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My question, motivated by idle curiosity while sitting in LaGuardia airport, is the following. You've just proved nice result A and it is time to write the paper. What is the real goal of the paper?

(a) to get the reader understand why result A is true.

(b) to convince the reader result A is true.

In some cases it may be feasible to do both. But often there is some tension between (a) and (b). To understand a result one needs global understanding. One may need to do some ugly computations in coordinates or deal with pictures that LaTex doesn't handle.

On the other hand to check the truth of a paper it is often easier to have local understanding because humans can only keep so much info in their brain at once. So if (b) is your goal you will work hard to break things up into smaller certifiable statements. That way once the reader has been convinced that Lemma X is true they can work with the statement and forget why it is true. To make it easier to check a proof one is often led to invent new formalisms or language and find coordinate free arguments. This can affect negatively (a) because you may not see the forest because of the trees.

I'd like to get the community's opinion. Usual Community Wiki rules are applicable.

Edit in hope of reopening. Perhaps if I make things more specific I can get the kind of answer I wanted and things would be less subjective. Suppose I have 2 proofs that finite sets $X$ and $Y$ have the same cardinality. One is proof is a relatively easy computation of the sizes of each set using known identities with binomial coefficients, Stirling numbers, etc. Any decent referee would follow it. The other proof is an involved bijection between $X$ and $Y$ whose details would be involved to check. Space considerations in the journal do not allow for both proofs. Which one should I submit?

Benjamin Steinberg
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    Is it really hard to do both at once? In my experience, heuristics that clears up things at the expense of verifiability tends to actually obscure things when read by somebody not already familiar with the subject. – darij grinberg Dec 10 '11 at 00:47
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    Vote to close as subjective and argumentative. This is by an large a matter of style/taste. Answers might make a nice/interesting read, but questions like this on MO cause too much direct and indirect difficulties. –  Dec 10 '11 at 01:40
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    @darij, of course sometimes one can do both. But have you never had the experience of reading a paper where you understood at each step why line x followed from x+1 but you had not clue how anybody could have thought it up?

    @quid, I definitely agree it is subjective. I hope it is not argumentative (that wasn't my intention).

     – Benjamin Steinberg Dec 10 '11 at 04:39
  • It may be the answer depends on result A to some extent. Take the odd order theorem in finite group theory. One doesn't need to really understand why it is true in a deep way to apply it and perhaps very few people do understand it. So (b) may be the best route for such a theorem. Other results may be more important because of an insight they provide than because they are useful to prove other results and so (a) would be more important. – Benjamin Steinberg Dec 10 '11 at 04:45
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    My answer: both (a) and (b). An author who doesn't try to do both has failed. (there, I'm both subjective and argumentative!) – Greg Martin Dec 10 '11 at 07:18
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    It also somewhat depends on the field, I guess. As a mathematical physicist, having a correct proof of a statement is far from enough for me (the fact that the statement holds having already been established with reasonable certainty by theoretical physicists). What I very much care about is really understanding why the statement is true, and that's the main purpose of a proof for me. So, a paper which can only be checked line by line, without ever leading to a full picture is useless, as far as I am concerned (as are proofs relying on long, unenlightening computations). – Yvan Velenik Dec 10 '11 at 11:09
  • @Benjamin Steinberg: I just said 'subjective and argumentative' since this is what the closing reason is called. What I really mean is that this question rather will lead to a discussion (as opposed to answers) and in addition it will be rather subjective one, because as already documented by what is around POVs depend a lot on context and precise interpretation in addition to personal style. Which in general is fine, but just for MO is rather not a good fit. –  Dec 10 '11 at 12:43
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    The analytic philosophers I know refer to this distinction as "the 'why' of justification versus the 'why' of explanation." – JSE Dec 10 '11 at 17:47
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    Meta thread at http://tea.mathoverflow.net/discussion/1244/ –  Dec 13 '11 at 18:53
  • The goal of the paper is to provide important and or interesting mathematics for a journal to publish and readers to enjoy or use. Your changes do not provide enough information because it is not clear if the binomial identity exercise is important, or the idea behind the bijection (never mind the proof) or the result itself. You can recast the goal of a paper in terms of your a and b, but that is an author-centric choice, and arbitrary. Reader-centric and publisher-centric views are just as arbitrary. Gerhard "But My Viewpoint Is Important" Paseman, 2011.12.13 – Gerhard Paseman Dec 13 '11 at 19:01
  • Is there a difference between (a) and (b) -- is one meant to indicate a formal proof while the other is not? Or is the difference somehow more subtle? – Ryan Budney Dec 13 '11 at 19:07

4 Answers4

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Papers are written so that their author(s) can forget their content and move on to other things. Therefore when you write you should be very careful to put in enough of the big picture and enough of the details so you'd be able to reconstruct your thoughts 10 years later if you'll need to, assuming you'll forget everything but retain some familiarity with some basic principles of mathematics.

