Math and Science Are Not Cleanly Separable

One of the hot topics of the moment is the E. O. Wilson op-ed lamenting the way math scares students off from science, and downplaying the need for mathematical skill (this is not news, really– he said more or less the same thing a few years ago, but the Wall Street Journal published it to promote his upcoming book). This has raised a lot of hackles in the more math-y side of the science blogosphere, while some in less math-y fields (mostly closer to Wilson’s home field of evolutionary biology) either applaud him or don’t see what the fuss is about.

The split, I think, comes from the fact that Wilson’s comments are coupled to a larger point that is basically unobjectionable: that math alone is not sufficient for science. Scientists working in a particular field need to have detailed knowledge of that field in order to even known what math to do:

In the late 1970s, I sat down with the mathematical theorist George Oster to work out the principles of caste and the division of labor in the social insects. I supplied the details of what had been discovered in nature and the lab, and he used theorems and hypotheses from his tool kit to capture these phenomena. Without such information, Mr. Oster might have developed a general theory, but he would not have had any way to deduce which of the possible permutations actually exist on earth.

Over the years, I have co-written many papers with mathematicians and statisticians, so I can offer the following principle with confidence. Call it Wilson’s Principle No. 1: It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations.

This imbalance is especially the case in biology, where factors in a real-life phenomenon are often misunderstood or never noticed in the first place. The annals of theoretical biology are clogged with mathematical models that either can be safely ignored or, when tested, fail. Possibly no more than 10% have any lasting value. Only those linked solidly to knowledge of real living systems have much chance of being used.

That’s absolutely fine, and the same can be said of a lot of physics. The mark of a useful physical theory is that it accurately describes reality, and that requires math to be constrained by empirical observations. To the extent that the much-ballyhooed “crisis” in physics exists, this is the root of the problem: high-energy theorists have not had the data they need to constrain their models, and that has impeded real progress.

What I, and many other physical scientists, object to is the notion that math and science are cleanly separable. That, as Wilson suggests, the mathematical matters can be passed off to independent contractors, while the scientists do the really important thinking. That may be true in his home field (though I’ve also seen a fair number of biologists rolling eyes at this), but for most of science, the separation is not so clean.

As much as I agree with Wilson’s statement about the need for detailed knowledge to constrain math, even in physics, there is also some truth to the reverse version of the statement, which I have often heard from physicists: If you don’t have a mathematical description of something, you don’t really understand it. Observations are all well and good, but without a coherent picture to hold them all together, you don’t really have anything nailed down. Big data alone will not save you, in the absence of a quantitative model.

Of course, that’s physics, which Wilson exempted from his comments at the beginning of the piece, so maybe we’re just oddballs on the boundary where math shades into science. But the close marriage of math and science pops up even in the life sciences. There’s no small irony in the fact that one of the other big stories of the week in science is a study showing that many neuroscience studies are woefully underpowered, in a statistical sense. This is a hugely important paper, because it calls into question a lot of recent results, and common practices in the field.

It also shows up the problem with Wilson’s contract-out-the-math-later approach. Because, after all, the problematic studies are doing essentially what he talks about– they’re out in the field, making observations of phenomena, and thinking about mechanisms to explain them. The problem is, many of these observations turn out to be of questionable value, because they didn’t do the math right. They didn’t have enough test subjects to reliably test the things they were trying to test. And this has very real negative consequences for the field, as people waste time and resources trying to duplicate results that turn out to be a statistical fluke. To say nothing of the career risks for an early-career scientist who plans to build on one of these results, who not only can’t replicate it, but can’t publish the failure to replicate.

As a general matter, science and math are just not cleanly separable in the way that Wilson asserts. If there’s an exception here, it’s his field, not the handful he airily waves off as inherently mathematical. You need observations to constrain mathematical models, yes, but you also need math to know what observations you need to do, and to determine the reliability of your results. The notion that the two can be cleanly separated, and the scary math bits farmed out to somebody else is not just faintly insulting to mathematicians, it’s flat out wrong. And that’s why people are annoyed.

16 thoughts on “Math and Science Are Not Cleanly Separable

  1. As a mathematician who has been working with evolutionary biologists over the past couple of years, I’d say Wilson is wrong about his own field too. Over most of his career, perhaps what he says is OK, but the huge amount of genetic data available to analyse over the last decade has meant a lot of work for mathematicians, statisticians, and computer scientists. And the amount of data is growing faster than Mooore’s Law. Evolutionary biologists are so desparate that they’re even employing physicists.

  2. Sometimes you have to be “an island”. Several years ago I came across a class of mechanical systems expressible in non-positive matrices. I appealed to numerous mathematicians for help and was uniformly advised that they were unaware of any maths dealing with the area. So I had to hunker down and do it myself. Collaboration is like communication – both sides must be engaged.

