The Foundations of a Non-Naive Mathematics

[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 definition (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.