How much math do scientists need to know?
I’m catching up with reading the “big data news” this morning (via Gil Press) and I came across this essay by E. O. Wilson called “Great Scientist ≠ Good at Math”. In it, he argues that most of the successful scientists he knows aren’t good at math, and he doesn’t see why people get discouraged from being scientists just because they suck at math.
Here’s an important excerpt from the essay:
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.
Given that he’s written many papers with mathematicians and statisticians, then, he is not claiming that math itself is not part of great science, just that he hasn’t been the one that supplied the mathy bits. I think this is really key.
And it resonates with me: I’ve often said that the cool thing about working on a data science team in industry, for example, is that different people bring different skills to the table. I might be an expert on some machine learning algorithms, while someone else will be a domain expert. The problem requires both skill sets, and perhaps no one person has all that knowledge. Teamwork kinda rocks.
Another thing he exposes with Wilson’s Principle No. 1, though, which doesn’t resonate with me, is a general lack of understanding of what mathematicians are actually trying to accomplish with “their equations”.
It is a common enough misconception to think of the quant as a guy with a bunch of tools but no understanding or creativity. I’ve complained about that before on this blog. But when it comes to professional mathematicians, presumably including his co-authors, a prominent scientist such as Wilson should realize that they are doing creative things inside the realm of mathematics simply for the sake of understanding mathematics.
Mathematicians, as a group, are not sitting around wishing someone could “make use of their equations.” For one thing, they often don’t even think about equations. And for another, they often think about abstract structures with no goal whatsoever of connecting it back to, say, how ants live in colonies. And that’s cool and beautiful too, and it’s not a failure of the system. That’s just math.
I’m not saying it wouldn’t be fun for mathematicians to spend more time thinking about applied science. I think it would be fun for them, actually. Moreover, as the next few years and decades unfold, we might very well see a large-scale shrinkage in math departments and basic research money, which could force the issue.
And, to be fair, there are probably some actual examples of mathy-statsy people who are thinking about equations that are supposed to relate to the real world but don’t. Those guys should learn to be better communicators and pair up with colleagues who have great data. In my experience, this is not a typical situation.
One last thing. The danger in ignoring the math yourself, if you’re a scientist, is that you probably aren’t that great at knowing the difference between someone who really knows math and someone who can throw around terminology. You can’t catch charlatans, in other words. And, given that scientists do need real math and statistics to do their research, this can be a huge problem if your work ends up being meaningless because your team got the math wrong.
mathbabe 
This reminds me of John Donne’s comment on the human condition, “no man is an island.” I think E. O. Wilson’s comment has broader application. We all have blind spots and areas of strength and area’s of weakness. It has really helped me to collaborate on projects with coworkers with different skill sets in an environment where we could help each other dispel our wrong ideas and produce something that was as good as possible given the available resources. The product was always better than the sum of the individual contributions.
The key is to assemble the proper group who will create an environment where it is safe to say “I don’t understand” or “I need another perspective on this” and to learn to give and welcome receiving constructive criticism. Note that this is different than using mathematicians, statisticians, or analytical chemists as a service. This is especially important now, given the rapid growth in the number of books and papers available. It really helps to have some help in finding the cream of the crop to read,
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I have mixed feelings about Wiison’s article. On the one hand, I agree with and support his position that people, especially younger students, shouldn’t be intimidated and avoid the sciences just because they don’t view themselves as having a talent for mathematics. In fact, I often think that science classes put too much emphasis on “doing math” and pay too little attention to helping the students see how observations move to theoretical descriptions that are eventually put into mathematical terms, thus showing how math is really another method of expressing our experience of the world rather than an end in itself.
Some years ago I had to be treated for Hodgkin’s disease and had a lot of time on my hands while resting at home in between radiation treatments. (Basically, I spent three months feeling like I had a low-grade flu.) I had left graduate school a few years earlier; so I dug out some of my old physics texts and decided to work through A.P. French’s Newtonian Mechanics, but looking for ways to use Mathematica usefully when solving the problems. What I realized was how many problems appeared to have been set up to force the student to identify some mathematical “trick” or simplifying relation to find the solution. In short, Mathematica didn’t offer much advantage. But I felt that with software like Mathematica, students could tackle more realistic problems that didn’t have to be designed to use some math gimmick. Instead, the problems could be based on data sets created by an imaginary lab assistant, which the student would have to analyze to understand the question and derive an answer. The graphic capability of the software would also enable students to graphically solve systems and then work out the underlying mathematical expressions.
