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Today I watched a YouTube video interview of Pierre Deligne. I have some confusion about a statement given by Pierre (video timing: 56:33 to 58:20). Here is the outline of the statement in the video:

Interviewer: Would you like to write letters to other mathematicians?

Pierre Deligne: Writing a paper takes a lot of time. Writing it is very useful, to have everything put together in a correct way, and one learns a lot doing so, but it’s also somewhat painful. So in the beginning of forming ideas, I find it very convenient to write a letter. I send it, but often it is really a letter to myself. Because I don’t have to dwell on things the recipient knows about, some shortcuts will be all right. Sometimes the letter, or a copy of it, will stay in a drawer for some years, but it preserves ideas and when I eventually write a paper, it serves as a blueprint.

What type of letter is he talking about and why does he write a letter to other mathematicians?

I mean, there are many different types of letters out there. I mean, some love letters, friends letters, etc.

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    I think such letters are essentially like papers about mathematics, except that they begin like a letter ("Dear ..., I hope you are doing well. How is your family?"), end like a letter ("Best wishes ..."), and omit explanations of things that the recipient already knows. I believe Paul Erdös sometimes skipped most of the non-mathematics ("Dear X, Let p be a prime and suppose ...."). – Andreas Blass Jan 28 at 22:34
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    @AndreasBlass, that is probably an answer. – Buffy Jan 28 at 22:35
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    One of the best ways to know whether you really understand something is to try to explain it to someone else. – Buffy Jan 28 at 22:40
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    "some love letter , friends letter etc..." A math letter. Like if you wanted to describe the math you were working on with a friend, rather than the entire field of mathematics. – Bryan Krause Jan 28 at 22:42
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    @AndreasBlass: I tracked one example down. You are exactly right, and they are on the most bare-bone side of the possible variations you mention - "Dear Looijenga, I mentioned your problem to Tits and Zagier, and they solved it for me.." (insert math) "Best, P. Deligne." homepage.ruhr-uni-bochum.de/christian.stump/… – gnometorule Jan 28 at 23:38
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I just did something like this last month...

Well, with modern technology available, I didn't write a letter; I wrote up some notes, scanned them in, and attached them to an e-mail. Keep in mind, though, that Prof. Deligne started his career in the days before e-mail, and telephone calls (from rotary dial phones attached to landlines) were expensive (like a dollar a minute.)

I had an alternate approach to proving a theorem in a paper I had read and thought perhaps the approach could also generalize beyond their theorem. I wrote up a sketch of the alternate proof (for which some details I have not actually proved) in 3 pages of notes. I scanned them, and attached them to an email to the authors that read something like "Dear X and Y, I hope you are well. We talked about this at Online Conference Z and I'm attaching some notes. I'm happy to meet by Zoom and talk. Best, -me"

I didn't write up a paper because it would take 10 pages and at least a couple weeks of work to give precise definitions of everything, and any paper would only be publishable in a fairly low-tier journal (since the result isn't new and the approach also isn't new, having been applied to other problems before), which at this point in my career doesn't count for anything.

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    "which at this point in my career doesn't count for anything." this is a very weird attitude... – Jannik Pitt Jan 29 at 8:27
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    @JannikPitt But all but unavoidable: if your career is dependent on doing certain tasks, then spending significant amounts of time on other tasks which are not beneficial to your career is actively harmful to that career. – user3482749 Jan 29 at 12:18
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    @JannikPitt: The paper also wouldn't benefit the research community. The average math paper is cited less than once. In fact, not counting the authors and reviewers, it's probably read less than once. In this case, the only people who might have read the paper are the people I've sent the e-mail to. Why should I waste the time of an editor and reviewers for some write-only journal, when their time is needed to evaluate the papers of people who actually need such papers published for their tenure and promotion? – Alexander Woo Jan 29 at 15:56
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    @JannakPitt - And I don't find it worth my time to write it up. And even if I did, I would just put it on the ArXiv without submitting to a journal, because it's not worth the peer reviewer's effort. (Peer review is breaking under the load, in case you didn't know.) It's not that the default is writing it, and I'm not doing it just because it doesn't benefit my career. Rather, the default is not writing it, and unlike people in certain career positions, I don't have some perverse incentive to change the default. Fifty years ago, no one would have even thought of writing this up. – Alexander Woo Jan 29 at 21:38
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    @AlexanderWoo I have no reason to doubt your judgement in this particular case, but this general phenomenon is how in my area we ended up with a lot of important facts and examples that are known only to "experts" (read: the people who happened to participate to the right coffee break at certain conferences), and every grad student has a real uphill struggle to get familiar with the "basics" which are basically trade secrets – Denis Nardin Jan 30 at 10:04
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I am a mathematician and I have written exactly this type of letter before. It was to a collaborator explaining what I thought could be the basis for our next project. As it happens, this collaborator and I are very close friends in addition to being collaborators.

In terms of format it was a letter to a friend. “Dear friend...” then four pages of math where I only had to explain things he didn’t already know and finally best wishes to him and his spouse.

