Most of us are familiar with the Math Subject Classification (MSC), a coded index attempting to classify all mathematical research areas by topic. The MSC, devloped jointly by the Math Reviews and Zentralblatt, is used by most journals and many grant institutions, such as the US National Science Foundation, as a way of grouping mathematical work into topic categories. The MSC codes were recently updated from the year 2000 codes to the current 2010 Mathematics Subject Classification. These codes are organized hierarchically, first dividing into broad research areas, then into sections and finally into more specific research categories.

Question. How well do these codes describe the natural divisions of research in mathematics? Could they be improved in some way? How should they be revised?

Most of us, when submitting a research article for publication, have to decide on the most appropriate codes for that particular work. My own experience is that usually there there is a natural code or two codes that fit very well, which aptly describe the research topic of the article. Sometimes I use two or more codes in a situation where the work doesn't really fit well into either of them alone, so that it isn't really a primary/secondary classification for me, but rather a classification into the union of two categories. Increasingly, however, I find myself stymied by the classification scheme, frustrated in my newest projects that perhaps four or five subcategories are involved, with none of them truly apt, except for the unhelpful "None of the above, but in this section" category. In such cases, I feel that the MSC has failed me.

I recognize that this may simply mean that I sometimes favor offbeat topics, and so perhaps this is my problem rather than the MSC's problem. Or perhaps my problem is that I would like my research to be categorized by the bottom level of the hierarchy, but I should be content just with using the middle level of the hierarchy.

At the same time, I recognize that the mathematical community has a specific interest in encouraging research that crosses the boundaries between established areas, perhaps cross-pollinating or unifying them or at least transferring methods and techniques from one area to another. In time, therefore, we expect subject classification boundaries to migrate or split in various ways. Indeed, perhaps some of the most valuable mathematical work tends to destroy the old classification scheme for precisely this kind of reason. Presumably, this is part of the reason why the MSC is somewhat regularly updated (every ten years I think). So I suspect that there may be many people who share my frustration.

How would you revise the MSC?

Let's have standard community-wiki rules; please provide just one group of changes per post.

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    Perhaps also someone could briefly explain the relationship between the MSC codes and the Math arXiv codes. – Joel David Hamkins Jun 16 '10 at 2:26
  • It's perhaps worth mentioning that some universities use these codes or their own variants for their performamce/research metrics. For instance, I was once asked as a PhD student to choose from a list of (somewhat ill-fitting) codes for data being collected on the university's research strengths. – Q.Q.J. Jun 16 '10 at 13:13
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    @Joel: The arXiv has a brief statement on this at – Terry Tao Jun 16 '10 at 15:53

I would add a top-level subject code roughly corresponding to the arXiv math.QA. Currently my field (quantum groups, knot invariants, TQFT, monoidal categories, etc.) is listed under 16T, 17B, 20G42, 58B32, 81R50, 57R56, 81T, 57M, 56L37, and 18D. But there's no top-level classification that's appropriate, I end up having to describe myself as 17 "Nonassociative rings and algebras" which is terribly misleading. Math.QA is perfect, and the MSC just doesn't have anything like it.

  • Thanks, Noah, this is exactly the kind of answer I am seeking. – Joel David Hamkins Jun 16 '10 at 3:04

I think MSC is a historical anachronism, often useful for bureaucratic purposes, but mathematically indefensible. For one, it is built as a tree with some weak "for xyz see ..." connectors, while a better form would be some kind of poset of subareas. Also, the reason arXiv seems better is because 1) it was invented later and 2) it has only large areas. If arXiv has sub-areas, 15 years later it would be just as bad.

Some years ago I was distressed by how Wikipedia treated the subject as well and completley rewrote/restructured the Combinatorics article, which is still more or less in the way I have made it. Based on that, let me comment only on MSC 05 (Combinatorics). Here is what we have:

05A Enumerative
05B Designs and Configurations
05C Graph Theory
05D Extremal
05E Algebraic

Now, 05A is a fine category as long as you don't try to look at its sub-cats. For example, 05A40 is "Umbral calculus". Quick, show of hands for those who think this sub-area is comparable with 05A05 which is "Permutations, words, matrices". But let's not go there - more trouble is coming up.