  3. The need for active collaborators is a good point. Also, I suspect it’s vastly easier for the eminent Professor Wilson to get help from a mathematician or statistician than the unknown Assistant Professor Nosliw…

  4. The question I have is whether there’s a place in science for people who aren’t particularly mathematically proficient? Are we losing potentially brilliant scientists because they are averse to mathematics, maybe because they went to one of the many schools with terrible math programs?

  5. I can certainly understand where Wilson is coming from. In the old days a kid with keen curiosity and above average intelligence could make it in science. Now you have to have top 1% math ability, and put in intense study of math in order to make it. Many people who were prominent scientists a hundred or two hundred years ago, if reborn today would be washed out. Its a new (science) world, and the particular constellation of native abilities needed to succeed has changed.

  6. Even if you “contract out” the mathematics, you need some mathematical knowledge in order to effectively be able to communicate. Vice versa, the mathematician needs some basic knowledge of biology. Separate the science and the math too much and the biologist and the mathematician will be working on completely different problems.

    Take a cow, assume it is spherical…

  7. As an (ex) mathematician, My take is that, while this may be true, it doesn’t really matter. Most (or at least MANY) mathematicians don’t CARE if their theorems relate to the Real World ™. The assumption that math has no value unless it is related to biology is kind of insulting.

  8. Am I to conclude from the fact I publish in English that English and science aren’t cleanly separable?

    There is plenty of biology that involves about the same amount of day-to-day math as eating at a restaurant. Snarky commentary aside, biology is science.

    Also, GENOMIC (not genetic) data is analyzed by all kinds of scientists. The BLAST algorithm/program, for example, was developed by a team of people trained as: a computer scientist, a medical doctor, 2 mathematicians, and a molecular biologist. And it’s an amazing tool that requires almost no math to use (perhaps “is this E value smaller than that E value?”).
    Most molecular biologists do contract out the big math. Even if there are 100 different algorithms for sequence alignment, most of the time you want to use a flavor of BLAST that’s on NCBI. Because scientific fields do actually benefit from standardization of methods, and also because you want to have the time to work on the actual biology, and also because BLAST is pretty dang great.

  9. rork’s rule #1: The math nerd can learn the biology better/quicker than the biologist can learn the math.
    #2: Biologists sometimes became biologist cause they didn’t want to study math. They are prone to thinking “I don’t need to know that” – something the math wonk hopefully never says about the biology.

    As a statistician doing cancer research, I can testify that we need lots more math wonks to come into the trenches in the war on tumors (and other things like hypertension). I also testify that I need the docs and other researchers to have much better math skills, just to be able to understand analysis of experiments, to realize when they need help with experimental design (hint: almost always), and simply to be able to read the papers properly. Folks, I’ve got scientists with trouble understanding why I want to take logs, what ascertainment bias is, and spotting when a demonstration is crap for lack of cross-validation. Even ordinary docs need much better quantitative skills these days to aid decision making. We must do much better.

  10. Am I to conclude from the fact I publish in English that English and science aren’t cleanly separable?

    As I am at pains to tell students every year, when they complain about having to write lab reports, ability to communicate effectively is an essential skill for science and engineering. You will not have a successful scientific career in the US if you cannot speak and write in English. The very best scientists I have worked with are also generally excellent communicators– they have to be, to get others to pay attention to their results and recognize their importance.

    Now, it’s true that, in contrast to the situation Wilson describes with math, somebody is apparently telling students that science is a great place to work if you have poor language skills. But that doesn’t make communications skills cleanly separable from science any more than Wilson’s assertions make math skills cleanly separable from science.

  11. Am I to conclude from the fact I publish in English that English and science aren’t cleanly separable?

    If your command of German, French, Russian, or Chinese is sufficient, you can publish in one of those languages. However, your audience will be limited to people who are able to read that language. And in any case you must be able to communicate in some language, or at least have somebody on your team who can communicate in that language. Today the default language of science is English, but that was not always true (German predominated until about 1940) and may not always remain true.

    As a scientist I routinely encounter English written by people for whom it is a second (or third, fourth, …) language. The good scientists make an effort to write in understandable English, and of those I have met, they always try to speak understandable (if accented) English. They don’t always get it right (Chinese and Japanese, for instance, don’t have articles, so native speakers of those languages don’t always use them properly), but they do better with their English than I would with their native language, so I don’t have grounds to complain.

    Most molecular biologists do contract out the big math.

    In a sufficiently large collaboration (although it is not as common as in some areas of physics, I have seen biomedical articles with more than 100 co-authors), it makes sense to have a mathematics specialist for this kind of thing. But if you’re trying to run a small lab, you may not have that luxury. You definitely need to know statistics well enough to be able to do that yourself.

