## May 28, 2009

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# What Kinds of Math Do We Need?

Biologists are debating how much quantitative analysis their field needs; at Language Log, Mark Liberman pivots to linguistics:

The role of mathematics in the language sciences is made more complex by the variety of different sorts of mathematics that are relevant. In particular, some areas of language-related mathematics are traditionally approached in ways that may make counting (and other sorts of quantification) seem at least superficially irrelevant — these include especially proof theory, model theory, and formal language theory.On the other hand, there are topics where models of measurements of physical quantities, or of sample proportions of qualitative alternatives, are essential. This is certainly true in my own area of phonetics, in sociolinguistics and psycholinguistics, and so on. It’s more controversial what sorts of mathematics, if any, ought to be involved in areas like historical linguistics, phonology, and syntax…

Unfortunately, the current mathematical curriculum (at least in American colleges and universities) is not very helpful in accomplishing this — and in this respect everyone else is just as badly served as linguists are — because it mostly teaches thing that people don’t really need to know, like calculus, while leaving out almost all of the things that they will really be able to use. (In this respect, the role of college calculus seems to me rather like the role of Latin and Greek in 19th-century education: it’s almost entirely useless to most of the students who are forced to learn it, and its main function is as a social and intellectual gatekeeper, passing through just those students who are willing and able to learn to perform a prescribed set of complex and meaningless rituals.)

My thoughts are still inchoate on this, so I’ll throw it open — is calculus 1) a waste of time for 80-90% of the folks who learn it, 2) unfairly dominating of the rest of useful mathematics, 3) one of the great achievements of the modern mind that everyone should know about, or 4) all of the above?

More to the point — what kinds of maths (as they say in the UK) have you found to be most valuable to your later life, work, thinking, discipline, whatever?

And looking to the future - I don’t think we have a mathematics entry as such in the New Liberal Arts book-to-come; but if we did, what should it look like?

## Comments

The big indicator is that people who don't use it, lose it -and my calculus is starting to go...

With things like Wolfram Alpha hitting the market, knowing enough math that you can ask a certain type of question will be more important than computing it (you have to know to ask about a derivative when you want the rate of change..WA will compute it).

Going forward, statistics seem like the critical branch for the lay person, but that usually means knowing some calculus (at least about distributions and what an integral is).

For isntance, I generally trust Atul Gawande, but even he writes things like:

"More than seventy per cent of physicians in high-cost cities referred the patient to a gastroenterologist, ordered an upper endoscopy, or both, while half as many in low-cost cities did."

Unless you understand the possibility for ambiguity you can't make good decisions about public policy. Atul Gawande's evil twin could easily use this same sentence to onfuscate the truth as opposed to exposing it.

Being a scientist obviously skews my answers, but I will say that DiffEq is constantly useful (not that you have to have it at your fingertips, but it's great to know what book to look in for the answers), and I am surprised by how often linear algebra comes up (again, having a high level understanding more than knowing all the axioms off the top of your head).

So that's my take, as a professional nerd. However, as a general thing I do think everyone should have at least a basic understanding of what calculus is for. There is a huge family of problems that can be solved easily with calculus, whereas without it you would probably convince yourself there is no solution and give up.

Anecdotally, I can remember having a long debate over whether Zeno's Dichotomy Paradox was worth any thought (the answer is no), where having a basic understanding of calculus was the key issue. Which is not to say that this anecdote proves calculus is invaluable, but it does come up more places than you might think.

On statistics: of course it is good for everyone to know at a basic level. But it is also of course more often abused than used properly. I think it's fair to say that statistics are

moreubiquitous in everyday life than they ought to be, in contrast to the rest of math.One point I often carp on is that 99% of summary statistics quoted would be better replaced with tiny graphs of the actual distribution of data. In any online/digital story/report/etc., it would in fact be better to have a tiny widget allowing dynamic replotting of any data cited.

One point I often carp on is that 99% of summary statistics quoted would be better replaced with tiny graphs of the actual distribution of dataI think it's important to distinguish between statistics ("the numerical facts or data themselves") and statistics ("the science that deals with the collection, classification, analysis, and interpretation of numerical facts or data, and that, by use of mathematical theories of probability, imposes order and regularity on aggregates of more or less disparate elements.") I am deeply bitter about the fact that I went to college right next to one of the world's best statistics departments and had

no ideawhat they were upto, and no thought to take classes from them. In 9 years of reading and socializing and asking questions, later, this deficit has bubbled up to my top three didactic gotta-fix-its. (No joke, in three weeks I'm going to start the calculus-dependent course sequence for majors.) In my mind, this deficit exemplifies the kind of yawning gap I'd hope the New Liberal Arts would prevent.I don't actually like the dictionary.com definition above. I prefer something simpler--

Statistics: how to make and understand probabilistic models explaining data. And at some level, is there any more important concept to port from the sciences to general, critical thinking? Possibly, but it's a top contender.Look at the facts. Come up with some models. Understand how likely those models are to be true.We don't need the educated public to know the details of computation, but the conceptual understanding is invaluable.I originally refrained from commenting on this post because I was writing a final exam for a calculus-based physics course, and I was feeling far too tyranically pro-calculus to say anything lacking bombast. Also, I'm incredibly biased towards computational biology--between my sister, my boyfriend, and most friends its gospel for me that biology thrives with quantification. But the calculus. . .not sure how necessary that is for biology. I hated calculus the first two or three times I learned it, and every time after n = 3, I probably only love it for the even n's. But the master calculus teacher I work with likes to insist he is teaching his students how to think, not math, and I think he's right: understanding how to consider change, or lack there of, the most and the least, the cumulative and the total, the ability of smoothing over rough edges to give a decent answer---in what fields are these not useful?

Analytic geometry and linear algebra--also invaluable. I agree you don't have to know how to compute things with great skill--but you have to have had the experience of computing them with a little skill so that you understand them conceptually, in your bones.