### E. O. Wilson vs. Math

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I certainly agree that theoreticians often find it hard to find empiricists who will make use of their equations. But whose fault is that? Seems to me that the fault often lies with empiricists who stick with their intuitions come hell or high water, and who actively resist the discipline that mathematics imposes on their groundless daydreaming. Intuition is great–as long as it’s only a starting point, and as long as you’re prepared to give it up when it’s proven wrong,

Via Tim Poisot, I see that E. O. Wilson has taken to the pages of the Wall Street Journal to explain why “great scientists don’t need math” (that’s the title of the piece). (correction: As pointed out by a commenter, that’s the title on the tab at the top of the web page; the headline on the piece is “Great scientist ≠ good at math”)

Like everyone, I have huge respect for E. O. Wilson. He’s a great scientist and then some–if you cut him in half, you’d have two great scientists. But this piece isn’t one of his better efforts. Indeed, I think it’s really poor. It’s basically Wilson overgeneralizing from his own example, and indeed I think actually misdescribing his own example. And the piece has the potential to do some damage (at least, as much as any single editorial ever can, which may not be all that much), because of Wilson’s prominence, the prominence of the venue, and because it includes advice for students. Really *bad* advice. So I’m going to push back, hard.

Wilson claims that strong math skills are relevant only a few disciplines, like physics. Elsewhere, great science is a matter of “conjuring images and processes by intuition”. He goes on to say that discoveries arise from disciplined daydreaming, from writing field notes, struggling to explain something to a friend, or eating lunch alone. And that making important discoveries requires “hard work and focus.”

I’m sure Wilson is describing his own approach here, and it’s worked for him. But I have to say, it’s surprising to find someone as famous for his breadth of knowledge as E. O. Wilson generalizing so unthinkingly from his own example. I wonder what his late collaborator Robert MacArthur would think of the notion that intuition alone is enough. I wonder what Bill Hamilton would think. Or R. A. Fisher. Or J. B. S. Haldane. Or Robert May. Or John Maynard Smith. Or George Price. Or Peter Chesson. Or Dave Tilman. Or lots of other great ecologists and evolutionary biologists I could name off the top of my head. Would Wilson seriously argue that none of those people were great scientists, or that they never made any great discoveries, or that the great discoveries they made arose from intuition unaided by mathematics? And as for the implication that theoreticians don’t daydream, don’t ever struggle to explain things to friends, don’t eat lunch alone (?!), don’t work hard, and don’t focus, the less said, the better (and if that’s not the intended implication, what is the point of this passage?)

Wilson does go on to say that following up the initial intuition often requires technical mathematical skill, and that’s where collaborators come in. But he makes clear that he sees this as a purely technical exercise. The important part, the creative part, is already done before the mathematical collaborator is brought in. Indeed, he emphasizes how easy it is for empiricists to find mathematical and statistical collaborators, compared to how hard it is for mathematicians and statisticians to find scientists able to make use of their equations.

Again, I wonder what Robert MacArthur, or indeed any mathematician or statistician, would think of the notion that collaboration with an empiricist is a routine and purely technical exercise. Or of the notion that good mathematical collaborators are a dime a dozen.

Having said that, I certainly agree that theoreticians often find it hard to find empiricists who will make use of their equations. But whose fault is that? Seems to me that the fault often lies with empiricists who stick with their intuitions come hell or high water, and who actively resist the discipline that mathematics imposes on their groundless daydreaming. Intuition is great–as long as it’s only a starting point, and as long as you’re prepared to give it up when it’s proven wrong, even if there’s no better intuition to replace it with. Unfortunately, that’s really hard to do.

Wilson goes on to say that “possibly no more than 10%” of theoretical studies in biology have any lasting value. I’m happy to agree–in fact, I bet it’s lower. *But it’s just as low for empirical studies*. The vast majority of *all* studies in science are of little lasting value, since the vast majority not only don’t report any major new discoveries, they’re hardly read or cited. Wilson thinks he’s arguing for the irrelevance of much theoretical biology here, but he’s actually arguing for the irrelevance of pretty much all everyday science not reporting a major breakthrough.

Wilson says that theory is only useful when it describes the actual world. To be useful, theory has to concern “the possible permutations that actually exist on earth”.