The effect would be quite different from traditional textbook approaches, since the tension between assignment time and mathematical complexity can be greatly relaxed. I began to think of a course that was more case study oriented, presenting the students with the sort of facts and data that typically confront scientists and then using a guided approach to enable the student to develop the understanding and math in a more realistic and natural way. I did some research on this idea, and came across a book by James Conant (the second of Harvard’s chemist-presidents) presenting a case-study approach to teach the sciences to non-scientists. I believe I can use Conant’s ideas with my own to make a much more interesting science course. In fact, I broach this idea to an assistant dean at Harvard’s Ed. Schools a few years ago and received quite a bit of interest. Sadly, I haven’t had the time or opportunity to actually work on the details.
In short then, this could be a great time for those with less confidence in their math skills to get into science.
On the other hand, I also recall my freshman year of college at Chicago when I had a work-study job for a couple of faculty members in the high energy physics group. All of the graduate students I worked with gave me the same advice, often quite earnestly: Don’t take the math-for-scientists-and-engineers courses; take the real math classes instead. I followed their advice, to the detriment of my GPA and self-esteem. But I found a big payoff after the agony–I actually had a foothold in understanding mathematics itself, and not just knowing how to solve equations. When I went of to grad school in organic chemistry (Harvard), I was one of a few who could talk as easily to the theoreticians as to the bench chemists. Taking the math enabled me to understand a wider range of disciplines and let me take on a more varied set of projects.
So, while I agree that students who are insecure about their talents in math should not abandon hope of entering the sciences, especially given the computer aids available today, I would also encourage them to do what Wilson eventually had to–learn the math. Sure eventually, we all have to focus on a particular area of study and therefore work in groups of experts to solve complex problems, but having the exposure to math really pays off. Some years ago, Bell Labs did a study of researcher productivity, and found that the most productive researchers were not the company’s many Nobel laureates but those lesser souls who could communicate across disciplinary groups–the “pollinators”–that enabled communication across disciplines and teams and facilitated the interactions among the staff.
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Great comments, thanks!
I agree about the “gimmicks” one sees in a lot of math problems. I’ve often wondered how to (perhaps quantitatively) define what it means for a given problem to be gimmicky. If you have any ideas I’d love to hear them. I generally think that the more gimmicky a math class is, the more it leaves the average student with the feeling that they don’t “have it” when it comes to problem solving in math.
Cathy
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That’s a tough question, but I tend to find problems “gimmicky” when the solution turns on some obscure mathematical identity or definition, rather than being more intrinsic to the underlying subject matter. I think this is why so many students get turned off in their early science classes, thinking that science is only for those with the mathematical talent to spot the “trick” quickly. Of course, the instructors end up frustrated with a class full of math wonks who have no feeling for the science.
As Mr. Damiani, one of my math teachers used to say to lighten up class tension, “Math is nothing but cheap tricks and bad jokes!” 🙂
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“But I felt that with software like Mathematica, students could tackle more realistic problems that didn’t have to be designed to use some math gimmick.”
Through seeing some fifty years of attempts at using math (whether it’s the largely mysterious quantum electrodynamics or idiotic applications of gaussian diffusion in fluid mechanics to financial derivatives), I opine that the reason “gimmicks” are so often appealed to:
Reality is mostly nonlinear and analytically unsolvable, therefore only an approximation can give a result compatible to mandatory and impressive high-speed digital calculation. The linear models of the easier problems of the past were very successful within their domains of validity, and involve very powerful methods of high specialization. By making the right series of assumptions to simplify the description of how a model is relevant to a hard, nonlinear problem, these old, proven, powerful techniques can be converted into gimmicks that are cut and pasted into massive digital programs.
The validity of the simplifying assumptions are often hand-waved away if the results in a suitably moderate, linearized domain look reasonable: “Look, it works, so why worry”. Eventually reality must win out, however (especially with things like Black Swan shock waves) and the futility of the assumptions becomes obvious to those without a vested interest in the gimmickly impressive but fallacious model.
Mathematica does have promise in modeling (and has already rewritten old and faulty “cook-book” handbooks of complex integrals), but the domains of validity must always be factored into its powerful approach to new problems.
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On the contrary. I’ve found that a formal foundation in mathematics, including probability & statistics, has been the key to opening up new fields to me. The only way I could have made the leap from (bio)physics to applied statistics to some pretty advanced topics in machine learning is through the common language of math.
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Selected comments on E.O. Wilson’s WSJ editorial:
1. I’m pretty sure that E.O. Wilson’s article is a pure example of selection bias where E.O. was “selected” in a way that makes his own experiences inapplicable to vast majority of undergraduates today.
