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I was always wondering why is the impact factor of mathematical journals "often" lower than the impact factor of journals in other disciplines? I come up with some guesses that I would like to know your opinion about them.

  1. Mathematics papers are so hard to read and follow (or so specialized) that only a few people can read and apply their results.

  2. There are relatively a large number of subfields and branches of mathematics and only a few people are working in each branch.

  3. The number of journals and papers in mathematics (divided by the number of active mathematicians) is relatively higher than other disciplines.

  4. A mixture of these.

Please, share your insights about this issue.

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    Why isn't the answer obviously "because mathematics papers are cited less often (and less quickly)"? – JeffE Jan 15 '13 at 0:58
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    @JeffE: That just reduces the question to "why are mathematics papers cited less often and less quickly?" It's clear that they are, and the question seems worthwile (if only to reassure mathematicians that the answer isn't necessarily "because they are less interesting or important"). – Anonymous Mathematician Jan 15 '13 at 1:20
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    Mathematics papers are cited less often because the mathematicians are citing less often – user2173836 Jul 3 '17 at 12:48
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    Because citations to maths papers can start and continue to accumulate decades after the paper is published, when the real importance of the work is apparent, rather than in the weeks after it appears and others publish papers jumping on the bandwagon, and then basically stop after a few years. – David Roberts Feb 4 '19 at 0:33
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There are a lot of different factors, and I know of no reliable way to determine which is the best explanation. For example, one theory is that mathematics simply has less impact (in the non-technical sense) than most scientific fields. I don't believe this, but it's hard to give a principled refutation. The whole subject strikes me as a little silly, with lots of opinions and numbers with no clear meaning.

One important factor is clearly that mathematicians write fewer, often longer papers than most scientists. Another is the two-year cycle mentioned by walkmanyi (citations after two years do not count for impact factor, which is incompatible with both the time lag in mathematics publication and the time required to carry out research in mathematics in the first place).

Another factor is the size of the field. The highest impact factors should occur in an enormous field with some incredibly important research and also a ton of less important papers that cite the great ones. Mathematics is just not that large a field (compared with biology or medicine, certainly), and it furthermore fragments into a lot of subfields it's difficult to move between. When someone makes an amazing discovery in algebraic geometry, you aren't going to get a flood of mathematicians from other areas rushing in to take the next steps, because algebraic geometry requires a lot of background. I don't think that's a bad thing for mathematics as a whole (the things the would-be algebraic geometers are doing instead are probably as valuable as following the latest trends would be), but it cuts down on the opportunities for amassing citations quickly.

Ultimately, I doubt there's any conclusive or satisfying way to determine how much of a role each of these factors plays.

For some published commentary on impact factors in mathematics, see Nefarious Numbers by Arnold and Fowler and Impact Factor and How it Relates to Quality of Journals by Milman. The first paper focuses on the flaws of impact factors and their abuse/manipulation, while the second explains how impact factor calculations relate to mathematical publication practices (and some of the incentives for journal editors). Neither directly answers the question here, but they both shed some light on it.

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    Size of the field: In my original answer I wanted to highlight this, but then decided not to as I cannot argue as IF is a ratio cited vs. published so it shouldn't be influenced by the size of the field. But if you are looking for arguments in this direction, there's quite some support in bibliometric maps of science. See e.g., here which is taken from here. There are more maps like that. Clearly, mathematics is a relatively small field in comparison to other scientific disciplines. – walkmanyi Jan 15 '13 at 9:16
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    In fact, it seems to me that size of the field should be related to the variability of the number of citations, rather to the the average number of citation by paper (see my answer). However, it makes me wonder whether the feeling that math journals have lesser IF than, say biology journals could be untrue, mostly based on the comparison of the highest IF journals in each field. This would make the size of the field a better explanation for that perception, in my opinion. – Benoît Kloeckner Apr 24 '15 at 7:15
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I must admit that I tend to disagree with the previous answers: while the description of the specifics of the mathematical community are accurate, I do not see why this should affect the impact factor (except for the time needed before an article is cited, whose influence is clear). In particular, size of the field does not have any impact per itself on the average impact factor of papers. In fact, papers are less cited mainly because papers cite less.

