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This is a question about practice and publication of research mathematics.

On the Wikipedia Page for Experimental Mathematics, I found the following quote:

Mathematicians have always practised experimental mathematics. Existing records of early mathematics, such as Babylonian mathematics, typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing results in a final, formal and abstract presentation. The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten.

My question concerns the last two sentences. I've heard several of my professors complaining about precisely this: in many mathematics papers, one can read the whole paper without ever understanding how the authors came up with the arguments in the first place.

Questions: 1.) (Historical:) Why did mathematicians stop publishing motivational steps when publishing mathematics? (For example, were there particular schools of mathematicians who actively advocated for this?)

2.) Is there any movement amongst mathematicians today to change this tradition? (By this, I do not mean movement on a personal level; I know that many mathematicians motivate results in their publications with preliminary calculations they have performed when they initially thought about the problem. Instead, what I am looking for is a movement on the community level to initiate a change)

3.) There seems to be a lot of disadvantages to the above practice; by communicating how one thinks about a problem, others would be able to copy the author's ways of thinking which will add to the knowledge of the greater mathematical community. Is there any practical benefit for not motivating results in papers?

Clarification. By "publication", I mean everything that one makes available to the greater mathematical community (so anything that one makes available on one's webpage is included (e.g. preprints) as well as anything uploaded to the ArXiv.

4.) Is the assertion made in the last two sentences in the quote accurate. (Thanks to YCor for pointing this out.)

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    $\begingroup$ This is a quote, which you put in the title as an unquestionable universal assertion. What about asking about accuracy of that assertion? $\endgroup$ – YCor Sep 12 '20 at 8:00
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    $\begingroup$ I would interpret "motivating a result" as explaining how it relates to other Mathematics (or maybe, sometimes, other sciences) and what potential applications it has there (and maybe what new insight it might give there). This is very field dependent. Personally, I don't really care what steps an an author had to go through to arrive at their result, as long as the proof is is correct and clearly written. $\endgroup$ – Geoff Robinson Sep 12 '20 at 8:43
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    $\begingroup$ “Clearly written” is the crucial element here. $\endgroup$ – Lubin Sep 12 '20 at 14:48
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    $\begingroup$ A small comment: often the first proof of a result is not the "right" one, and the "correct" way of thinking about the problem (that is, the one which properly "motivates" it and makes clear the proof) is only understood much later. $\endgroup$ – Sam Hopkins Sep 13 '20 at 15:44
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    $\begingroup$ Mathematical research is an activity which involves as much competition as cooperation. I wonder if some mathematicians don't keep for themselves some of the ideas behind their solutions also because they feel their way of thinking as an advantage which might let them get into other new results first than their competitors. $\endgroup$ – Anderson Brasil Sep 14 '20 at 4:06
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Others have given excellent answers. In general I do think that the pendulum has swung too far in the rabbit-out-of-a-hat direction, and it would help the mathematical community if there were more sharing of the process of discovery and not just the final result. But let me try to answer question 3 and suggest that it would not necessarily be an unqualified good for the pendulum to swing the other way and for everyone to be expected to give an account of how they arrived at their results.

For many of my results, the honest answer to the question of how I came up with them is, I don't know. I mean, I can usually say how the problem came to my attention, and if I were forced by some bureaucrat to report how many hours I spent working on it and what I did during those hours, I could come up with something. But if you're interested in the answers to questions such as, "How did you come up with that argument?" then often I will have no idea how to answer that question. In a few cases I might be able to quote chapter and verse for where I saw a similar argument in the literature. More often, the argument, or at least the line of attack, seems obvious to me, and I wouldn't know how to explain (to someone who doesn't find it obvious) why it seems obvious. Other times, if the idea isn't obvious, I still won't have any explanation for where it came from. This may be especially true if it's a collaborative effort—was it something my colleague said that triggered a thought in my brain? Again, usually, I don't know.

Even in the cases where I am able to answer such questions, being able to give an account of the process of discovery that is helpful to other people is as much of an art as any other kind of writing. What do you write down and what do you leave out? How do you organize your narrative? Writing down something like, "Well, I spent three weeks chasing down this dumb idea before finally realizing that I had misremembered a certain theorem and what I was hoping for couldn't possibly work" might provide emotional comfort and reassurance to insecure junior mathematicians, but do we really want to read detailed accounts of every single dead end in every single mathematical paper? What might help one person might not help another, and it's not a simple business to craft an account that will help a significant number of people.