Any actually time sensitive communication was already done by email and if necessary the phone. So this letter was a deliberately slow method of communication. It’s primary value was for me to sit down and think through the entire idea. That I put it into the mail is something of a secondary effect.

So to answer your “what kind of letter” it is probably best to imagine it as an informal business letter between colleagues who have known each other for a while and can include friendly chatter as well (but the chatter probably happens through other channels anyway.)

Just an oddly relevant piece of experience.

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I do research in applied mathematics and computer science and, whenever I have an idea, I write an email explaining it and send it to myself.

I found that writing such emails helps me find holes in my reasoning (things I missed or skipped over) but also helps me synthesize my ideas (which sometimes leads to simplifications of the resulting concept). This emails also prove useful when I want to email other people to exchange on my ideas.

Furthermore, I write new developments as answers to the first email which gives me a centralized place to see the evolution of the idea and be able to come back to my previous positions on the subject.

The main downside is that, at the moment, I am the sender of most of my unread emails...

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The lengthy and fruitful collaboration of G. H. Hardy and J. E. Littlewood was the most productive in mathematical history. Dominating the English mathematical scene for the first half of the 20th century, they produced a hundred joint papers of great influence, most notably in analysis and number theory.

(From a transcript of https://www.youtube.com/watch?v=asXHAvibq1g )

They did that joint work almost exclusively by correspondence, even when they were at Cambridge together.

From https://moleseyhill.com/2010-03-22-hardy-littlewood-rules.html :

Hardy Littlewood Rules

Axiom 1: It didn't matter whether what they wrote to each other
         was right or wrong.
Axiom 2: There was no obligation to reply, or even to read, 
          any letter one sent to the other.
Axiom 3: They should not try to think about the same things.
Axiom 4: To avoid any quarrels, all papers would be under joint
          name, regardless of whether one of them had 
          contributed nothing to the work.
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    I think, it is widely accepted, that both Hardy and Littlewood were brilliant mathematicians. But I cannot wonder if such rules would not sit well with publication and research ethics nowadays. – Oleg Lobachev Jan 29 at 20:15
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    @OlegLobachev There's an SE post on this: academia.stackexchange.com/questions/5179/… . – Ethan Bolker Jan 29 at 21:01
  • @OlegLobachev Regarding Axiom 4, that's not the original phrasing. In fact the information comes second-hand via Harald Bohr anyway, but according to Bohr it is "it was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name" (this version can be found in Littlewood's Miscellany). I would translate this as "It is not necessary to remove authorship if one of us contributes much less than the other to a joint paper". Hardy and Littlewood still published single-author papers. – Robert Furber Jan 30 at 8:10
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Imagine a time before fax machines were standard, before the Berlin wall came down, before TeX. Travel to see other mathematicians was not always possible, and about the only way to communicate mathematics that involved intricate diagrams was to write and draw with pen and paper, and put it in the post. Perhaps someone the age of Deligne will post an answer, but at least I have worked with Mathematicians born about that time so I know a bit of their habits.

By 1972, I imagine that photocopiers were easily accessible at L'Institut Des Hautes Études Scientifiques so the author of a letter could easily keep a copy. One could write another mathematician with a partial result, a conjecture, perhaps the outline of a new theory. Knowing the recipient personally was decidedly optional.

Mail to another country could take one to twenty weeks, so it often made sense to write letters that went on for many pages. If one knew the background of the recipient, one could skip that bits about notation and definitions and also ignore the realities of technical typists.

Sending by fax become affordable to academics circa 1990, and one could write shorter letters as one could expect a timely reply. However, drawing something as simple as a 3D commutative diagram was really hard except drawn by hand.

The last time I received such a math letter, albeit in the form of a fax, was probably in 1995. The first such letter I ever wrote was in 1985, I think, on special lightweight paper so it could go by air from the US to Romania.

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but often it is really a letter to myself

In engineering, this is a key concept as a part of the review process. Of course a reviewer provides a second pair of eyes to spot mistakes by the designer. However there is an secondary well-recognized effect as well: having to explain your design to someone else can allow you to see flaws in it which were not obvious during the design process.

The same thing happens with fault-finding. It is not at all uncommon that in the process of explaining a confusing problem to someone, you realize what the cause might be.

This is commonly known in software engineering as "Rubber Duck Programming", on the principle that explaining the problem to an inanimate object (such as a rubber duck) can be as useful as another human being.

The author in this case is finding the same thing. Whilst his correspondents may have useful insights, what matters are the insights he finds himself whilst writing that explanation.

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Rubber ducks, and then drawers

It's a form of rubber duck debugging where you preserve the conversation in the form of paper/letters.

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There is a rather touching book by Steven Strogatz, the famed chaos theorist, entitled The Calculus of Friendship chronicling decades of correspondence between him and his high school math teacher, Joff. Usually, Joff would write with a puzzle or general question on something and Steve would respond with the solution in addition to some banter or small talk. There are some very interesting proofs/discussions of nontrivial (at least to me) topics (e.g. general method for deriving closed-form solutions for recurrence relations with shift operators, the convergence point of sum(sin k / k) for k=1,inf) explained in understandable ways, its fun to read and probably an enjoyable thing to do so it is not unheard of as many have already testified.

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