May 28, 2009
What Kinds of Math Do We Need?
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?