Let me back my point, first assuming we are looking in a field that is closed (only cites itself and is only cited by itself) and stationary (no evolution of the number of papers published or the average number of references per article). Consider the publication graph of a given year : it is a bipartite graph, whose vertices are papers published year 0 (first partition) and years -1 and -2 (second partition) and whose edges are citations from the first ones to the second ones. Then the (article) average of impact factors AIF in this domain is the ratio

AIF = (#citations from year 0 papers to year -1 and -2 ones)/(#year -1 and -2 papers)

which is equal to (#edges)/(#papers published in two years), since the field is assumed to be stationary. This is also twice the average number of references to the two preceding years that a paper in the fields has.

So the article average impact factor of a closed and stationary field is solely governed by the references habits in the field. In particular, this is not affected by the overall size of the field (e.g. math as opposed to biology).

Given the distribution of references, an expending field will tend to have bigger impact factor, as will a field that is often cited by other ones. I do not feel that speed of expansion is an important factor for math compared to other fields, but fondamental mathematics are probably seldom cited outside itself. This has little impact if one consider maths against the rest of the world, though, since math papers seldomly cited outside the field too.

Another factor can be the distribution of papers among journals: for example, if a field has only two journals, one very large and one very small that only gets the very top articles, then the (unweighted) journal average IF will be extremely high. I doubt this explain much of the difference between math and the other fields, since mathematics have a strong hierarchy of journals.

So, what we really have to explain is why math papers cite less papers in the two-years range than papers in (most) other fields. This will explain why they are less cited.

Then the answer seems quite clear: maths papers are often long to read, and take time to be digested. The core of a biology paper is usually easy to understand and such papers are more easy to cite. There is also a small subfield effect: mathematician can work on problems that involve few previous papers. This is different from the size of fields, because it is more about the degree of specialization.

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    Hmm, interesting. I agree that math papers cite less within the two-year window for field-specific reasons, but I don't see any evidence they cite less otherwise. I guess what this suggests is that the distribution of citations per page in mathematics must simply be more uniform if it is less highly peaked? I agree that the average number of citations per paper can't scale with the size of the field, but I don't think that's what anyone observes. Instead, it's the number of citations per good paper, and that probably does scale. – Anonymous Mathematician Jan 15 '13 at 13:42
  • (I.e., I doubt the number of breakthroughs scales with the size of a field, thanks to diminishing returns. It presumably increases too, but a little more slowly, and all the other papers are citing the breakthroughs.) – Anonymous Mathematician Jan 15 '13 at 13:50
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    I fully agree with this answer, and want to add the point that mathematics has few review articles compared to a field like biology, for example. Review articles often contain hundreds of references and thus push impact factors greatly. – silvado Jan 15 '13 at 16:16
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why is the impact factor of mathematical journals "often" lower than the impact factor of journals in other disciplines?

Besides your observations 2 and 3, my take on this would stem from the observation that the pace of work in mathematics tends to be a longer shot than in disciplines, such as biology where often there would be several competing groups working on a very close subject. The impact factor is calculated as "recent", but in disciplines with slow pace of development, sooner than a paper gets cited, it already falls of the considered recent period (2, or 5 recent years).

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    Rather than the "pace of work", I would put this down do the appalling culture, in Mathematics, of taking forever to peer-review journal papers. If it takes more than a year for a paper to get accepted, it takes more than two years for references to said paper to appear. – Pedro Jan 14 '13 at 23:57
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    @Pedro: I don't find it appalling. Journals simply aren't the primary means of communication in mathematics (preprints are). Instead, we've got a hybrid system of instantaneous communication of the latest results combined with a slow and careful refereeing process for long-term archiving. Of course not everyone's as quick or efficient as they could be, and editors are always trying to find ways to speed the process up, but there's nothing wrong with a system where it takes ten times as long to check the proofs in a math paper as to check that a biology paper uses a reasonable methodology. – Anonymous Mathematician Jan 15 '13 at 0:41
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An interesting perspective here:

Antonia Ferrer-Sapena, Enrique A. Sánchez-Pérez, Fernanda Peset, Luis-Millán González, Rafael Aleixandre-Benavent, The Impact Factor as a measuring tool of the prestige of the journals in research assessment in mathematics, Research Evaluation 25 Issue 3 (2016) pp 306–314, https://doi.org/10.1093/reseval/rvv041

Stating the obvious:

A great difference among the journals that publish pure mathematics and the specialized journals in other sciences is that there are very prestigious mathematical journals that publish a little number of papers per year. In other sciences, a typical leading journal publishes hundreds of papers per year.

