# Is discrete mathematics mainstream?

Recently, the Department of Mathematics at our University issued a recommendation encouraging its members to publish their research in non-specialized, mainstream mathematical journals. For numerical analysts this will make an additional obstacle for their promotions. But even for discrete mathematicians this recommendation is causing concerns.

For several top mainstream journals I checked with tools offered by MathSciNet what percentage of discrete mathematical papers they published in recent years. Some statistics indicate that in some journals the number of papers with primary MSC classification, say 05 or 06 decreased significantly in the past 30 years. There are several possible explanations to this fact.

• The quality of research in DM is dropping.
• The majority of research in discrete mathematics is so specialized that it is of no interest for the rest of mathematics
• Some discrete mathematics journals attract even the best work of discrete mathematicians.
• Some top journals may be biased against discrete math.
• Maybe discrete math is no longer part of mainstream mathematics and will, like theoretical computer science, eventually develop into an independent body of research.

But the key issue is whether discrete math is nowadays perceived as mainstream mathematics.

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This is going to become discussion-y quite fast! :P In any case, your Department issued such a recommendation with what purpose? –  Mariano Suárez-Alvarez Mar 5 '10 at 21:52
This looks like an opinion thinly veiled as a question. The question should be community wiki, if open at all, in my opinion. –  Boris Bukh Mar 5 '10 at 21:59
So far this is only a recommendation for younger colleagues. However, the idea is to require for a promotion a certain number of papers published in high-quality mainstream journals. –  Tomaž Pisanski Mar 5 '10 at 22:20
Number theory and combinatorics are included in discrete mathematics. Some significant problems from number theory are always perceived as mainstream mathematics. –  Sunni Mar 6 '10 at 0:55
I think that the following paper by J. F. Grcar in Notices raises even broader concern: ams.org/notices/201011/rtx101101421p.pdf –  Tomaž Pisanski Nov 30 '10 at 16:30

There are two different questions here, one objective and one subjective. I will try to give my view, for what it's worth. Bear with me.

First, you are asking what is the publication history of discrete mathematics? (Even if I suspect you know this much better than I do). Well, originally there was no such thing as DM. If I understand the history correctly, classical papers like this one by Hassler Whitney (on coefficients of chromatic polynomials) were viewed as contributions to "mainstream mathematics". What happened is that starting maybe late 60s there was a rapid growth in the number of papers in mathematics in general, with an even greater growth in discrete mathematics. While the overall growth is relatively easy to explain as a consequence of expansion of graduate programs, the latter is more complicated. Some would argue that CS and other applications spurned the growth, while others would argue that this area was neglected for generations and had many easy pickings, inherent in the nature of the field. Yet others would argue that the growth is a consequence of pioneer works by the "founding fathers", such as Paul Erdős, Don Knuth, G.-C. Rota, M.-P. Schützenberger, and W.T. Tutte, which transformed the field. Whatever the reason, the "mainstream mathematics" felt a bit under siege by numerous new papers, and quickly closed ranks. The result was a dozen new leading journals covering various subfields of combinatorics, graph theory, etc., and few dozen minor ones. Compare this with the number of journals dedicated solely to algebraic geometry to see the difference. Thus, psychologically, it is very easy to explain why journals like Inventiones even now have relatively few DM papers - if the DM papers move in, the "mainstream papers" often have nowhere else to go. Personally, I think this is all for the best, and totally fair.

