Why do pure mathematicians tend to start publishing good papers a bit late?

Why is it so common for pure mathematicians to start publishing "good" quality paper a bit late? Maybe towards the end of PhD or start of Post-doc. Whereas in some fields people start publishing research papers as early as in their masters. I know there are some undergrad research programs, for e.g.- REU, but only in some cases do they get published; but not in some good quality journals. Most of the times they maybe just arxived or in some cases not even that. I know some people, from different fields, who had 2 or 3 publications in their masters itself, while some of them are in their 2nd/3rd year of PhD and already have around 10 publications. Maybe such a thing is also possible in mathematics, but there must be some good reason why mathematicians or graduate student, along with their supervisor, agree to publish good results in good journals, a bit late.

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A big issue with mathematics is that it is a very old field. For most courses we teach/take in undergraduate, the theory was developped in the 1700s or 1800s, and in the last 2-300 years these subjects have been studied soo much that there is not too much left for a student to do. For most mathematicians, to do reearch you need first to familiarize yourself with the newest theory, which is done in graduate school either via classes or most of the time via reading (note here that many if not most undergraduate students do not have the background needed to do this).

Some other factors not yet mentioned (which depend on country/school): in math masters work is often focused on learning rather than research, math grad students often have to spend a lot of time teaching, and the publication process is slower in math than many fields.

@Mehta Not having a thesis advisor for several years in grad school seems fairly exclusive to the US. In the UK, for example, you have a supervisor from the very start.

3 Answers 3

One fundamental distinction between mathematics and more-obviously experimental scientific and engineering disciplines is that, in math, there is no analogue of "reporting on several months of experiments". There's no analogue of "keeping experiments running, and whatever turns out, is a paper".

That is, while there is in fact a large experimental aspect to mathematics, it is not the same sense of "experiment" as in other sciences, and the conventions are such that this does not generate papers.

Yes, the ambient pressures do push PhD students in math to try to arrange at least one or two publications prior to PhD completion. Not entirely a bad thing, but definitely not a convivial atmosphere.

Edit: and it may be worthwhile to observe that most of prior mathematics does not become obsolete or wrong due to new discoveries, unlike sometimes happens, and is always possible, apparently, in more experimental sciences/etc. Thus, mathematicians cannot too much disregard prior work. of which there is a great deal. Many excellent (and not-so-excellent) results known prior to 1900 are rediscovered on a regular basis, and do indeed show evidence of insight!, but are not "publishable".

This "problem" is the reason I tell myself, and my research students, to not even think in terms of "verifiable novelty", because it is just a mess, for mathematics. Better to follow a good, natural line of inquiry, and leave the appraisal of "novelty" till later. (It is admittedly hard to ignore academic-administrative pressures. and, yes, this is corrupting academic mathematics, among other disciplines. )

@FreePawn, I would not claim to really know the current conventions in CS, but/and I suspect that there is a large range, from the theoretical end to the engineering end.

@FreePawn Yes, and even on the "theoretical end" it's probably easier to still come across a publishable idea than in pure math (albeit far from trivial). Benefits of a younger field.

There's big variation within math, in particular, combinatorics (one of the most CS-like fields in math) is reasonably similar to a lot of computer science in terms of people publishing shorter papers more often and often doing so earlier on in graduate school.

@freepawn, my company has a program where CS students can work half with us, and study the other half of their time. There definitely are unis (at least in Germany) where at least some part of their contents are current (e.g. modern topics like cloud security, CI/CD, blockchain etc.), and if the students are lucky, their profs are interested in driving technology forward. In this case, all possibilities are open I guess - while most progress is probably directly driven by companies (and open-source), there are plenty topics of theoretical nature which those will probably not tackle much.

@FreePawn CS has both a practical ("applied") research aspect and a theoretical research aspect. The practical/applied side has new areas opening up all the time and has publishable experimental aspects (and is funded/employed by industry). The theoretical side is more like theoretical math, but since it is a few hundred years newer, it's not as hard to find the frontiers.