Writing a paper is already a difficult task, and it would be burdensome to impose an additional requirement, on top of existing expectations, to provide a "account of discovery process" section. I know you weren't asking for such a requirement to be imposed, but I just want to sound a note of caution to be careful what you wish for. Before calling for the community to move in a certain direction, do you have a clear idea of what the destination is, and would it necessarily be better than where we are now? Maybe the way things are now is fine—people can share the process of discovery if they think it would be illuminating, but don't have to if they don't.

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  • $\begingroup$ Of course research reports should be justified in some sense, maybe because the result was a breakthrough and the proof and its key ideas surprising enough to qualify as a historical event in mathematical research. $\endgroup$ – Manfred Weis Sep 12 '20 at 13:24
  • $\begingroup$ @ManfredWeis : See my comment about Aristotle's distinction between efficient cause and final cause. $\endgroup$ – Timothy Chow Sep 12 '20 at 13:26
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There is a small ambiguity in the expression motivate a result.

You seem to use it for (A): "explain why the authors came out with certain arguments, definitions, methods etc, in order to prove the result".

But it can also refer to (B): "explain why they think the result is interesting/important; what is it aimed for; what is the reason to do such a research, why we should buy it".

It is true that the issue (A) is sometimes neglected in written papers (maybe it's more present in seminars, in anecdotical form); possible good reasons are:

  • mental paths that lead to the truth in some cases may help, yes, but in some other may be convoluted and distorted and of no help; the final point of view may be far shorter, clearer and simpler for understanding the logical structure.

  • the editorial issue of saving room in an article.

  • another (maybe less good) reason, a bit of vanity: remove all scaffolding and leave an aesthetic, shining and unintelligible object -- and let you think the author is genius.

I'd say issue (A) becomes important, both from the historical and pedagogical side, later, once (B) is agreed and the result accepted in the mathematical community.

Artists have always been protective about their tools and methods; it's their bread. Here are some historical examples.

  • I think the habit of vanity mentioned above was quite common in the European mathematical style of a century ago or more, and hopefully has been reduced in favor of a more pedagogical American style.

  • In Tartaglia's time, mathematicians would not even give you a proof, just the plain result.

  • Everybody would like to know how Gauss reached his neat conclusions, for instance. But the introduction of a paper is not necessarily the best moment and place.

  • Archimedes wrote the computation of the surface area of the sphere in perfect style and rigor, by means of the exhaustion method. But the explanation of how he arrived to "four times the largest inscribed circle" is not there. He first computed the volume of the sphere, by means of his favorite tool, the lever. He explained this later, in a letter to Erathostenes, The Method, a magnificent piece of scientific communication.

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    $\begingroup$ Good point about the distinction between what Aristotle might have called (A) the efficient cause and (B) the final cause. $\endgroup$ – Timothy Chow Sep 12 '20 at 11:27
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    $\begingroup$ I'd so much prefer (A) to the current (B). I feel like (B) is an explanation to the editor/reviewer why this article should be published, while I as the reader already know I am interested in the topic if I am reading the article at all. I've always seen (B) as waste of reader's time. $\endgroup$ – Džuris Sep 13 '20 at 8:49
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    $\begingroup$ good point, the current (B) is often just advertising $\endgroup$ – Pietro Majer Sep 13 '20 at 10:57
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    $\begingroup$ While I agree that (B) can degenerate into mere advertising, I don't agree that that's its only function. Perhaps it's because I'm more of a generalist than a specialist, but I frequently find myself skimming papers in areas that I don't know too much about. A priori, I might not know how interested I am in the paper. If the paper makes no effort to connect itself with something I already know something about, I'm likely to ignore it. And yet maybe the paper actually contains something that would interest me if only someone took the time to point out the connection with my interests. $\endgroup$ – Timothy Chow Sep 13 '20 at 15:44
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    $\begingroup$ Note that the Discrete Analysis journal includes editorial introductions. I suppose that is more (B) than (A). $\endgroup$ – J W Sep 13 '20 at 16:24
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C.F. Gauß may have been one of the earliest proponents of that "ideal"; if I remember correctly, E. Galois once said that Gauß is like tricky fox that wipes its trail with its tail. Also the motto "pauca sed matura" supports the assumption that Gauß played an important role in that development.

