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Topic: Problem understanding proof of Jordan Curve Theorem
Replies: 13   Last Post: Jul 12, 2013 9:25 AM

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Paul

Posts: 393
Registered: 7/12/10
Re: Problem understanding proof of Jordan Curve Theorem
Posted: Jul 11, 2013 11:20 AM
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On Thursday, July 11, 2013 3:53:03 PM UTC+1, dull...@sprynet.com wrote:
> On Thu, 11 Jul 2013 00:53:42 -0700 (PDT), pepstein5@gmail.com wrote: >On Wednesday, July 10, 2013 5:02:59 PM UTC+1, dull...@sprynet.com wrote: >... >> >> It's clear. >> >> >> >> The proof I'm about to give is "perfectly clear" in my book. It >> >> may strike you as something that he should have written >> >> out in detail because it's not obvious. I disagree - it >> >> was immediately obvious to me, and the paper was >> >> written with a certain audience in mind... >> >... > >But I never said anything remotely critical of the author or the paper. I made the factually correct statement that the lemma was not clear _to me_. > What you've done is to arbitrarily come up with a statement that you disagree with, and then to explain your disagreement. Sigh. Why do you spend so much time on this sort of thing, debating nuances of who said what about who said what about who said what? I said you _may_ disagree that the argument I was going to give was not clear. Where by "clear" I meant that it really was sufficient for the author to simply state that the existence of a maximmizing disk was clear, without giving any explanation. One reason I thought you might feel that way was various things from the past, where you've said something was not clear and insiisted that the author should have provided more explanation. Another reason is that the assertion _was_ perfectly clear! The author says the existence of a maximizing disk is clear. You say it's not clear to you. The proof I gave didn't involve any sublety or cleverness - it was the sort of thing where when I ask myself how I'd prove that I saw the proof immediately. So if you say the assertion is not clear to you then either (i) the proof I gave was not clear to you, the possibility I mentioned, or (ii) you didn't see that proof until I pointed it out. The reason (i) seemed likely to me is that I didn't see how (ii) was possible! You start with what you're trying to prove, think about it for a second, and you have the proof. >I could follow in your footsteps by saying that I disagree with the statement that English is the most commonly spoken language in Singapore. You _really_ think that's a fair analogy? >On the maths side, the help given by yourself and Rupert is greatly appreciated and I now fully understand why the remark in lemma 3 that I referred to is correct. Which remark? There's no "remark" that there exists a maximizing disk. There _is_ a remark to the effect that the existence of a maximizing disk is _clear_. The reason I ask is actually to help you out, whether you believe it or not. You're trying to learn some math. That's good. If the existence of a maximizing disk is not "clear" then you really need to get better at finding proofs of trivialities. Which is why skipping most of the exercises in Rudin was not a good idea. It's not too late... > >Thank You, > >Paul Epstein

David,

Thank you for this reply. I agree with almost everything you (and Rupert) have said on this thread. A few points to clarify where I stand:
I did not initially see the value to your comments about the clarity of the point I was stuck on. Those comments did not seem helpful (but do now seem helpful in the light of your recent post). Apparently, the purpose was to alert me to the fact that I need to understand some of the underlying theory/ proof techniques better -- I didn't initially realise that you had that purpose.

A fact is that I do enjoy digesting the proofs of classical theorems (like JCT) but do not usually enjoy trying to solve maths problems or exercises. (Not wanting to do the exercises would be a worrying trait for a maths academic or student but I don't have either role.) That explains why I didn't do many exercises in Rudin. I may rethink that approach if it means I'm unable to understand a lot of the classical results I'm trying to learn the proofs of -- like JCT, Stokes Theorem and Brouwer's theorem, the isoperimetric theorem in two dimensions etc.

Previously, when I said "remark", I meant the mathematical statement which I asked about at the beginning of this thread. You are correct that I should not have used the word "remark". Perhaps I should have said "claim".

I'm steadily making progress on the Tverberg paper. I very much hope not to get stuck again. If I do get stuck, I'm sure that you or others will be kind enough to help me fill in the gaps, as the sci.math community has always made an excellent job of doing in the past.

Thanks again,

Paul Epstein





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