Dror Bar-Natan
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    10 years? I envy you! – Mariano Suárez-Álvarez Dec 10 '11 at 01:19
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    This purpose seems more easily accomplished by blog posts (I understand this is what Terence Tao uses them for, for example). – Qiaochu Yuan Dec 10 '11 at 01:21
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    I should like to vote several times this answer. When I come back to my own articles, two decades all, I often feel lost... – Denis Serre Dec 10 '11 at 15:33
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    Papers are written without any (explicit) regard for readers other than the author himself/herself? :( – Kevin H. Lin Dec 15 '11 at 01:52
  • I so far like this answer best. But this would definitely lead me to (a) since I would like to remember how I thought it up. I would if writing for myself write something close to te raw intuition and forget all the tricks that I came up with to make it easier on the reader. – Benjamin Steinberg Dec 16 '11 at 05:59
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    Surprisingly for me, I have heard similar answer from Yuri Manin, he told me he writes papers keeping in mind himself after 10 years as a reader. Actually I do not quite agree with such position - since I guess not so many people as smart as him that probably means it might would be difficult to understand his papers... On the other hand hearing this for the second time, makes me think that may be I am not wise enough to take accept the truth :):) – Alexander Chervov Dec 19 '11 at 20:12
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    Amusing... I didn't know of the Manin opinion. Mine is a variant of something I heard from someone who heard it from Shlomo Sternberg, who supposedly "writes books to forget their content". – Dror Bar-Natan Dec 20 '11 at 02:35
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    @KevinH.Lin (Almost four years later...) one way to read this is as more of a minimal requirement: if the paper is written in such a way that the author wouldn't understand it ten years from now, then what hope does the poor reader have? (Another is the straightforward principle that the math we once did will not be lost to us in the future -- this seems to gain in importance as we get older. Even the great Erdos is said to have sometimes found papers written by his younger self almost impossible to follow!) – Todd Trimble Aug 13 '15 at 01:59
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I would add

(c) To convince the reader that I had a good reason for caring that A is true.

This has seemed sadly lacking in many of the papers I've refereed (sample bias, probably). Of course this goal is not easy to achieve, but I do sometimes wish authors would make more effort.

Yemon Choi
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    Good point. I think that is compatible with both (a) and (b). My point was whether the goal is to be understandable at an intuitive level or at a technical level. – Benjamin Steinberg Dec 10 '11 at 04:41
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    I should perhaps clarify that the "I" in (c) means the author. That is, I see too many papers which don't even tell me why the author wanted to prove that theorem, let alone start to convince the reader that he or she should care. I have probably been guilty of this fault myself. – Yemon Choi Dec 10 '11 at 05:38
  • Why he proved the theorem? Because he wanted that postdoc position or other and needed x number of papers to be relevant on the academic job market? No.. that can't be it. – santker heboln Jul 03 '14 at 08:25
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As you say, one goal of a paper is to certify that something is true. But the author should be more concerned with the result's certifiability than any reader. The author, being human, also requires the result to be broken up into smaller and more easily verified pieces. I can't count the number of times that I have proved a statement modulo certain details, and upon writing them up carefully I discovered that a new case emerged or the statement of the theorem needed to be altered in some (perhaps material) way.

The introduction of a paper is a great place to sketch an overview or give heuristics ... most readers will not venture beyond it anyway.

Xander Faber
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Put (a) in the introduction, and explanations between theorems and lemmas. Then use proofs to achieve (b).

timur
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  • This can work sometimes, but ofttimes I think it does not unless the intuition is still close to the final proof. Maybe the original proof was via explicit computation with chain complexes, but after you knew the proof you could use fancy homological algebra to avoid the messy stuff. – Benjamin Steinberg Dec 13 '11 at 17:53