05B. This would be a coherent choice if this was "codes, block designs and sphere packing" or something like that. However, by looking at some sub-areas you quickly realize this category has no structure at all. Essentially, anything can be "configuration" in the opinion of the MSC authors. For example, 05B includes "Matroids" (large area) and "Polyominoes" (not an area at all, sort of similar to "unit cubes"), which are and should be viewed as distant parts of Geometric Combinatorics. It also includes "Matrices" (hello?) and "Difference sets" (huh?). The most beautifully titled category is 05B99 "None of the above, but in this section", and given that 05B has no common theme, anything goes, I guess...

05C. In contrast with 05B this is a very clear and coherent category. It is also very popular and on a permanent quest for independence (the suggestion being that it becomes 07 which for whatever historical reason is missing in MSC).

05D. Shouldn't this be "Extremal and Probabilistic Combinatorics"? So if a paper proves a new result in Ramsey theory using probabilistic method, is it 05D40 (Probabilistic methods) or 05D10 (Ramsey theory)? Anyway, this is a clear category which should be linked to 60C (Combinatorial probability), which itself needs to be renamed "Discrete Probability".

05E. This would a clear category if the areas were more connected. As it stands, 05E30 (Association schemes, strongly regular graphs) should really go into 05B. Similarly, 05E40 is an important sub-area and should really go into MSC 13, or perhaps replaced by a better named sub-cat (every part of Algebra now has "combinatorial aspects" - should we list them all in 05E?), or even become a separate part of 05 (or of both 05 and 13, if MSC becomes a poset). Most strikingly, another important area 05E45 (Combinatorial aspects of simplicial complexes) should be taken outside of 05E (what exactly is algebraic about it?) and made into a separate part of 05 titled "Topological Combinatorics".

Ugh... In summary, this mess has to be largely redone. New parts of 05 need to be created (we have 21 letters F..Z remaining for "Geometric", "Topological", "Analytic", "Arithmetic", etc. - see Wikipedia page). What is left in each category would then be more coherent and searching for such cats in MathSciNet would actually make sense.

  • Thanks for this thoughtful answer, Igor. I agree with your general assessment, and similar issues arise in section 03. – Joel David Hamkins Jun 17 '10 at 0:29
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    Speaking of Wikipedia, its data structure of "categories" (still hierarchical, but no longer a tree, and even with an occasional directed closed loop) is superior for organizing material if the ease of searching is the goal. – Victor Protsak Jun 17 '10 at 1:09
  • @Victor: that's right. Another important WP feature is that it allows a quick and easy update when things change. Unfortunately, if you make changes to MSC too often that also diminishes the value of the classification system. Some kind of middle ground works best in this case. – Igor Pak Jun 17 '10 at 3:14

To be honest, I have never in my life paid any attention to this stuff (except for its sieving application when reading postdoc job applications). I read papers because of the title/abstract, or knowing what the content is about. I pick a couple when forced to by journals, and didn't think anyone actually cares. It's like the "Keyword" stuff I am forced to create (which again I never look at for papers I read). Are those actually used for something?

I agree that for NSF logistics and job applications it is useful. But the question is focused on its role in research papers, and that is where I don't get it, and is what my puzzlement is about. I never look for papers in my areas of interest by searching for MSC numbers, and I never look for papers by searching for a "Keyword". I am really baffled by the tradition of putting these things in research papers.

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    MSC was created in order to categorize or sort the papers being reviewed. As with many other things, it acquired life of its own... – Victor Protsak Jun 16 '10 at 4:32
  • @Victor: Oh, so its purpose is to make life easier for the folks at Math Reviews (e.g., to funnel the paper to the correct editor there)? And the Keywords help them to more easily pass the paper on to suitable reviewers? That makes a lot of sense. So it is like the story of US Social Security numbers. – Boyarsky Jun 16 '10 at 5:06
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    There have been a small number of times when I've found keywords and the MSC useful. About 7 years ago, I decided I wanted to know about everything that was known about Witt vectors. It was pretty easy to find lots of new papers by doing MSC and keyword searches, and I found certain veins of knowledge, probably much earlier than I would have otherwise. More generally, the MSC and keywords are probably useful whenever you want to know what people know about a very specific topic that cuts across many different fields and which has an accepted name. – JBorger Jun 16 '10 at 6:17
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    @Boyarsky: It's unfair to the Math Reviews staff to characterize the classification that way. They do need a fairly stable and hierarchical classification scheme in order to assign papers to editors and then to contact appropriate reviewers. But the scheme is also needed by users to retrieve information over many decades. It's easy to pass suggestions along to MathSciNet for changes, but these occur slowly in collaboration with Zentralblatt and have to preserve the archival search framework. I've persuaded them over the years to refine the classification in areas of interest to me. – Jim Humphreys Jun 16 '10 at 12:01
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    From my point of view, the reviews at Mathscinet are of limited value, too, so I am not at all convinced by the argument "MSC codes are useful because reviews". – Federico Poloni Jul 25 '16 at 17:59