You cannot be serious! Indeed, this is so obviously wrong that I’m sincerely mystified how someone as sharp as E. O. Wilson could possibly say this. The claim that only descriptions of our actual world are useful, everything else being an irrelevant hypothetical, will come as news not only to every theoretician, but *to every empiricist who’s ever conducted a manipulative experiment*. I mean, come on! Does E. O. Wilson really need a snarky blogger like me to explain why experiments are useful? Does he really need* *me to explain that the way you learn why the world is the way it is, is by studying how it *isn’t*? Does he really need me to point out that the way you learn how the world works is by *manipulating it so that it works differently than how it actually does*? Because that’s what an experiment does–it creates *unrealistic* conditions, conditions that quite literally *would not have occurred if not for th**e intervention of the experimenter*. That’s the whole *point* of experiments. And* it’s the whole point of piles of mathematical models too*. When Fisher famously asked “Why does most every sexual species only have two sexes?”, he answered that question by modeling species with three or more sexes, *a condition never observed in nature*. When Fisher asked “Why is the sex ratio usually 1:1?”, he answered that question by modeling *what would happen if it wasn’t 1:1*. And so on–I could keep giving examples like this all day. But E. O. Wilson can’t possibly be unaware of this–so why did he write as if he is?

Wilson goes on to advise students that, while they ought to learn more math if their “competence is low”, they can “do outstanding scientific work” with they math they have, so long as they’re careful to avoid specializing in fields requiring “close alternation of experiment and quantitative analysis”, such as “most of physics and chemistry, as well as a few specialties in molecular biology”. It’s totally unclear why students should bother learning more math if they can do “outstanding scientific work” without learning more math, but leave that aside. To students reading this who rightly admire E. O. Wilson and so might be tempted to take his advice here seriously: *he has just given you terrible advice*. *Every* field of biology requires “close alternation of experiment and quantitative analysis”. This has been true for decades. I have no idea what E. O. Wilson is talking about here. Seriously: you are *not* going to make any major discoveries in any field of biology if you don’t know mathematics or statistics, and don’t plan on conducting experiments or quantitative analyses! And Wilson’s claim that deep interest in a subject, combined with deep immersion in masses of data, is sufficient, because hey, it worked for Charles Darwin, is utter rubbish. First of all, just because it worked for Darwin (or Wilson) doesn’t mean it will work for you, and just because it worked in the 19th century doesn’t mean it will work in the 21st. If for no other reason than that there are plenty of people out there, in *every* field, who not only have a deep interest in the subject and an encyclopedic knowledge of the data, but who *know a lot of mathematics and statistics*. Second of all, the notion that Darwin got his ideas by just compiling masses of observational data and then thinking hard about it is just false. Darwin read very widely, including outside of his field. The idea for evolution by natural selection was inspired in large part by his reading of an economist, Thomas Malthus. Darwin also considered the relevance of lines of evidence from fields as distinct (even in his own time) as geology, embryology, and animal husbandry. And he spent the last 40 years of his life running experiments (and if statistics had existed back then, he’d surely have used it to help him design and analyze his experiments).

Wilson concludes with what he calls “Wilson’s Principle No. 2: For every scientist, there exists a discipline for which his or her level of mathematical competence is enough to achieve excellence.” This is question-begging, because it doesn’t specify when a scientist’s “level of mathematical competence” is determined (is high school mathematics enough? How about just middle school mathematics?) And it’s false.

Wilson says he’s dismayed by talented undergraduates turning away from science because they don’t know math. I’m dismayed by a professor catering to their math phobia rather than trying to overcome it. Make them do the math! Someday, they’ll thank you for it.

UPDATE: Tim Poisot’s own thoughts are here. And the Math-Frolic blog weighs in.