Now, your second question is whether DM is a "mainstream mathematics", or what is it? This is much more difficult to answer since just about everyone has their own take. E.g. miwalin suggests above that number theory is a part of DM, a once prevalent view, but which is probably contrary to the modern consensus in the field. Still, with the growth of "arithmetic combinatorics", part of number theory is definitely a part of DM. While most people would posit that DM is "combinatorics, graph theory + CS and other applications", what exactly are these is more difficult to decide. The split of Journal of Combinatorial Theory into Series A and B happened over this kind of disagreement between Rota and Tutte (still legendary). I suggest combinatorics wikipedia page for a first approximation of the modern consensus, but when it comes to more concrete questions this becomes a contentious issue sometimes of "practical importance". As an editor of Discrete Mathematics, I am routinely forced to decide whether submissions are in scope or not. For example if someone submits a generalization of R-R identities - is that a DM or not? (if you think it is, are you sure you can say what exactly is "discrete" about them?) Or, e.g. is Cauchy theorem a part of DM, or metric geometry, or both? (or neither?) How about "IP = PSPACE" theorem? Is that DM, or logic, or perhaps lies completely outside of mathematics? Anyway, my (obvious) point is that there is no real boundary between the fields. There is a large spectrum of papers in DM which fall somewhere in between "mainstream mathematics" and applications. And that's another reason to have separate "specialized" journals to accommodate these papers, rather than encroach onto journals pre-existing these new subfields. Your department's "encouragement" to use only the "mainstream mathematical journals" for promotion purposes is narrow minded and very unfortunate.

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Integer partitions are a good example of a subject which seems borderline between discrete and non-discrete. There are many papers that prove partition identities bijectively, which is a very discrete thing to do, I think; on the other hand some people attack partition problems via modular forms, which doesn't feel discrete at all. (Of course you know this, Igor; this is a comment for people who might not recognize it.) –  Michael Lugo Mar 6 '10 at 17:44
The way I see DM can be explained in two ways: There are two main paradigms in science: discrete and continuous. Both were known already to Greek thinkers. In my view DM is mathematics seen though discrete paradigm. If you do not know whether some math concept should be classified as discrete or continuous, think of its source. If the motivation comes from classical physics it is very likely to be continuous, if it comes from CS it is probably discrete. –  Tomaž Pisanski Mar 6 '10 at 22:50
Well, finite fields are pretty discrete; yet the corresponding part of algebraic geometry seems to be rather far from discrete math. –  Mikhail Bondarko May 2 '11 at 20:24
On a pop culture response to your comment that 'most people would posit that DM is "combinatorics, graph theory + CS and other applications"', there's a Numb3rs episode where a character, when asked what combinatorics is, responded with "It's a branch of computer science." ::cringe:: –  Gwyn Whieldon May 14 '11 at 20:46
I somewhat revised my views on the issue: wp.me/p211iQ-4O –  Igor Pak Aug 20 '12 at 8:16

Here is a slightly frivolous answer, but only slightly. Suppose we were to divide mathematicians into two classes: those who could in principle happily pursue their research without knowing what cohomology is, and those for whom that would be completely unthinkable. (Of course, there is a spectrum in between, but let's not worry about that.) Now people in the latter class can be found in many many areas, from topology and geometry to algebra and number theory. There is a certain sense in which mathematicians of that kind have something very important in common, despite their differences, and form a mainstream from which discrete mathematics is mostly excluded.

However, it is also true that nowadays discrete mathematics is much more accepted by members of that mainstream as being an important subject. One sign of that is that there are papers appearing in Annals that almost certainly would not have been accepted twenty-five years ago. Another is that top universities tend to want at least some discrete mathematicians in a way that they used not to. I don't know whether we have got to the point where one could speak of an alternative mathematical mainstream, but I think that discrete mathematics is a well-established area that is broadly respected by more traditional mainstream mathematicians. (Perhaps another sign of that is that Lovasz is president of the IMU.)

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Your answer may be slightly frivolous, but it's memorable, because I think it's easy for most mathematicians to decide which side of your cohomology gap they're on. –  Michael Lugo Mar 6 '10 at 12:50
If I ask "What kind of cohomology?" am I giving away which side of the gap I stand on? –  Pete L. Clark Mar 6 '10 at 13:56
Googling for "combinatorics and cohomology" gives 217,000 hits, which confirms my personal experiences that many discrete mathematicians know and love (co)homology. I bet discrete mathematicians use more (co)homology than say analysts. –  Igor Belegradek Mar 6 '10 at 15:40
Igor, Googling "combinatorics and cohomology" gave me 60,030 hits and Googling "combinatorics or cohomology" gave me also 60,030 hits. Possibly Google's AND is the same as Google's OR. –  Gil Kalai Mar 6 '10 at 15:40
Funny, since hardcore Analysts (the quintessentially non-discrete people) don't use cohomology either. –  Felipe Voloch Mar 6 '10 at 20:47

