Staring at one of the graphs in a recent post on the bit-player blog ("Googling the lexicon"), I was struck by a downward trend in the frequency of the phrase "catastrophe theory". That led me to do a few searches of my own on Google's n-gram viewer, which is one of those insidiously fascinating Internet gadgets that suck away so much productivity. (Who am I trying to kid? It's Christmas break; how productive was I going to be anyway??) Being a closeted mathematician, I first did a comparison of three recent "fads" (perhaps an unfair characterization) in math: catastrophe theory, chaos theory and fractal geometry.
Then I took a shot with a few terms from optimization: genetic algorithms, goal programming, integer programming and linear programming.
I find it a bit surprising that chaos theory gets so much more play than fractal geometry (which has some fairly practical applications, as Mike Trick noted in his eulogy for Benoit Mandelbrot). Two other things (one I find serendipitous, one less so) struck me about the graphs.
The serendipitous note is that references to catastrophe theory are tailing off. I recall sitting at a counter in a chain restaurant during my graduate student days, scarfing down breakfast while trying to chew my way through a journal article on catastrophe theory (which was rather new at that time). Differential equations (especially nonlinear PDEs) were not my strong suit, to put it charitably, and the article was heavy going. A few years later social scientists heard about catastrophe theory. As I recall, the canonical image for it was a trajectory for a PDE solution on a manifold that "fell off a cliff" (picture your favorite waterfall here). Social scientists are in the main clueless about differential equations (let alone PDEs on manifolds), but waterfalls they get. The next thing I knew, all sorts of political science and maybe anthropology papers (not sure about the latter) were saying "catastrophe theory this" and "catastrophe theory that", and I'm pretty sure they were abusing it rather roundly. So I interpret the decline in references as at least in part indicating it has reverted to its natural habitat, among a fairly small group of mathematicians who actually grok the math.
The less serendipitous note is that all the graphs are showing downward trends. Since the vertical axis is in percentage terms, I cannot attribute this to a reduction in the number of published books, nor blame it on the Blackberry/iPod/whatever technical revolution. My fear is that interest in mathematics in general may be on the decline. Perhaps what we need now is for the next installment of the "Harry Potter" series to have Harry defeating the Dark Lord by invoking not incantations but mathematical models (but not catastrophe theory, please).