Are you afraid of equations?

by James Thorniley

Jellymatter is, we claim, not afraid of equations, but apparently scientists are. A study in PNAS claims to have found that theoretical biology papers are cited less when they are densely packed with mathematical language. The authors argue that this impedes progress, since empirical work needs to be backed up and commensurate with some theory to have deeper scientific meaning.

I think it’s a very interesting point. Mathematics is often said to be the language of science, but actually, contrary to perceptions that many people might have, most scientists aren’t big fans of maths. But using maths is one of the ways that scientists try to make the meaning of their work precise – without it, scientific theory is too vague and subject to interpretation. In that way, maths should be an aid rather than a hindrance to communication.

Maths also gives you a way to be sure of what logically follows from the statements you make. As a simple example, think of the simultaneous equations you learn in school – if you are given that x+y=2, and x-y=0, you can show that as a logical consequence of these assumptions x and y must both themselves be equal to 1. Much of the use of maths in scientific theory comes down to derivations like this, and it allows you to be sure that your results follow logically from your propositions.

Yet in spite of its advantages, it seems maths turns off a lot of scientists. Actually probably the most influential work in theoretical biology, Darwin’s On the origin of species, contains essentially no maths. By contrast, I’ve recently been reading Norbert Wiener’s Cybernetics, which is again an influential work, but it relies on a lot of maths, and I wonder if perhaps this puts people off really appreciating Wiener’s arguments.

Finally, at the end of the story on this paper here, one of the authors is quoted arguing that maybe a good middle way is not to exclude maths from theoretical work, but to make sure to add more explanatory text. This is clearly a sensible suggestion. We also perhaps shouldn’t generalise too much – maybe maths does put off some scientists, but as I have pointed out, it also has some strong advantages. While highly mathematical papers might not be the most popular or influential, they do still have a place and a utility.

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4 Comments to “Are you afraid of equations?”

  1. Personally, although I’m certainly not afraid of equations, I do find an excess of mathematical language to be distasteful. I find that excessively formal language is sometimes used to disguise a lack of clear thought about the work. There’s a kind of trade-off, I think: mathematical language allows you to express your thoughts in an extremely clear way (at least, it does if the thoughts were quantitative in the first place), but it also demands a certain amount of effort on the part of the reader, and if you jump straight in with the “we define \mathfrak{R} as a hypermorphic polybongo set of degree \mathcal{R}” then it’s hard for the reader to assess whether that effort is worth investing. For me the ideal paper expresses its ideas clearly in natural language, and uses an appropriate amount of mathematical formalism only where such precision is required. Edwin Jaynes’ papers are often excellent examples of this style.

    Anyway I have to go now because there seems to be some kind of insect in the room HELP IT’S A BEE

  2. Yeah, I think that’s the point being made partly. Also, you have to distinguish between actual *maths* papers (where it may be interesting to the reader to just define some abstract entity and derive something from it), and scientific theory papers, where the reader is expecting there to be some kind of meaningful point, in terms of something they could empirically test.

    • Even in purely mathematical papers there’s a need to explain the ideas clearly in natural language before launching into the formalism. Unless they’re really really pure maths papers with no possible practical application I guess.

      I find it to be a particular problem with probability theory and information theory. A lot of papers in that field like to define everything in the most general possible way using measure theory. But this makes it difficult for mere mortals to understand, and that’s a shame because if information theory was as widely understood as classical hypothesis testing statistics, we’d all be a lot better off.

  3. Oh yes, overgeneralisation is a pain. Exactly at what point is it too general or not general enough? It’s hard to know, but as you say I think to most “mortal” statisticians the basic concepts are pdfs, cdfs, means, variances etc, rather than the measure theoretic basis of all that which of course is an interesting mathematical entity, but not always needed (though of course it could be needed sometimes).

    Lebesgue integrals, are, I’m told, more general than Riemann integrals. But I couldn’t tell you off hand why, and for most of the work I do, I just use “integrals”, as in, y’know, the thing you learn in school or university that’s the opposite of differentiation…

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