Question: http://mathoverflow.net/questions/56104/first-two-author-math-paper

Apart from the dubiousness of the topic, my main issue with this question is that it is unverifiable. I've voted to close, and I see that at present there are three outstanding votes to close. The comment thread is long enough to warrant a meta discussion so if anyone thinks that this question should stay open, here's your chance to persuade the rest of us.

Andrew, I'm puzzled by your notion of unverifiable and how it applies here. It seems to me that the corpus of mathematics is a definite finite object. Sure, the corpus is evolving, mostly with new stuff but also some occasional discoveries of older materials. I don't think the fact that the corpus is evolving is a valid objection since MO is evolving too. Maybe you think the corpus is too large? I don't think that the fact that the corpus is large is a valid objection since MO is a great place to pool limited observations.

Back to the question itself. I agree with Todd that the word "important" makes the question subjective and hence inappropriate for MO. However, this could be corrected with a simple edit. I also think the question should address the issue of defining the corpus of mathematics, perhaps limiting it to the relevant corpus. (In antiquity, authorship as we know it today was not a major concern so it's probably irrelevant to go that far back.) In summary, I think the question is inappropriate but salvageable.

Perhaps the underlying issue is whether the history of mathematics is off-topic for MO. My personal view here is that the history of mathematics is a perfectly fine topic for MO. However, I suspect that many find some areas of the history of mathematics to be off-topic.

Voted to reopen. I would remove the word "important" from the formulation of the question also, but one word does not make question bad. The question of collaborative math is very natural and the question of why we have much more of it now than, say in 19th century, is natural too. The fact that there are more mathematicians now does not seem to be satisfactory. I think that this part of the question has not been answered yet.

Ideally one should also clarify what it means by a published paper. Do you want to count Euclid's elements, much of which being a collection of results proven by other people, as a published paper? Or do you want to arbitrarily restrict your attention only to papers published in the Western notion of a scientific journal, which arised only after the advent of the printing press and more-or-less after the founding of learned societies, with a more modern definition of authorship? In which case, one can do worse than starting from volume 1, issue 1 of the Philosophical Transactions of the Royal Society back in 1665 and go foward until one sees a jointly authored mathematics paper (or perhaps also Hist. Acad. Sci. Berlin...)

I just took my first look at this question and its answers, and noticed a strange effect of the way voting plays out: the listed papers are in nearly reverse chronological order, exactly the opposite of what should be the case. (Make of that what you will; I'm not interested enough personally to endorse a course of action.)

My issue with this question is that I don't have a clear definition of what constitutes a math paper. Didier Piau's answer is interesting but it is far from clear to me that a publication that old constitutes a math paper as the term is used nowadays.

Henry, I dispute the contention that "history of mathematics" is a subfield of mathematics. I did not realise that anyone thought so and so thought that the statement "History of mathematics is off-topic" was unnecessary to be said. Since you ask it, I shall say it:

I think that questions on the history of mathematics are off-topic.

When I referred to the history of mathematics as a sub-field, of course I was being sloppy. In fact, it's a meta-field of mathematics. In any case, the link is close enough for the arXiv:math advisory committee to include History and Overview in their subject classification.

I'm just glad to have prompted Andrew to admit that his objection derives from a deeper objection to the whole topic of history of mathematics. Todd's, Qiaochu's and Daniel's comments also seem to lean in this direction. I hope we won't need to have this discussion every time a history-of-mathematics question is posted.

Henry, your logic is backwards. My objection to this question does not derive from a deeper objection to the whole topic of history of mathematics. Rather, I see that my reason for objecting to this question would apply equally to almost any question on "history of mathematics". My stated reasons for objecting to this question is also my deeper reason for objecting to almost any question of historical nature. Therefore, I feel I can support the statement "Questions about the history of mathematics are off-topic.". As with any sweeping statement, there will be exceptions, but they will be rare. My reasons are:

1. Off topic: the skills needed to answer a history of mathematics question reliably are not the same as that of a mathematical question and cannot be assumed to exist in the majority of the user base.
2. Not a real question: the skills needed to verify an answer to a history of mathematics question are not the same as that of a mathematical question and similarly cannot be assumed to exist in the majority of the user base.

I hope also that we won't need to have this discussion every time a question failing one of these two tests arises. But to claim that I have some irrational, subconscious fear of "history of mathematics" and that my vote-to-close this question is prompted by that is laughable.

History of mathematics is, in my opinion, a good topic in MO, like it is a good topic in ICMs. It is certainly of relevance and importance to a working mathematician. Many mathematicians are interested in historical questions about mathematics. Of course, the majority of our users are not experts on history questions but this is also true for most and maybe all subfield of mathematics. (As a general rule I think we should find ways to give incentives to people to answer in their area of research and expertise.)

I think we have a long tradition of accepting history-of-mathematics questions. Therefore, it will not be right to vote to close a question just because this is the topic.

I did not understand the verification issue at all.

I like this question and think it should stay open.

I'm having trouble imagining a historical question that someone would ask on Math Overflow that a historian would have the special skills needed to answer it, but a mathematician would not. If somebody asked a vague question like "How did the needs of Renaissance commerce lead to the rise of symbolic algebra?" then sure, we would be hard-put to judge the accuracy of the answer. But the historical questions here have always been more specific than that.

Alexander has put into words exactly what I was trying to say. Thank you. I find that argument far more compelling than, to paraphrase, "I find it interesting" or "This authority says that they are related".

I am speaking from ignorance here, but I could certainly imagine that this question might take historical expertise to answer. It could be that the historical issues are clear-cut, but it could also be that they turn out not to be.