A key difference between pure math and many other areas is that there are typically only 2, maybe 3 authors. The papers you cite with Masters students are often in larger collaborations wherein the Masters students do some of the menial tasks: data collection, data analysis, but not necessarily theory development that require more years of training. In pure math, these sorts of papers are quite rare: You have to understand the theory if you're writing with just one or two co-authors, because otherwise there is nothing for you to contribute. As a consequence, there is just no opportunity for students in pure math to be part of authors earlier in their career.

I've seen some stark examples of this. I went to a talk by an undergraduate student who was included as one of many authors on a experimental physics paper and she was unable to answer basic questions about the experiment or seemed to have no knowledge of any of the workings of the experiment on which the article was based.

@Tom yep, typically with experimental physics collaborations it's simply "if you're in the collaboration you're on the author list". I'm an author on a paper about a laser system that I haven't even looked at the optics table setup of (if I can avoid being in a room with an extremely powerful pulsed Ti:sapph I will), because I worked on the other end of the experiment. Not ideal but it avoids a lot of arguments and potential discord within the group

@Tom: Same with some of the stuff I did in what might be called computational biology. I could have done my end without knowing anything about the biology (though I learned a good bit just out of curiousity), but the work couldn't have been done without someone with my computer expertise.

Also I had a friend who was one of five authors on a journal article. When I asked how much of a contribution they had made or how much of their MSc thesis had been used, they said they didn't know as they hadn't read the article! I would at least want to have read the actual article all the way through if I am classed as one of the authors.

Folks (@Tom @llama), it's not necessary to disparage other disciplines here. Good science is done outside mathematics as well, they just have different strategy of who is included in the author list.

The other answers do a great job explaining why math is quite different from experimental sciences. But I don't think this is the full answer, as computer science, statistics, and theoretical physics are "mathematical sciences" and it's less clear why they should be any different than math. In fields where there's substantial overlap (e.g. there are people working on tensor categories in math and in condensed matter physics) I think there's also different publishing conventions in terms of what level of originality is expected of publications. For example, working out an example that any expert could do and writing it up is much more acceptable in physics than it is in math. There's advantages and disadvantages to both approaches (in particular, math ends up with far too much "folklore" that is hard to learn without social connections to experts), and I think a lot of this difference is cultural. For example, the culture in math departments is that the main goal of a PhD is to have a substantial result, while in CS the expectation is that you will have multiple less substantial papers. There's similar cultural differences within mathematics, e.g. combinatorics is more amenable to short papers and homotopy theory or Langlands number theory are more amenable to people who publish more rarely in larger papers. I think these cultural differences (like many inter-departmental cultural differences) are going away as there is increased pressure for mathematicians to publish more and publish earlier even if that means not doing as substantial work.

Finally, I should point out that most of the humanities is even more extreme than math in terms of focusing on a large substantial work, and the expectation is that your PhD should be a publishable book which you will still be editing several years after graduation. Again, I'm not saying this means math is "better" than other mathematical sciences, or that history is "better" than math, there's advantages and disadvantages to both approaches.

@FreePawn, unsurprisingly, quantity is easier to determine (by administrators) than quality. Even for hiring committees, the number of entries is vivid. while the quality of each is most likely completely obscure except to experts.

@FreePawn, in my observation, yes, when the decision-making committee for awards has many people from high-publication-rate fields, the math people have a hard time competing on paper-count basis. Not that there's a conscious bias, but just that it's somehow inconceivable (to some people) that grad students wouldn't publish 3-4 papers each year, etc.

Another related convention which keeps down math publication numbers is the convention that advisors not typically coauthor material from their students thesis (even if the advisor’s contribution is substantial enough to merit coauthorship). I don’t think that convention is sustainable.

CS and physics apply maths, they don't have to advance maths, they use them as a tool. You don't have to understand everything, you have to know your tools. Many physicists don't even understand the proofs or the limits of their tools, but get great results anyway. I even know applied math works, that are more programming and developing a clever numerical scheme then real maths work in the sense of finding a theory and proving it.

@FreePawn Not quite. The point of an REU is to do research, but the point of your first couple years of grad school, in the US at least, is to learn more mathematics.