There's a curious sense in which almost no one really feels comfortably mainstream, regardless of how they stand with respect to the cohomological divide, or even of their community status. The Grothendieck phenomenon is rather an obvious example, but there are many others. If we venture outside of mathematics proper, Noam Chomsky, often referred to as the most cited intellectual alive, frequently speaks of himself as an outsider. (Specifically in relation to his linguistics, not his politics.)

Of course, it’s tempting to speculate about the honesty of such self-perception, but I tend to think of it as largely reflective of the human condition. It may also be that this kind of view goes well with a sort of rebellious energy conducive to creative intensity. For people who like literature, the sensibility is wonderfully captured in the novella `Tonio Kroeger’ by Thomas Mann. The irony is that almost anyone who reads the story is able to relate to the loner, as is also the case with the typical rebel in simpler dramas.

Why go far? Here we have Tim Gowers, an enormously respected mathematician by any standard, apparently presenting himself as a spokesman for the tributaries. In his case, I take it as the prototypical gentlemanly self-effacement one finds often in Britain.

At the very least, the whole picture is complicated.

The point is it’s probably not worth spending too much energy on this question. Administrative constraints, classifications, and selections are a real enough part of life within which we have to find some equilibrium, but serious mathematics has too much unity to be divided by the watery metaphor.

David Corfield once (good-humouredly) misquoted me with regard to the perceived distinction:

'Which do you like better, the theorem on primes in arithmetic progressions or the one on arithmetic progressions in primes?’

The original context of that dichotomy, however, was a far-fetched suggestion that there should be a common framework for the two theorems.

Added: The more I think of it, the more it seems that the original thrust of cohomology was very combinatorial, as might be seen in old textbooks like Seifert and Threlfall. The way I teach it to undergraduates is along the lines of:

space $X$ --> triangulation $T$ --> Euler characteristic $\chi_T(X)=V_T+F_T-E_T$ --> $T$-independence of $\chi(X)$ --> dependence of $V_T$ etc. on $T$ --> 'refined incarnation' of $V_T, E_T, F_T$ as $h_0$, $h_1$, and $h_2$, which are independent of $T$--> refined $h_i$ as $H_i$.

The emphasis throughout is on capturing the combinatorial essence of the space.

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Thanks for your time. I am sure the "mainstremness" is a multi-dimensional question and your answer covers many of its dimensions. I agree with your ideas. However, I think that mainstream in discrete maths is not necessarily the same as the mainstream in "continuous" maths. –  Tomaž Pisanski Mar 8 '10 at 20:15

In your list of reasons why the percent of discrete math papers in these journals is decreasing, you ignore the possibility that the percentage of mathematicians doing DM is also decreasing.

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I must admit that this idea did not cross my mind since in Slovenia the DM group is very strong and numerous. –  Tomaž Pisanski Mar 7 '10 at 0:12

If a "mainstream" area of mathematics is an area that any good research mathematics department would be embarrassed not to include among the specialties of their faculty, then discrete mathematics is definitely not mainstream.

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It's not clear to me that any area of mathematics (depending on how broad an "area" may be) is mainstream by this definition. –  Mark Meckes Mar 6 '10 at 18:29
@David: let's check this claim. Without getting into particular universities, I took the definition to be "a tenured faculty at the mathematics department" and used my own sense of what is DM. I quickly checked from memory the top 40 departments on this list: math.scu.edu/~eschaefe/grad.html I found that only 11 departments do not have a single person whose expertise includes DM, with more than half of these having excellent DM people in applied math, CS or engineering departments. So I think there is both half-full and half-empty way to look at this. –  Igor Pak Mar 6 '10 at 21:32

You mention numerical analysis explicitly. From my point of view, numerical analysis has a completely different set of top-flight generalist journals than traditional mathematics. People here care to publish in SIAM Review, in Inverse Problems, in the other slightly less generalist SIAM journals (just to list a few).