UPDATE #2: This post seems to have won the intertubes. There are now discussions going on on Reddit (here and here) and Metafilter. Wide range of views, of my post and Wilson’s original piece. The strongest defenses of Wilson basically take the line Terry McGlynn and others take in the comments here: Wilson’s piece was autobiographical, and it’s unfair (or at least uncharitable) for me to read it as making more general claims. My reply to this line is in the comments in response to Terry, and I won’t repeat it here. Also saw some other good comments, such as the point that there’s a difference between understanding math and being able to do it, or do it easily. It’s quite possible to get by in science, even these days, without being able to do much math (at least not without a textbook by your side), but much harder or impossible to be able to get by without understanding it. And the point that, for all of Wilson’s protests about how poor he is at math, that depends who you’re comparing him to. I think that’s a good point. In his autobiography, he notes that he’s been taught how to solve partial differential equations and been walked through the mathematics of quantum mechanics. He emphasizes that he quickly forgot how to solve PDEs and how quantum mechanics work. But you know what–*I* don’t know how to solve PDEs or how quantum mechanical math works, and never have! Wilson may have come to math late, and may not enjoy it, but he eventually did a reach a level of understanding and competence that I’d say is typical or even a bit higher than that of the average empirically-oriented ecologist or evolutionary biologist. And even if he’s not actually doing any math himself, I am sure that his understanding of it is broadly helpful to him, in particular in his interactions with his collaborators. And so while Wilson considers his own example inspirational to non-mathematical biology students, I can tell you, based on long experience teaching undergrads myself, that many of them would find even his level of mathematical competence intimidating! Even very sharp undergrads often feel this way. One of the very best students I’ve ever taught at Calgary, someone who is *really* good at the mathematical side of ecology (far better than I was as an undergrad) insists continually that not only doesn’t she like math, *she isn’t good at it and never will be*. I truly believe that Wilson’s piece, even read as pure autobiography, is not actually the way to help most of the students he’s trying to help.

UPDATE #3: Razib Khan of the Gene Expression blog over at Discovery Magazine weighs in with a great post. Lots of good context about Wilson (including the fact, noted by Terry in the comments here, that one reason many people (not me) are so upset with Wilson’s editorial is that he recently put his name on a very controversial and heavily-mathematical Nature paper on the evolution of eusociality despite almost certainly not understanding the math). Also good stuff about other great biologists who likely would take a different view of the role of mathematics in biology. For instance, did you know that Bill Hamilton was self-taught in mathematical population genetics, and that formally-trained theorists initially were very skeptical of his models?

UPDATE #4: And over at Psychology Today, Jonathan Wai shares his research on the mathematical abilities of scientists working in different disciplines, as compared to the general population, and as compared to verbal and spatial abilities. In some ways the results aren’t surprising (biologists are, on average, far more able at mathematics than the average person, though not as able as physicists or engineers). But it reinforces the point that, while some biologists certainly are more mathematical than others, it is not true that you can succeed in biology these days no matter what your mathematical ability, if only you find the right subfield. Wai also notes that, on the evidence of his own eloquent writings and self-descriptions, Wilson himself is probably more skilled verbally and spatially than mathematically. Wai makes the interesting suggestion that Wilson himself may be mixing up evaluation of his own mathematical abilities *relative to his own abilities in other areas* with evaluation of his own mathematical abilities relative to those of other biologists, or of students. I think this is an important general point (whether it’s correct in Wilson’s specific case, I don’t know). I suspect it’s part of what’s going on with the very sharp undergraduate of mine I mentioned in a previous update. Relative to other students, she’s extremely good at math. But math feels harder to her than other subjects do, and she enjoys it less. Which may be part of what leads her to feel (incorrectly!) that’s she’s not good at math as judged by some standard external to herself.

UPDATE #5: Graduate student Manu Saunders of Ecology Is Not A Dirty Word weighs in, defending Wilson’s point of view. It’s a good piece, though I don’t agree with all of it. I think that, like Wilson and many other empirically-oriented ecologists, she has too limited a view of the ways in which mathematics can help us learn about nature. Manu, if you’re reading this, I highly recommend Bill Wimsatt’s great old piece on false models and how they’re often empirically useful precisely *because* they’re false. And Terry McGlynn of Small Pond Science has followed up his comments here with his own post. Terry’s an ant guy who I believe has met E. O. Wilson, and knows Wilson’s work much better than I do. So he read Wilson’s piece with rather different eyes than I did (though not necessarily approving eyes). Terry’s post starts out by talking at length about intellectual “tribes” in science, the benefits and costs of joining one (and the difficulty of *avoiding* joining one), and about interactions within and between tribes. He notes, correctly in my view, that E. O. Wilson himself recently helped start a big, unproductive intertribal battle over the evolution of eusociality. He goes on to suggest that Wilson’s piece in the Wall Street Journal was at least in part calculated to start similar intertribal war between empiricists and theoreticians. And that in reacting as I did, I basically gave Wilson what he wanted–a predictable, boring fight between artificial camps (artificial because hardly anybody, certainly not Wilson and certainly not me, is purely theoretical or purely empirical). He goes on to talk about what he sees as the real issues here, the ones that members of both tribes ought to be able to discuss productively (not agree on how to resolve, discuss productively). Go read the whole thing, it’s long, but it’s well worth your time. And then you can come back here and check out some old posts I’ve done on why theoreticians and empiricists often talk past one another, and some theoreticians’ eloquent attempts to explain themselves to empiricists.