Roughly the same subject came up some time ago at the Secret Blogging Seminar (search for MSC).

One missing item is optimal transportation, that should probably go inside functional analysis. With all the recent development of this topic, I find very surprising that it was not added in 2010.

I suspect that any attempt to produce a hierarchal top-down style tree of mathematical subjects is bound to be problematic. You might make the majority of the people happy at the time you create the tree (like the arXiv now) but long-term it's likely to have the same problems that people see with the MSC classification.

The history of mathematics is that deep connections are found between fields that are at earlier times perceived as distant. So fields glom together. Similarly, fields drift apart (like math.GT and math.AT now, in the 50's there was just topology) for various reasons, evolution of techniques being one of them.

My initial guess would be that the best long-term solution would be to have an MSC classification that is as "flat" as possible. So that when you pick your MSC classification, it's like choosing flavours at an ice-cream shop.

My experience has been that the editors of Math Reviews pay attention to suggestions about revisions of the classification system. I'm not saying that they implement all the suggestions (especially because the suggestions sometimes contradict each other), but they do listen and, a while before the once-a-decade revision of the system, they ask some people (including me on some occasions) about both the proposed changes and any other ideas we might want to contribute. The revision process is non-trivial, partly because of the need (or at least desire) to coordinate with Zentralblatt, and partly because a major revision of any section has a big downside (the loss of backward compatibility) and must therefore have a big upside to make it worthwhile. But big revisions have happened, and they may well happen again. The editors also watch for classification areas that have gotten either unpleasantly small (so they might be merged into other areas) or unpleasantly large (so they might be split, if natural dividing lines can be found).

  • Based on my own experience for MSC2010 I can confirm that editors do listen to suggestion, even from people they did not ask. There was a public call for suggestions, and the process of revision (at least in part) was public as well. The webpage for this (I think so far it did not get mentioned) is still visible ; it was advertised when the procsess was current on MathSciNet and/or Zentralblatt MATH. My experience contributing to an (unsolicited) suggestion was very positive and in the end it was (in part) implemented. – user9072 Aug 30 '12 at 18:06

The top-level categories in the IMU list include one for Lie theory.

First, we should ask what purpose MSC type codes serve. After all modern papers can be indexed in full text and, even if paywalls make full text search impractical one can do a full text search on abstracts at mathscinet. It seems to me they offer two important features.

  1. MSC codes let one track all papers in a given are even when no particular search terms would be sufficiently specific without leaving anything out. This allows paper archives to present their contents in a hierarchical manner or searches to be restricted to a particular subject.

  2. MSC codes (at least in theory, I've never used them for this purpose) distinguish papers working with particular mathematical objects/approaches that can't easily be identified using keyword searches. For instance even if I restrict my attention to papers in computability a search for "admissible set" is likely to turn up to much (admissible, like good, is overused) and miss some instances that use other terminology.

The first usage calls for a basic hierarchical description of major areas of mathematical research. Each area should be well populated and each paper should fall into only one or at most a handful of areas. These areas should be developed enough that they won't disappear with changes in research focus.

The second usage calls for a plentiful list of canonical tags for particular objects, approaches or questions. Each paper might fall under arbitrarily many such tags, the more the better. As far as this use is concerned there is no real harm if research directions change and papers falling under some tag stop being published. Codes should be generously added for every conceivable object/approach/question constrained only by the requirement that every such concept have a canonical code (no codes that are synonyms). Search engines could then maintain a list of terms associated with each such code so one could do a search in computability theory for papers that mention both "model" (in the model theory sense) and admissible ordinal.

I submit that this naturally suggests two systems of codes. One hierarchical that roughly corresponds to the first two digits + letter, e.g., 03A, of MSC2010 and the other a much more numerous database of tags and their synonyms.

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