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The author of an answer on math.se remarked that much of mathematical progress is a factor of the accrual of both, theorems, which other mathematicians can use in their proofs, and more efficient proofs of those theorems, which enables, students, and mathematicians, to apprehend them faster than they would have otherwise apprehended them.

In the paper A Beginner's Guide to Forcing, Timothy Chow introduces the phrase open exposition problem, which he uses to denote all theorems for which there is no proof such that any person capable of apprehending the premises would, after s/he read the proof, disbelieve that s/he could have inferred the conclusions him/herself.

Considering that, it seems to me that mathematicians appreciate how succinct expositions affect the progress of their discipline.

I study undergraduate math and philosophy. Next year, I'd like to submit an essay on the importance of solving open exposition problems in philosophy to an undergraduate journal. Before writing it, I'd like to read what mathematicians have written on either, the effect of the simplification of proofs on the progress of mathematics, or how to simplify proofs. Have mathematicians written on either topic? If so, could you recommend some titles.

Thank you,

-Hal

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    $\begingroup$ This is interesting but I'm having trouble getting my head around the phrase open exposition problem. Could you give an example? $\endgroup$ – Karl Mar 30 '15 at 19:11
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    $\begingroup$ For an example of an exposition problem, see this post by Gowers. Also, I would remark (IIRC...) that Polya (1945) recommends scaffolding problems by asking questions that the student feels she could have thought up herself; such a concept fits in well with Vygotsky's idea of ZPD. $\endgroup$ – Benjamin Dickman Mar 30 '15 at 21:47
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    $\begingroup$ @Karl, I think this is the simplest way to put it: An open research problem is a hunt for a certain theorem. An open exposition problem is a hunt for the proof of a theorem that's the easiest one for people to understand. $\endgroup$ – Hal Mar 30 '15 at 22:14
  • $\begingroup$ Got it. So the hunt for an easy to understand version of Fermat's last theorem for example? $\endgroup$ – Karl Mar 30 '15 at 22:23
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    $\begingroup$ @Karl Yes. I think the idea is that the problem is solved when there's a version that's, for all intents and purposes, impossible not to understand after reading the proof once. That is, provided the reader understands the theorems/concepts that the proof builds on. $\endgroup$ – Hal Mar 30 '15 at 22:54
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Thurston, William P. "On proof and progress in mathematics." New Directions in the Philosophy of Mathematics (1998): 337-55. (arXiv abstract link).

I think that Thurston's famous essay supports the notion that simpler proofs are a mark of progress in mathematics, because simpler proofs lead to easier communication and deeper understanding, and

"The measure of our success [in mathematics] is whether what we do enables people to understand and think more clearly and effectively about mathematics."

An example is deBranges 1984 proof of the Bieberbach Conjecture, which was long and complicated but eventually simplified to a four-page proof by Weinstein in 1991 (and further "simplified" by use of Zeilberger's computer WZ method).

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