UPDATE #6: And now Paul Krugman (!) weighs in. He uses Wilson’s piece (which he mostly likes) as a jumping off point to talk about something he often emphasizes, the way he uses simple models (“simple” by the standards of modern economics research) to undermine our pre-mathematical intuitions about economics. He also notes that what’s worked for him won’t work for everyone, and that everyone thinks the optimal amount of math to know is “exactly the amount you personally happen to know”. The big differences between Krugman’s reaction, mine, and Terry’s are a nice illustration of how people’s reactions to any piece of writing are shaped by their own experiences and knowledge. Krugman, like Wilson, has always been keenly aware of his own limited math skills relative to the most mathematically-oriented people working in his field. And so he’s inclined to like Wilson’s piece, with the caveat that he does see simple math as being hugely important and thinks you’ll make serious mistakes if you know some basic math. I read Wilson’s piece as an ecologist, who has worked hard to purge the field of widespread, serious mistakes arising from failure of too many ecologists to understand even very simple mathematical models. I also read Wilson’s piece in light of other recent pieces from prominent senior ecologists, lamenting that ecology is losing its natural history roots in favor of mathematics. And I read it as an instructor who works very hard to help math-phobic biology students appreciate how math can help them do biology better, and convince them that they *can* get it. Terry reads Wilson’s piece as someone who’s read a lot of Wilson’s stuff, has met Wilson briefly, and who’s talked to a lot of people who know Wilson well. Which causes him to read Wilson’s piece as deliberate bomb tossing, an intentional provocation to theoreticians. None of which is to say that any of us is totally right, or totally wrong. All of our readings have objective support in Wilson’s text itself, but the text itself isn’t sufficient to narrow the range of defensible readings down to a single one.

UPDATE #7: And from beyond the grave, JBS Haldane disagrees with Wilson!

https://twitter.com/JBS_Haldane/status/321725967179608065

HT to Terry McGlynn for injecting a bit of levity into the discussion by pointing this out.

UPDATE #8: Jag Bhalla weighs in with a guest post at Scientific American, and he’s been kind enough to link to us and to Terry over at Small Pond Science. Jag offers some historical perspective, tracing back to Galileo’s faith that “the Book of Nature is written in the language of mathematics.” He pushes back against this faith, arguing that the sort of math physicists use has permeated all fields, to the detriment of those fields because that sort of mathematics isn’t good at dealing with stochasticity, and conditionality. I enjoyed Jag’s post, though I found it a bit too vague and sweeping (maybe the argument he’s trying to make is too big to be easily compressed into blog length. There are huge areas of math (and programming) for thinking about (and simulating) highly complex, stochastic, nonlinear, non-equilibrium systems with many players behaving in their own unique ways, and those are precisely the sorts of systems that are hardest to think about *without* the aid of mathematics. And conversely, as Paul Krugman (linked to in an earlier update) notes, even highly complex, stochastic, systems with many players behaving in complex ways often can be partially understood in aggregate via very simple mathematics, in ways that demonstrably improve over the verbal narratives for which Jag argues. At the moment, a very strong case can be made that massive damage is being done to the global economy, and to the lives of many millions of people, on the basis of verbal narratives about how economics works. Narratives that are both intuitively appealing, and false. So while I actually agree with Jag that there are indeed huge, important chunks of the social world we currently can’t (and quite possibly never will) be able to fully describe with mathematics in a sensible way, his post can be read as an argument for different sorts of math, rather than as an argument for the limits of math.

UPDATE #9: Homologous weighs in, coming down broadly on Wilson’s side on the grounds that overreliance on mathematical models in finance and other areas has led people badly astray. And Larry Bartels of political science blog The Monkey Cage chimes in, agreeing with Krugman’s point of view. Larry’s work used to be more mathematical, but he found that “sophisticated” math was subtracting more than it was adding. Theory based on fancy math often makes only qualitative predictions that are hard to test. He finds that basic mathematical intuition and statistical estimation techniques, focused on magnitudes of effects, are actually more valuable for quantitative analysis of real data.