Another reason may be rooted in publishing mathematical results in journals; for one part there are often limitations on article length and for the other part articles that are published in such journals tend to be influential, which in turn may leave on young mathematicians the impression that being able to strip down publications to the absolutely necessary is a precondition to fame.
Or maybe it is also owed to mathematician's passion in abstracting (from latin "abstrahere", pull away, strip off) so that leaving only the bare bones of a mathematical result to the generations to come may be a special joy to certain mathematicians.

There are however signs of hope that mathematicians that are more in the vein of L. Euler provide insight into the stories behind their results via accompanying online publications as indicated in the answers to my question Examples of Mathematical Papers that Contain a Kind of Research Report

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    $\begingroup$ Euler is wonderful. Most of us are not Eulers. We are lousy writers, even less competent to write about what you call the "motivation" than we are to write up the "bare bones", and the result of our attempt would be boring and unilluminating. Moreover, for some of us, the process of solving a problem consists of forgetting the various false steps and bad ideas, so that by the time we are ready to write the paper, there is nothing left but the "bare bones", and we have no better idea than anyone else how they were found. $\endgroup$ – bof Sep 12 '20 at 7:37
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    $\begingroup$ I think there is value in having published papers being written somewhat "efficiently," so that a reader can quickly find the important results. On the other hand, sharing the motivation is also important and makes the math more human. One way to do both is to have a blog and write a less formal summary with your papers which gives a more complete story. $\endgroup$ – Gabe K Sep 12 '20 at 12:24
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    $\begingroup$ The arXiv is very recent compared with the history of mathematics. Even TeX is only a few decades old. Mathematical typesetting used to be a specialized and expensive skill, making journal pages a limited resource. Mathematical style takes a long time to evolve. (Heck, people still swear by illogical relics of physical typesetting like "<quote>," or "<quote>." when the punctuation isn't part of the quote.) $\endgroup$ – Noam D. Elkies Sep 12 '20 at 13:12
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    $\begingroup$ @NoamD.Elkies This is a bit off-topic, but why did that come from physical typesetting? I didn't know that. $\endgroup$ – Will Sawin Sep 12 '20 at 14:00
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    $\begingroup$ @WillSawin style.mla.org/punctuation-and-quotation-marks $\endgroup$ – Danu Sep 12 '20 at 15:16
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1.) I think I remember hearing that Gauss and Cauchy both liked to present results 'from nowhere'. Rumors, not history, but maybe it's right. I wouldn't bet my life on it.

2.) I think a lot of this goes on at seminars and in personal interactions. As Markus Land repeatedly stressed to us, mathematics is a social activity. That said, I don't know that I agree that this practice is extremely widespread. Maybe it depends on your subfield?

3.) Some referees and editors don't like to keep too much of that stuff in, or so I hear. I do have a cool example, though: Justin Noel, who recently left academia, used to include explanatory notes, motivation, and more detailed computations as comments in his TeX files, which you could find on the arXiv!

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    $\begingroup$ Why do referees and editors not like to keep too much motivational stuff in papers? $\endgroup$ – user676464327 Sep 12 '20 at 6:59
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    $\begingroup$ From a recent peer review on one of my papers: "The proofs are as expected and boring." So I guess the referee would like a shorter presentation because he/she feels that the details are unneccessary $\endgroup$ – Alexander Schmeding Sep 12 '20 at 7:15
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    $\begingroup$ The arXiv versions of most of my papers are longer than the published versions. Referees ask to cut things for the published versions, most often examples, but I keep them in the arXiv version. I suspect many people do that. $\endgroup$ – Ben McKay Sep 12 '20 at 7:31
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    $\begingroup$ @BenMcKay so he is no more working as a professional mathematician, though it seems that he still likes the topic. I apologize again for my perhaps pedantic comment (and I do not intended to be rude): it is only that in my opinion, Mathematics is first of all passion, therefore it is very difficult if not impossible to "leave it". $\endgroup$ – Daniele Tampieri Sep 12 '20 at 12:38
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    $\begingroup$ @DanieleTampieri : Indeed, according to some, I "left mathematics" in 1998, even though I still usually list my occupation as "mathematician" and occasionally publish research papers in mathematics. $\endgroup$ – Timothy Chow Sep 12 '20 at 16:20
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I don't have any data or evidence of the following ... the following is just my opinions and perspectives on the matter.