E.O. Wilson became famous by thirty and at Harvard probably had no shortage of talented mathematicians/statisticians to work with. In addition, he became famous at the perfect time: mathematical models of evolution had been developed by Fisher in the 1940s. However, basically no one knew to how to use them until the the 1950s, they did not become widespread until the 1960s: by which time Wilson had already attained tenure/massive status for some of his cool ant studies. By the 1970s that “I’ll learn calculus by thirty-two” bs wasn’t going to fly for the median academic that didn’t bring *a lot* to the table in biology.
2. 10% of the papers in mathematical biology are useful and relevant to the phenomena they’re modeling?! That’s way, way more than the number of useful models in finance. They must be doing something right.
3. I’m huge fan of the “learn math as you need it for specific applications” approach, as is every entrepreneur evar and the next columnist to write something Wilsonesque in the WSJ will find that path yielding and pleasant.
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“I agree about the “gimmicks” one sees in a lot of math problems. I’ve often wondered how to (perhaps quantitatively) define what it means for a given problem to be gimmicky. If you have any ideas I’d love to hear them. I generally think that the more gimmicky a math class is, the more it leaves the average student with the feeling that they don’t “have it” when it comes to problem solving in math. ”
I’ve read that, as a matter of aesthetic preference Grothendieck apparently used to throw out/shelve any proofs that relied on “tricks” that weren’t obvious previous developed theory.
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“Wishing that somebody could make use of their equations.” This leads to a more general point that is not math-specific, but is true. In science, there are “systems people” and “techniques people.” The techniques people parachute into problems with their technique, make a quick hit, have some fun, and go onto something else. They either need to read a lot of literature or be very collaborative with people who understand the problem. Systems work is centered on the problem, and is a damn sight slower. I was a techniques person for most of my career as a scientist (now long passed.) I wasn’t a mathematician; I was an experimentalist who knew how to do time-resolved spectroscopy and read a lot of collateral literature. I had a lot of fun and a few productive quick hits. But I think that systems people, in the end, are what science is about.
Wilson’s basic point, or course, is correct. Math is but another technique in science, albeit the most important single one. Having no math is a disadvantage, but then again, so is being a klutzy experimentalist. Both disadvantages can be transcended. My thesis advisor, for example, was a klutzy lab rat (none of his grad students let him within 10 meters of the instrumentation), but an insightful designer of experiments.
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I guess what I’m saying is that, inside of mathematics, there are also both technique and systems people. You only see the technique people when you are a scientist, but that doesn’t mean the systems people don’t exist.
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“Math is but another technique in science, albeit the most important single one” – I have to disagree. Math is but another language, and by far the most precise and transparent one. It is very hard to say “When I use a number …”
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Agreed. It’s easy to say “I speak math-ese,” though.
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Wilson may have had a valid point in there somewhere, but he totally comes off as a pompous asshole who has little respect for, or understanding of, what mathematics is all about. If I were one of his mathematician co-authors, I’d be feel feeling pretty dissed. He’s essentially saying, “Hey, I’m the one doing groundbreaking work over here. There are tons of math people who can help with the details.” And to some extent, that’s probably true, but there are very few Wilsons out there. Joe Blow biologist may be doing solid work, but there is probably not a gang of solid math people lining up to work for him to do the dirty work that he is ill-prepared to do himself (and there will be fewer still if students start following Wilson’s advice). His advice really only applies to the parts of science that are 90% qualitative. It’s worth noting that that is a small fraction of “science” that gets smaller every day.
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It’s difficult to read Wilson’s article and not be reminded of those wannabe entrepreneurs who’ve “got an idea for the next Facebook and just need a programmer to implement the details”.
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How Much Math? Enough to know when thy are being lied to.
It is an interesting exercise to calculate mathematical probabilities of so-called “conspiracy theories”. The mainstream media and their cadre of online gatekeepers use the term “Conspiracy Theorist” (CT) as a derogatory label for those who seek the truth. According to the media, there are never conspiracies. But they avoid factual discussions based on the scientific evidence.
These myths are promoted non-stop in the mainstream media.
– Oswald acted alone in 1963 – with a magic bullet and defective rifle.
– Bush won Florida in 2000 and had a 3 million “mandate” in 2004.
– Nineteen Muslims armed with box cutters who could not fly a Cessna, hijacked four airliners and outfoxed the entire U.S. defense establishment – while Bin Laden was on dialysis, near death and hiding in caves.
But the media can’t refute the mathematics that proves beyond a reasonable doubt that there is a massive conspiracy to hide the truth of these events from the public.
Scientific notation is necessary to express the extremely low probabilities of the following events. For example, the probability P that at least 15 material witnesses would die unnaturally in the year following the JFK assassination is 0.000000000000006 or 1 in 167 trillion. There are 15 zeros to the right of the decimal point (represented in short-cut scientific notation as 6E-15).
much more…
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