The notion of a 'mathematical paper' has changed. It is not entirely clear whether various 16th century works should be considered a 'mathematical paper' or not. For that matter, it's not clear what 'mathematics' is. I find it a reasonably compelling argument that there was no mathematics done basically between the ancient Greeks and the mid 18th century, considering that Newton, Leibniz, and the Bernoullis frequently saw no need to give proofs for their statements.

Furthermore, it is not clear what is meant by 'author'. The modern notion of an author didn't really become solidified until at least the middle of the Reformation, and one could argue until the early 19th century. Certainly people in the 18th century considered it perfectly normal to pay someone some sum of money to become the 'author' of their work (for example, L'Hopital's book on calculus and the Mozart Requiem). (It's interesting that we call it L'Hopital's book but Mozart's requiem!!!)

Possibly these issues don't actually end up mattering for this question, but they might. If they do, it seems to take someone with historical training to address them properly.

EDIT: Having read some of the answers, it is quite clear to me that comments indicate the status of the Bernoulli work, the Euler paper, and the Latin translation of Conics take some historical expertise to understand completely. I take no position in this post on whether such historical expertise is what should be expected of every educated person or instead sufficiently specialized to require special training in history (which many historians of mathematics have).

Ha ha, very clever Gerry.

I agree wholeheartedly with KConrad's comment. (And it is not just the invention of calculus that occurred in the period between the ancient Greeks and the mid 18th century; e.g. one has the solution of the cubic by Cardano, the invention of logarithms by Napier, and the invention of analytic geometry by Descartes and Fermat, just to mention three of the best known developments in post-medieval European mathematics predating the invention of calculus.)

On a related note, along with Gil Kalai and others, I hope that interesting history of mathematics questions will continue to appear on MO.

The question seems ill-posed to me. I think one could make an interesting question about the evolution of collaborative work in mathematics, but this one seems both too broad and too narrow. Too broad in that it just asks for early examples of mathematical collaborations with no restrictions on geography, timespan etc., and too narrow in that it demands that the results must have been published as a paper.

I am not a historian but it seem that the paradigm of new mathematical results necessarily being distributed as papers can't be that old, since there apparently was no need for a periodical publishing mathematical results until 1826. What kind of "papers" were there even before Crelle? There was certainly work submitted for Grand Prizes at the various european academies of science. What about if you just wrote down a new result and posted it in a letter to the other academies, is that a paper? Before the advent of scientific academies it gets even harder. And then there is also the question of authorship mentioned by Alexander Woo.

Probably you could make an interesting question about the history of mathematical papers as the dominating format for publishing, and you could make an interesting question about the history of mathematical collaboration, but I don't think that the two have anything to do with each other.

Dear Dan,

I don't see that it is necessary to take such a literal view of the question. If someone knows of an early case of a mathematical collaboration that resulted in work that was a book or something similar, rather than a literal paper, I'm sure they will post it; indeed, such an example has already been posted (although it turned out to involve collaboration on translation and editing rather than collaboration on new mathematics).

Regards,

Matthew

KConrad: I wouldn't personally say that what Newton and Leibniz did was not mathematics, but I think that is a defensible position. If Newton was asked why he knew calculus was true, he would ultimately give the argument that it correctly predicted the motions of the planets, not the argument that it could be proven (by the standards of his time) from generally accepted axioms. That to my mind makes him a physicist, not a mathematician. (It does not automatically follow that what he did was physics, not mathematics.)

For that matter, one could go to the (admittedly rather ridiculous) extreme that mathematics begins with Bourbaki, and everything before and much of what goes on today is some kind of pre-mathematics.

Looking at another discipline, there was at some point a few analytic philosophers who did claim that philosophy proper began with Frege and Russell. (I heard a story that at one point the Harvard philosophy department blocked off the middle of its hallway with a pile of chairs so that the analytic and continental philosophers would stop fighting each other over the other side's right to be considered philosophy.)

In general: I think we should take a broad view here of what constitutes mathematics, and include history of mathematics as well as mathematical modelling. One of the reasons mathematics as a discipline has done well in the past half century is that we have mostly avoided internal purity wars (i.e. no departments have had to block off their hallways) and been inclusive with respect to neighbouring disciplines even as we have acknowledged the possibility of epistemological differences.

With regards to this question: I think in general we should avoid asking questions which are driven by idle curiosity. The vast majority of questions on MO should meet 'the acknowledgement test'. If you can't imagine reasonable circumstances in which you would acknowledge MO in some paper you write for publication, then you should not ask it here.

@Gil: I don't necessarily regard such concerns as game-breakers either. Nor am I disputing the accuracy of the answers I've seen (leaving aside for the moment the question of what constitutes a 'paper'). But lacking a way of settling on a definitive answer, I still think (repeating myself) that this particular case amounts to a fishing expedition and therefore should be CW. The fact that other history questions were not CW carries no weight for me.

@Alexander: with regard to "With regards to this question: I think in general we should avoid asking questions which are driven by idle curiosity." -- as a general precept, I tend to agree, but probably too high-minded to expect people to go along with very much. If I am honest, I admit to sometimes liking idle-curiosity questions, and sometimes they lead in interesting directions.

Dear Todd, I made no comment regarding the CW issue and I am not sufficiently knowledgable with the CW matter anyway. What you write for why the question should be a CW seems very reasonable to me, but I did not give this matter much thought yet in this or other cases.

Dear Michael,

My understanding is that Euclid contains many original arguments, and (in that respect alone) is more than a textbook. Perhaps a research monograph is a good comparison?

Regards,

Matthew