UPDATE #10: If you think *I* was hard on E. O. Wilson, you should read this take from West Hunter (a collaborative blog between an anthropologist and a physicist). Includes a famous line from Darwin which I can’t believe I’d forgotten about, and which I haven’t seen anybody else quote until now:

I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense.

Which gets back to a point many commenters have made: it’s true (though probably much less true than it used to be) that one can do significant scientific research while knowing less math than many other scientists do. But it’s also true that one can do better research than one otherwise could’ve done by knowing more math. Yes, we want to encourage math-phobic students to go on in science. But surely we want to encourage them to become as good at science as they can be–which means helping them understand and value math, not encouraging them to avoid it.

UPDATE #11: Writing at Slate, mathematician Edward Frenkel says that Wilson is wrong. His take is very much in line with mine, arguing that Wilson is overgeneralizing from his own example. Wilson could’ve said “I’ve been fine without much math, I have other strengths and I’ve been fortunate enough to have wonderful collaborators. But you don’t have to be like me, and indeed it’s not clear that you’d want to be in this day and age, so let me help you overcome your fear of math.” But he didn’t.

UPDATE #12: Theoretical ecologist Will Wilson has drafted an op-ed in response to E. O. Wilson’s (no relation) and put it up on Ecolog-L. It’s a very nice piece. Will takes the same autobiographical approach E. O. Wilson took. He talks about how he came to biology via physics, because he didn’t like all the memorization in biology, and about how mathematical biologists are just as curious and question-obsessed as other scientists (echoes of Fisher here). A nice reminder that students get scared away from biology for more than one reason. Students start choosing their tribe very early, often without realizing they’re choosing a tribe, just because different students have different backgrounds and interests. I haven’t looked at the other related posts and comments on Ecolog-L (I tend to avoid Ecolog-L as a discussion forum), but there is discussion going on there if you want to check it out.

UPDATE #13: Financial blog (!) Above the Market weighs in. Emphasizes mathematics as a way to fight the many well-studied cognitive biases to which all of us (not just financial investors) are subject. And notes, correctly in my view, that Jag Bhalla’s “narratives” (see update #8) need not be seen as an alternative, non-mathematical way of understanding the world. Rather, by doing the math first, we’re often in a position to build better narratives, to tell ourselves *true, unbiased* stories. I’ve tried to do this myself, for instance using simple, intuitive analogies to convey mathematical results, thereby undermining the *bad* intuitions driving zombie ideas.

UPDATE #14: Over at the blog of the BEACON Center for the Study of Evolution in Action, grad student Luis Zaman says that “consultation ≠ collaboration”. He uses the interdisciplinary work that BEACON scientists do as an example of what truly collaborative work at the interface of mathematics, biology, and computer science looks like. It’s not a matter of biologists just roping in mathematicians when they need to formalize their intuitive ideas.

UPDATE #15: Brand-new ecology blogger Jennifer Wright has chosen the Wilson debate as the subject of her very first post. She relates her own struggles with math, and argues that we shouldn’t try to “force feed” math to students who struggle with it.

UPDATE #16: Ok, I’m drawing a line under this. But not before linking to one more reaction to Wilson, from Mike the Mad Biologist. He juxtaposes a sober version of his reaction:

Of course, scientists need think conceptually and broadly, and the overemphasis on models can be harmful (economists are the worst about this, where they attempt to convince politicians to change reality when reality violates the assumptions of their models). But in an era of declining funding, having a well developed set of technical skills can keep you gainfully employed as a scientist

with an entertainingly less-sober version:

BREAKING!!Senior tenured faculty member at Harvard and leader in his field can find others to do the technical bits while he thinks Huge Fucking Thoughts. Guess what roleyou’llplay?

Good to have someone on the internet who makes me look like the restrained, balanced voice of reason.

Changed my mind: let’s give the final word to Joan Strassman of Sociobiology (now there’s a good title for a blog if you’re going to post about E. O. Wilson!) A generous, balanced take on what Wilson gets right, and the many things he gets wrong. Also emphasizes a point not made sufficiently often elsewhere: “math” is actually all sorts of stuff, used for all sorts of purposes. Wilson unfortunately lumps together stuff as diverse as experiments, quantification, statistics, and mathematical theory of all sorts. This stuff does indeed all look the same to students brand new to it–it all involves numbers and symbols. But it’s surely our job as instructors to make clear what all these different sorts of “math” are for, when and why they’re useful, etc.

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