A limits problem (fractal initiator & generator)....

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Look said:
A proof without words is a self-evident true that uses symbols without the need of further verbal-symbolic definitions.
This is exactly the reason of why a proof without words is more elegant than verbal-symbolic-only proofs.
 
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micromass said:
For centuries, people thought it was self-evident that the Earth is the center of the universe. Self-evident doesn't mean true. There are many self-evident things in math that turn out to be false. This is why we need mathematical rigor.
Come on, we are not talking about inductive "self-evidence" , but about deductive self-evidence.
 
Here's another fun proof without words:

bLWK1.png
 
micromass said:
I own over hundred math books and I have read hundreds of research papers in math. Can you explain why exactly zero of them have proof without words?
Hundred math books or hundreds of research papers not cover math.

I think that you simply find what you search for (according to your philosophical view of math), no less no more.

Here is some stuff that you probably did not read:

http://www.maa.org/press/periodical...hout-words-and-beyond-proofs-without-words-20

http://link.springer.com/article/10.1023/A:1024312321077

http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Borwein897-910.pdf

http://link.springer.com/chapter/10.1007/1-4020-3335-4_2

http://web.stanford.edu/group/cslipublications/cslipublications/pdf/1575863243.pdf

https://www.researchgate.net/profile/Mateja_Jamnik/publication/7575141_What_is_a_proof/links/542bda5a0cf27e39fa91b493.pdf

etc...
 
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micromass said:
Of course you'll find papers that discuss visualization in mathematics. All of the papers you cited are exactly on the topic of visualization. None of the topics you cited actually produce a novel result.
Did you read all of them in ten minutes, understood them and then concluded that non of them "actually produce a novel result" ?

You must be some kind of superman.
 
micromass said:
Here's another fun proof without words:

bLWK1.png
My diagram is not only a picture, but it uses a linkage among visual AND symbolic reasoning.
 
micromass said:
Yes.
So you are some kind of a superman.
 
micromass said:
It's still not mathematically rigorous.
By your (probably) formal school of thought it is not mathematically rigorous, but this is no more than your philosophical view of mathematics, that has nothing to do with how mathematics actually can be done.
 
Look said:
By your (probably) formal school of thought it is not mathematically rigorous, but this is no more than your philosophical view of mathematics, that has nothing to do with how mathematics actually can be done.

Then I hope you realize that all modern mathematicians agree that this is not mathematically rigorous? I mean, if you have a different philosophical view on mathematics, that's fine. But you do realize you're minority, right?
 
micromass said:
Here's another fun proof without words:

bLWK1.png
Exactly as pictures-only can be misleading so is the case about verbal-symbolic-only reasoning.

Since my diagram is not less than a linkage among visual AND symbolic reasoning, you can't value it by using pictures-only or verbal-symbolic-only reasoning.
 
micromass said:
Then I hope you realize that all modern mathematicians agree that this is not mathematically rigorous? I mean, if you have a different philosophical view on mathematics, that's fine. But you do realize you're minority, right?
Since when being minority means that one does not doing valuable mathematics?
 
micromass said:
Then I hope you realize that all modern mathematicians agree that this is not mathematically rigorous? I mean, if you have a different philosophical view on mathematics, that's fine. But you do realize you're minority, right?
I hope that you realize that since there is minority, then it is not true that "all modern mathematicians agree that this is not mathematically rigorous".
 
Dear micromass, since ##a+b+c+d+...## is defined by "zig-zag" lines that all of them have the constant length ##2X##, then ##a+b+c+d+...=X## implies ##2X = X\sqrt{2}##.

If you disagree with this implementation then please show that ##a+b+c+d+...=X## doe not imply ##2X = X\sqrt{2}##.

Here is the diagram, which is not less than visual AND symbolic reasoning:

28314892783_93fe8f577f_z.jpg


Without loss of generality you can use the case ##X=1## (in that case ##1/2 + 1/4 + 1/8 + ... = 1## which implies ##2 = 1\sqrt{2}##).

Since you agree that ##1/2 + 1/4 + 1/8 + ... = 1##, then the diagram is a proof without (additional) words that ##1/2 + 1/4 + 1/8 + ... = 1## implies ##2 = 1\sqrt{2}##.
micromass said:
No, since I don't count you as a modern mathematician.
Now you fail by attacking me and not my argument. Is this an example of how to use rigorous mathematical reasoning?
 
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Look said:
If you disagree with this implementation then please show that ##a+b+c+d+...=X## doe not imply ##2X = X\sqrt{2}##.

I already told you: I can't show that. Nobody can show that. Gödel has proven that one can never show that something can not be proven.

Since you agree that 1/2+1/4+1/8+1/16...=1, then the diagram is a proof without (additional) words that 1/2+1/4+1/8+1/16...=1 implies ##2 = 1\sqrt{2}##.

And since ##2=\sqrt{2}## is false, and since I have already given a rigorous proof that ##1/2 + 1/4 + 1/8 + ... = 1##, that implies directly that your diagram proof is wrong.

Now you fail by attacking me and not my argument. Is this an example of your rigorous mathematical reasoning?

Sure. So why I am wrong. Why should you qualify as a mathematician? Do you do mathematics as your job? Do you have a degree in mathematics? Have you published mathematical papers? If you haven't, then it's not an attack, but a statement of fact. If you have, then I was wrong by not counting you as a mathematician.
 
micromass said:
And since ##2=\sqrt{2}## is false, and since I have already given a rigorous proof that ##1/2 + 1/4 + 1/8 + ... = 1##, that implies directly that your diagram proof is wrong.
No, all you did is to use verbal-symbolic-only reasoning, as currently used by the majority of, so called, "pure" mathematicians.

By using visual AND symbolic reasoning, as I did in https://www.physicsforums.com/threa...nitiator-generator.881023/page-5#post-5542100, I rigorously prove that the acceptances that ##1/2 + 1/4 + 1/8 + ... = 1## implies directly that your verbal-symbolic-only reasoning does not hold, since ##1/2 + 1/4 + 1/8 + ... = 1## implies ##2=\sqrt{2}##.

micromass said:
Have you published mathematical papers?
Yes.
 
Look said:
No, all you did is to use verbal-symbolic-only reasoning, as currently used by the majority of, so called, "pure" mathematicians.

By using visual AND symbolic reasoning, as I did in https://www.physicsforums.com/threa...nitiator-generator.881023/page-5#post-5542100, I rigorously prove that the acceptances that ##1/2 + 1/4 + 1/8 + ... = 1## implies directly that your verbal-symbolic-only reasoning does not hold, since ##1/2 + 1/4 + 1/8 + ... = 1## implies ##2=\sqrt{2}##.

Oh nono, You asked repeatedly in this thread and specifically a proof from real analysis. I provided you with exactly such proof. You never asked for visual reasoning.

Also, I proved the result by verbal-symbolic reasoning. Why would that method make it invalid. Where is the error in the proof?

Yes.

Care to show? It's alright if you don't want to, I believe you anyway.
 
micromass said:
Oh nono, You asked repeatedly in this thread and specifically a proof from real analysis. I provided you with exactly such proof. You never asked for visual reasoning.

Also, I proved the result by verbal-symbolic reasoning. Why would that method make it invalid. Where is the error in the proof?
One can't find this error in verbal-symbolic-only reasoning by using verbal-symbolic-only reasoning.

In order to find this error, one has to use visual AND symbolic reasoning, as done, for example, in https://www.physicsforums.com/threa...nitiator-generator.881023/page-5#post-5542100
micromass said:
Care to show?
Only if you show yours in this thread.
micromass said:
It's alright if you don't want to, I believe you anyway.
Thank you.