  12. rork- only if you don’t mind your mathematicians mixing up “genetics” and “genomics”. At least biologists KNOW they don’t know the math. Mathematicians honestly think they know the biology when… they don’t. I strongly suspect your understanding of M1/M2 macrophages is… as lacking as my appreciation for log/log plots.

    Chad- my point was not that *communication* skills are unimportant, simply that it verges on egomania to suggest science must be done in English to BE science. This is more or less what happens with physicists who are obsessed with calculus. It’s not that communication and quantitative skills aren’t both important. What I object to, rather, is automatically assuming it’s a sign of superior inteligence/rigor/science-essence to learn calculus at 16 compared to 32. Some people, including scientists, really don’t need it until they are 32 (or at all).

    Eric- in a sense, every molecular biologist who is using BLAST who did not in fact develop BLAST, is contracting out the big math. Co-authorship has different implications than ‘contracted out’. Not all research needs *novel* mathematical tools; most of uses existing mathematical infrastructure. Do we suggest chemists need to know plumbing because they wash glassware?

  13. You can absolutely get away with doing all your mathematical processing through software packages written by other people. There’s no absolute requirement that you understand all, or even any, of the underlying math. Of course, this is not without risks: specifically, as noted in the post, that a huge swath of research might turn out to be of questionable value because the people planning and carrying out the experiments didn’t realize that their studies lacked the necessary statistical power.

    Similarly, you can do complex financial transactions just by plugging numbers into programs somebody else wrote, as long as you don’t mind occasionally wrecking the entire world economy.

  14. I’m not a neuroscientist, so I can’t be sure what’s up with them, but I doubt it’s as simple as not understanding the math underlying their software. I do understand some of the constraints they are under. I’d guess that by and large, it’s not that they don’t have the time, intellect, or inclination to learn sufficient statistics to power their studies correctly. It’s that neuroscience requires brains. And sample size thus has incredibly crucial ethical implications. If you actually want neuroscience studies to be appropriately powered, talk to the IACAC (institutional animal care and use committees) and get statisticians on *them*.
    Note to any statisticians: all IACACs are supposed to have “community member” representation. Feel free to stack the IACAC deck with math-y types. If somebody wants to set up a petition to NIH to get a statistician on every IACAC, I’m down with that.

    We also can’t ignore incentives in the scientific hierarchy re: publishing- i.e. it’s easier to publish splashy findings in prestigious journals, so an occasional Big finding that turns out to be not true is probably better for your career than a lot of little findings that turn out to be correct… what would we *expect* to happen to statistical power given those incentives?? Similarly, you can scorn bankers for not understanding formulas, or you can notice that these people were simply playing games with *other people’s money*, and that the interests of large capital were somehow always protected. It’s only little people that get crushed from the “entire world economy” getting “wrecked”. The problem with derivatives isn’t that the math is hard. The problem is that *intentionally* confusing math was used to obscure bad products. It wasn’t that people buying and selling them didn’t realize the were junk, it’s just that the incentives are such that it’s easy to simply keep collecting your paycheck while hoping you aren’t the last one standing after the music stopped (and the ones with real money/power/influence already knew they had their chairs set).

  15. “This is a hugely important paper, because it calls into question a lot of recent results, and common practices in the field.”

    Common practices – i.e. those stemming from the deeply conceptually and technically flawed orthodox ‘inference’ – and the damage they’ve done and do to science, have been called into question in hugely important papers and books repeatedly during the last ~50 years.

  16. @Becca: One important thing about mathematical tools is that there are certain underlying assumptions under which they are developed, and if you have no idea what those assumptions are, you risk producing utter nonsense because the assumptions do not apply in your situation. I have seen theoretical physicists make this mistake: I have reviewed at least one paper where the plotted results seemed puzzling to me until I followed the citation chain back to the original reference for the equation they were using and saw that it was derived under assumptions that did not hold in the situation where the authors were using it.

    With statistics, one issue is the validity of the assumption that things follow Gaussian distributions. Many standard statistical methods depend on this assumption. Often you can make this assumption. But many interesting problems in physics, as well as finance (and this may also be true of biological sciences) involve non-Gaussian probability distributions. What does this do to your p-values? I don’t know, but it’s something that somebody with a better background in statistics might want to check. And they might be able to get a Wall Street I-bank to fund this research, because (Chad @13 alludes to this) part of what caused the 2008 financial crisis was the incorrect assumption that asset price movements followed a Gaussian distribution. When you are seeing 25 sigma deviations several days in a row, as the then CFO of Goldman Sachs claimed, there is something wrong with your model.

Comments are closed.