If I had written an excellent paper in my field, it wouldn't even cross my mind to submit it to Acta or Inventiones; I'd find it no more appropriate than, say, Nature or The new England J of Medicine. Surely it would strike me as peculiar if someone asked me to publish in those.

Actually I think many people I meet in conferences don't even know which are the best pure-math journals. Just for fun, I tried looking on MathSciNet for papers by past or present big names in NA published in top-flight pure math journals; it took me a few minutes to find even one of them.

Does this make us a completely different discipline that lives under the same roof (and competes for the same funding and positions) as pure mathematicians? Maybe, I don't know. It's a matter of definitions.

OK, now that I've written this, I'd better go and check scicomp.stackexchange.com --- there is not enough applied maths here. :)

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IMO this reccomendation in your department makes a lot of sense.

If you have a group in your department where they only publish papers in a specialized journal, it makes it that much more difficult to assess the quality of their research. The good thing about having a central core of competitive journals (like the Annals, Inventiones, Advances, JAMS, etc) is that you have a vetting place for ideas across fields.

To put it to extremes, if there's a group in your department that publishes in only one journal, and all the people that publish in that journal only publish in that journal, this leads to suspicion that the journal publishes pretty much anything (at least for that group of people).

Trying to publish in general journals, that's a first step towards getting some percentage of papers in that field into broader circulation. This is what we need if we want to be a community and not simply warring factions.

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@Ryan: I agree with you that publishing in a variety of high-quality journals should be recommended. However, I think that even broad and deep research that may span branches such as, say, combinatorics, geometry, logic and computer science will be much harder to publish in top mainstream maths journals than deep but narrow research in, say, analysis. –  Tomaž Pisanski Mar 7 '10 at 21:19

I often think of "the set of mainstream topics" as approximated by the large (strong) component in the directed graph of citations. In other words, mathematical work is mainstream if and only if most of the mathematical world is reading/using/citing such work.

Sorry to use a discrete structure in my definition... maybe this makes it biased. :)

Anyway, by my above pseudo-definition, is discrete math mainstream? I admit that most of it is not, but obviously the best of it fits very nicely into the rest of mathematics.

Here is how I see it.

Level 0: The worst discrete math uses no post-17th century mathematical ideas. Sorry, but this is the truth, sad as it may be.

Level 1: Discrete math at this level uses somewhat deeper mathematics, but still seldom gives back to the rest of mathematics. By the nature of the subject, this is the best many of us can hope for. I'll include myself here.

Level 2: The most mainstream discrete math directly connects with questions other mathematicians care about.

Sadly, even my "level 1" discrete mathematician may not, in fact, qualify as a mathematician. Here's an analogy: If you run to catch a bus, does that make you a runner?

Having trashed my own subject, let me right the wrong (in a sense) by saying that I find discrete math truly beautiful, and I am hooked for life. Who cares whether it is mainstream, and for that matter who cares about journals? What's better: crappy work in a "mainstream" journal, or amazing work in a specialist journal? The OP should put this question to his upper admin. Personally, all I care about is the beauty and utility of the work.

(I am a major Gowers blog fan, by the way, but I really don't like the cohomology dichotomy. There is no chance this can be the right litmus test. I'm lacking enough reputation to down-vote it!)

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This map of journals in all fields of science may interest you: vosviewer.com/maps If I understand correctly, it's visualized in a way the more citations between given two journals have, the closer they're on the map. I don't know how much citation directions are considered though. In any case, "prestigious mainstream pure math journals" form a clique-ish circle on the edge of the map. This might give a rough, pseudo-objective measure of how far discrete math is from the mainstream math circle in a way you described (except maybe less use of the "use" vs. "give back" information). –  Yuichiro Fujiwara Jan 9 at 11:04
I love this map. Thanks for sharing it! –  Peter Dukes Jan 9 at 19:55