  1. (The 'why') Mathematics is a field of rigour, sometimes to excruciating details. If a reviewer needs to verify facts, they often just want the statements and the argument and they can verify it - reading a story of how the facts came around are secondary or even a waste of time to some. Even at conferences, I feel that mathematicians spend a lot of time presenting a proof rather than telling a story of how they came across the results or what the implications of the result are. Also, publishing mathematics is rarely meant to be an activity for non-specialists to participate in. Since a lot of modern math publications are in sub-sub-areas of a particular field of math, the only people reading and/or reviewing that work are specialists in that area and so there is not usually effort put it to telling the story of how it came to be, or the importance of it in the current research landscape because you are preaching to the choir.

  2. I agree at seminars / conferences, I've seen little attempt at trying to appeal to a wider audience with stories of how the work arrived. It's not uncommon to see many titles at a conference simply be a re-statement of the main theorem of the result. I try to stray from that.. once, I was presenting at the Fields Institute and before I even started speaking, an audience member in the front row said my slide had "a soft title" and once I started and talked about motivations behind the work, he walked out for a smoke before I even got to my first theorem statement. (And this guy is a very notable individual of the research area.) There are a few people who will tell a good story at a conference and leave the proof details to the paper, but there is a common attitude that this is some kind of sign of weakness. I have sat through plenty of talks that were not exactly my area and 30 seconds was given to the slide with all the required definitions and the presenter essentially went through them with a "we all know that blah is defined to be this expression" and moves on faster than a person could read/absorb the definitions. Then proceeded to talk about proof details, which was a waste of time for anyone who didn't get the definitions. I don't know if there is a movement to change these attitudes. I've just decided that to go less math conferences and more computer science ones (or other fields).

  3. I agree - I've been asked to remove storytelling in math submissions yet in other venues, I was told that I needed to increase the background, motivation, implications, etc (much to my surprise at the time). I essentially began in a math field and migrated to computer science.

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  • $\begingroup$ I think you're being a little hard on people in #2. Some of that is really just a lack of skill in knowing your audience and in organizing your talk. I think I can say that I have never given a really good talk, but it's not because I try to tailor my talks toward experts. I'm just not that good… $\endgroup$ – Harry Gindi Sep 12 '20 at 7:31
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    $\begingroup$ @HarryGindi I don't know which one of is being more hard on people in #2... me saying that they present in a smug manner or you saying that they lack a skill in presenting :D $\endgroup$ – JimN Sep 12 '20 at 7:50
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Disclaimer: I am myself a junior mathematician who has done no real research (yet), but have read many publications.

At the beginning of my journey, I would have agreed with you. But now, especially as I learn more advanced topics, I find that my opinions have changed.

You say

...by communicating how one thinks about a problem, others would be able to copy the author's ways of thinking which will add to the knowledge of the greater mathematical community

Again, I once would have agreed with you, but now I think this claim is false. I think what you are saying here is that you wish for the author to "implant" their intuition on the subject into your brain. But I argue that this is impossible. It seems that there is always an intuitive way to think about a subject for any individual mathematician, but there is no single explanation that is intuitive for all mathematicians.

If you allow me to use an analogy, you are saying that you wish for the author to write down some algorithm which, if you hadn't been presented the answer, would have allowed you to come up with the proof on your own. But this is exactly what writing the proof without any motivation is, it is an algorithm (albeit a trivial one) which says "you could have read the references and written <insert proof here> and you would have produced the proof".

If we treat your brain and the author's like a computer, and the proof as a program running on it, the problem you are trying to solve is that, in your brain, there are a lot of functions and variables and types used in the program which are undefined, thus you are getting a "compiler error". However in the author's brain, the program compiles because all of these functions and variables and types are defined. That is a way to give more meaning to the phrase "the author's way of thinking".