And

The index of obsolescence of the papers in mathematics is very long, and the articles start their influence in the mathematical research later than in other sciences ( Bensman et al. 2010 ). A 10–20 years old paper may be completely in order for a research that is starting now. On the other hand, the papers ‘start to live’ later than in other sciences. In the first 2 years the articles may have no citations at all.

Mathematicians also don't disdain old sources

Mathematicians do not believe that only recent papers can be useful for their research interest. In fact, old papers and books appear in the bibliography of almost all papers in pure mathematics. Consequently, even the 5-year IF involves a too small evaluation period ( IMU 2008 ). This means that in practice, researchers take into account all the mathematical literature for their research, without considering that new references are in any sense better than old ones. Moreover, sometimes an old paper of a prestigious author is preferred to a new paper with similar results. Classical books in well-established mathematical disciplines are considered as primary sources and appear in the list of references of almost every paper. Also, bibliographic material that is never considered as primary sources in other disciplines—specialized books, doctoral thesis, (even unpublished) lecture notes—are usual references in mathematical papers. These sources of information are not considered for the computation of the IF. Summing up all these aspects, it can be said that mathematics are in a sense more similar to classical humanistic disciplines than to the scientific ones. In any case, the 2-year IF is considered in general as inadequate for researchers in mathematics ( IMU 2008 )

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While I am not familiar in detail with mathematical literature, there is a factor I often see when I compare different branches of Chemistry, Physics or Medicine. Depending on the nature of research, it can often be built heavily on other papers and cite them extensively. For example, it is not rare that a short Chemistry letter of 2-3 pages cites other 20 papers, which includes important experimental details, reviews of other experiments etc. Longer paper, of course, can include much more.

What I see often with theoretical papers that they rely much less on others papers, therefore often cite far fewer - in exchange they are also far less cited on average. I can easily imagine that the situation is similar to mathematics, where proofs are not built on extensively on others work.

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    " proofs are not built on extensively on others work." this is not the case. Proofs in mathematics rely on work dating back hundreds of years, depending on the field. But we don't cite Pythagoras, or Cauchy, or Cantor, or Hausdorff, or Atiyah and Singer, just because we use their techniques, unlike in biology where decades-old techniques that are in textbooks that everybody knows about still need to have their original papers cited (cf the-scientist.com/research/…) – David Roberts Feb 4 '19 at 23:16
  • @DavidRoberts "Proofs in mathematics rely on work dating back hundreds of years" and do you cite them? Do you need to cite them? I am talking about information that is not common knowledge yet needed for understanding the paper. This is even truer in interdisciplinary fields - the readership can be very wide. Your examples are more of exceptions than the typical. The vast majority of tenure-track researchers has papers with 50+ citations. Do you think all those papers are written decades ago and textbook knowledge? I think you take this too personal. – Greg Feb 5 '19 at 2:39
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I recently started working at the intersection between physics and chemistry. As a physicist, it struck me that the impact factor of chemistry journals of a given quality is always significantly higher than the impact factor of physics journals of similar quality. The explanation is simply down to different citation habits in these fields. Typical chemistry papers will contain about twice as many bibliographic references as typical physics papers of the same length.

In an ideally isolated field where papers only cite other papers within the field, the average number of citations a paper attracts must equal the average number of references that a paper contains. Obviously, this simple models does not take into account interdisciplinary citations, and the fact that IF only considers citations received during the last 2 (or 5) years, but it helps to explain why the IF of chemistry journals is much higher than the IF of physics journals.

In a field like mathematics, where I expect the specific subfields to be fairly isolated in terms of citations, I think that lower IF can be easily explained and understood in terms of short reference lists being the norm.

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