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What is a much better proof in mathematics, is it need to be a much more advanced one compared with the proof already available or a much simpler one?
I think you can challenge a proof in two different ways. Either we can find much generalised proof covering all cases as an extension for the previous proof using much advanced concepts in mathematics or we can suggest a much simpler proof. As an example how would you compare if a mathematician could able to prove FLT in less number of pages when comparing with the proof given by Andrew Wiles since after giving the full credit for the first valid proof there is no chance of withdrawing it later for a better one.
Edited ; I posted this because I think as mathematics educators we need to know how we have to decide which proof is much better and that kind of issue is much relevant to mathematics teachers rather than students. Hope you will give your fullest support to get the full use of this discussion with much understanding .

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    $\begingroup$ This seems to be more of a mathematics question than a mathematics education question. $\endgroup$
    – JRN
    Commented Oct 7, 2023 at 7:43
  • $\begingroup$ @JRN I think knowledge of evaluation of better proofs can be important for teachers as well as students. $\endgroup$ Commented Oct 7, 2023 at 8:23
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    $\begingroup$ I think knowledge of math can be useful for teachers as well as students. Oh wait. $\endgroup$ Commented Oct 7, 2023 at 9:51
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    $\begingroup$ Voted to close. It's not clear to me what precisely you are asking, and those parts of the question that I understand do not seem to be about math education (as already mentioned in other comments). $\endgroup$ Commented Oct 7, 2023 at 11:04
  • $\begingroup$ @JochenGlueck is it not important issue as for teachers how to evaluate the quality of a proof in other words how to compare proofs of different levels ? $\endgroup$ Commented Oct 7, 2023 at 11:38

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As an example how would you compare if a mathematician could able to prove FLT in less number of pages when comparing with the proof given by Andrew Wiles since after giving the full credit for the first valid proof there is no chance of withdrawing it later for a better one.

That would, of course, be a very impressive accomplishment.

But solving an unsolved problem at all, especially a long-standing well-known problem like FLT, is generally way harder than coming up with a simpler solution to an already-solved problem.

Which proof would be "better"? The simpler one.

Who should get the credit for solving FLT? Andrew Wiles.

Does that mean there's no point in striving for a simpler proof? No -- when you come up with a simpler proof, you're generally doing so by developing a better understanding of the "core" of the problem. And that better understanding can helpful for solving other unsolved problems, which is beneficial regardless of whether you're thinking altruistically (what's in it for the field of mathematics as a whole) or selfishly (what's in it for you in particular).

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Let me disagree with this dichotomy:

"Either we can find much generalised proof ... or we can suggest a much simpler proof."

I think there are other options, proofs that yield different insights, draw on different areas of math, make connections to different fields. There is not one "better proof"; "quality" seems an inappropriate measure.

An example is Euler's formula: $$V - E + F = 2 \;.$$

David Eppstein maintains a webpage with Twenty-one Proofs of Euler's Formula. I've found the variety quite useful for (college) students appreciating the result in the classroom. Different students may find different proofs more insightful, depending on their prior exposure to the relevant background concepts. Not unlike Tommi's viewpoint.

StereoProj

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For educational purposes, what you want is a proof that has different ideas in it. This is similar to teaching: you want pupils to present different ways to solve a given problem and then compare and contrast, to increase understanding of the problem, the concepts and the tools.

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