The task you are asking of the author is to predict which functions and variables in your brain are defined and only use those to produce the proof. This is clearly an impossible task for all mathematicians, as what one mathematician knows is never the same as another. It is also the purpose of references. By giving references, the author is saying "we have tested that this program runs if you have these dependencies installed (i.e. that you read or at least understand the references), but we give no guarantee that it will run without them". If you don't install the dependencies then you can't guarantee that the program will run. When you think about it, the references (and recursively their references) are probably close to the path the author took to come up with the proof (maybe over the course of their entire life).

If you ask for an algorithm which is non trivial and produces the proof (i.e. not one which simply says "read these references, then write this"), what you are really asking is for the mathematician to produce a proof of a more general theorem, or at the very least, a more amazing insight than the one which has already presented, which is an unreasonable expectation (since the author would have to again extend that result, and then extend that one, etc.).

Thus, it is up to the individual to find such a motivation that works for them, to fix the "compilation errors" in their own brain, by reading references, reading books on the subject, and asking questions on websites like mathoverflow.

It is not the researcher's responsibility to completely fill in your understanding to allow you to think about the problem in the same way that they do, just like it is not the responsibility of a software developer to manually install every dependency on the computer of every person that uses their program.

The researcher's only responsibility is to present factual information in a way that is not overly complicated, and to fix any "bugs" in their proof which are found.

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As to Question 1: Euclid's Elements is written in the style you deplore.

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    $\begingroup$ I think it does partially answer Question 1. It points out that omitting "motivation" didn't start with Gauss but goes back to Euclid. $\endgroup$ – Timothy Chow Sep 12 '20 at 14:15
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    $\begingroup$ The Elements was not a publication of research mathematics but a textbook based on earlier work (now lost) by Eudoxus, Hippocrates, Pythagoras and others. Why would anyone expect Euclid to know or explain the motivations of these predecessors who died before Euclid was born? $\endgroup$ – Matt F. Sep 12 '20 at 14:25
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    $\begingroup$ But the style of the Elements was generally seen as something to emulate if one strove for rigor (cf. Spinoza's Ethics). $\endgroup$ – gmvh Sep 12 '20 at 15:41
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    $\begingroup$ @Matt F. Parts of Elements are known to be based on work of earlier thinkers, e.g. Eudoxus and Theatetus, but it isn't possible to be confident that this was true of the entire work. The number theory, for example, doesn't seem to have known antecedents. The lives of Euclid and Eudoxus probably overlapped, if only slightly. Most present-day scholars of Pythagoras believe he did not do anything that could be called "research mathematics". It's true that, where there are predecessors, Euclid does not mention them, much less their motivations. $\endgroup$ – Will Orrick Sep 12 '20 at 15:59
  • $\begingroup$ @WillOrrick, Wikipedia suggests that Eudoxus died in 337 BC and Euclid was born c. 325 BC — where do you differ? $\endgroup$ – Matt F. Sep 12 '20 at 16:20
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Community Wiki answer, too long for a comment, addressing things one may do about it rather than the practice itself.

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A personal story. I once met T. Y. Lam's daughter, Fumei Lam, during a lunch. She works in graph theory. She was upset about taking months to find a counterexample to one of her own conjectures, plus an annoyingly simple example at that. I said (I must paraphrase, years ago) "But you know what happens next. You thought you had everything in one big lump. Now you have a little bit outside the lump. You continue to work on it; the big lump will shrink a little more, the little bit grow somewhat. If things go unusually well, you may produce a classification." I think I added something about others working on it. She did seem relieved at that.

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This is not in any way a comprehensive answer to your question, but it may be worth pointing out that while there are certainly several disadvantages to "erasing the trail" up to a result, there is also (at least) one major advantage: it lessens the "path dependence" and encourages others to think about possible applications of the result that would not have occured to the original discoverer.

For example, many people discover a result by thinking very carefully about one specific example. But the beauty of abstract math is that the same general result often applies to a huge range of concrete examples that differ hugely in their details, but share the minimal mathematical structure necessary for the result. The original discoverer may be so used to thinking about the result in the context of one type of example that they miss applications to other classes of examples. Presenting the detailed example that they worked through to motivate the result risks similarly locking other readers into the same mode of though. Whereas if a reader approaches a new result with a completely different example/application in mind, then they may be able to more easily extend the result to new corollaries that the original discoverer didn't think of.

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