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Topic: easiest proof of Jordan Curve Theorem as a corollary of Moebius
theorem #1982 Correcting Math

Replies: 7   Last Post: Aug 11, 2014 3:19 PM

 Messages: [ Previous | Next ]
 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
fight in math over Jordan vs. Veblen and why case studies is never a
proof #1986 Correcting Math

Posted: Aug 11, 2014 3:19 PM

On Sunday, August 10, 2014 3:49:12 PM UTC-5, Archimedes Plutonium wrote:
> Now unless I post this contentious issue of whether anyone proved the Jordan Curve theorem or that all of them were flawed, that future readers of AP will think I was making this up or exaggerating. But no, the JCT is questionable that anyone gave a sound proof. That is why my proof is all that more remarkable in that I take the Moebius Theorem proof and find the Jordan Curve proof as a minor corollary of Moebius.
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> --- quoting from Wikipedia on the contentious issue of whether anyone really had a proof of the Jordan Curve theorem or all of them were flawed ---
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> http://en.wikipedia.org/wiki/Jordan_curve_theorem
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> The Jordan curve theorem is named after the mathematician Camille Jordan, who found its first proof. For decades, mathematicians generally thought that this proof was flawed and that the first rigorous proof was carried out by Oswald Veblen. However, this notion has been challenged by Thomas C. Hales and others.
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> The first proof of this theorem was given by Camille Jordan in his lectures on real analysis, and was published in his book Cours d'analyse de l'École Polytechnique.[1] There is some controversy about whether Jordan's proof was complete: the majority of commenters on it have claimed that the first complete proof was given later by Oswald Veblen, who said the following about Jordan's proof:
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> His proof, however, is unsatisfactory to many mathematicians. It assumes the theorem without proof in the important special case of a simple polygon, and of the argument from that point on, one must admit at least that all details are not given.[2]
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> --- end quote ---
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Now searching in Google for "Veblen proof Jordan Curve" I run into these two hits early on:

Jordan's Proof of the Jordan Curve Theorem - Mathematics
https://www.math.sunysb.edu/~bishop/classes/math401.../HalesDefense.p...
by TC Hales - Cited by 19 - Related articles
Abstract. This article defends Jordan's original proof of the Jordan curve theorem. ... At the same time, Veblen's proof has suffered with the passage of time. His.

Jordan Curve Theorem
math.osu.edu/~fiedorow/math655/Jordan.html
Ohio State University
Jordan Curve Theorem Any continuous simple closed curve in the plane, ... The first correct proof of the Jordan curve theorem was given by Oswald Veblen in ...

This situation is funny and laughable, because we have math professors some of whom side with Jordan and some side with Veblen.

And the reason all the raucous is that many in mathematics such as Hales or Veblen or Jordan do not know and have never known that in Mathematics we can conduct a Analysis on a statement giving many case studies, or we can conduct an actual proof of a statement which shucks and removes the case study analysis. You see, Hales never learned that case studies is not a proof but only an adventure in getting to know the statement far better, but when it comes to proving the statement, we put aside all those case studies, roll up our sleeves and actually give a proof.

So much of topology and perhaps graph theory is riddled full of these silly obnoxious case study Analysis. Of course, the bards who love dealing in case studies like Hales, they want reward for their case studies and so they hide the fact that theirs is a mere case study analysis and announce they found a proof, when all they did was piddle paddle make a case study. They want and try to promote their case study as the proof. When it never was and never will be.

In modern times we have such worthless case studies Analysis promoted as proofs when they never were proofs. We have the Appel & Haken case studies of 4 Color Mapping. We have the Andrew Wiles promotion of case studies of FLT. We have the Hales promotion of case studies of Kepler Packing.

But Hales has really become the king of case studies, who does not seem to mind studying thousands of cases of a statement of mathematics, and then promoting that lousy case study as a actual proof.

Math needs true real proofs of math, not some silly lousy thousand pages of thousands of different cases. Cases are for learning the full aspect of the statement, but they do not serve as the proof of the statement. They never serve as the proof, because the number of cases is just a random haphazard number which is realized today, but maybe more cases tomorrow.

Now if you read the Wikipedia entry of the proof of the Jordan Curve theorem saying this:
"The Jordan curve theorem was independently generalized to higher dimensions by H. Lebesgue and L.E.J. Brouwer in 1911, resulting in the Jordan-Brouwer separation theorem."

Now the only problem with that is likely that Lebesque and Brouwer never defined or proved how high dimensions can go. If theirs is only a rise to 3rd dimension then they be alright in a generalization, but if higher then theirs is no proof either.

But Brouwer and Lebesgue were on the right track in wanting to "generalize the Jordan Curve theorem". But they failed to realize that the Moebius theorem was already that generalization and that they needed never to concoct a higher dimension generalization.

AP