial and sterile approach of standard Matt (oops, Math).

[/quote]

I can't, seeing how you've not said what you mean by "information form".

 

[Lama]

See my edit. Also, I find that you saying that your fractal analogy is better or less abstract than a rigorous mathematical proof to be absurd in the highest degree.

Your "rigorous" proof depends on excluded-middle black_XOR_white reasoning.

therefore you cannot deal with the complexity of 0.999... case.

Only Included-middle reasoning can deal with the complexity of this fractal.

In short, your "rigorous" proof is nothing but the image of your trivial black_XOR_white reasoning method.

 

[Lama]

Hurkyl,

also please read this http://www.geocities.com/complementarytheory/9999.pdf to understand why I call 0.999... a path of a fractal.

 

[Hurkyl]

To what proof are you referring?

 

[Lama]

Do you really have no simple ability to conclude that the base value expansion method is actually a fractal?

Please look at pages 3,4 of http://www.geocities.com/complementarytheory/9999.pdf

 

[matt grime]

the limit of a sequence is something that is NEVER reached. hence defined as such 1 IS the limit of .9+.09+.009...

presumably excpeting say, the constant sequences, and the eventually constant sequences then.

you shold stop, you're just showing up your ignorance.

 

[Lama]

Assume that .99.. < 1. Then 1/3(.999) < 1/3 (1), and .33... < 1/3. A contradiction...

There is no contradiction here beacue 0.333... < 1/3 for the same reasons that 0.999... < 1, which clearly can be understood here http://p071.ezboard.com/fthelanguageofmathematicsfrm2.showMessage?topicID=2.topic

 

[Hurkyl]

Well, then, here's a cute question:

If 0.333... isn't the decimal representation of 1/3, then what is?

 

[Lama]

A cute answer:

1/3 is not a fractal where 0.333... is a single path of a fractal.

They are not the same number exactly as 0.999... and 1 are not the same number.

When your logical system is based on an Included-Middle reasoning, the internal structural properteis of any mathematical element, cannot be ignored anymore.