Mathematical Fads
Dr. Johnson, one of my favorite profs at UH, gave me a bibliography of fractional calculus. This was prompted by one of my essays for the NSF fellowship: I asked him to write a recommendation for me, and I mentioned that in my first class, we were supposed to learn about fractional derivatives, but never did. I think it’s cool that my comment moved him to supply me with a starting point for looking into it myself.
Apparently, fractional derivatives have been motivated by the question: “Can the meaning of derivatives of integral order 
be extended to have meaning when 
is any number, real, complex, and irrational?” Nothing under the sun is new: Liouville published several memoirs on the topic beginning back in 1832.
It’s interesting how I’ve never heard about fractional derivatives from anyone other than Johnson. Perhaps, as he intimated, the course of mathematics is influenced more by fads than any more objective reason. But then again, what would such an objective reason be? The most obvious candidate, scientific application, is not really as objective as it would seem: physical problems can usually be approached from several different directions, with correspondingly different tools.
I’ve been thinking about current mathematical fads I’m aware of: chaos theory, game theory, wavelets. I’m sure there are more. The question worth considering is, which will survive their 15 minutes of fame?
December 2nd, 2005 at 4:00 pm
So, out of curiosity, which reference(s) did you get? All I know is that Emil Post wrote an article on the subject of generalized differentiation in the beginning of the 20th century, while an undergrad at City College. (Though apparently not published until much later.)
December 4th, 2005 at 6:26 pm
More fads: mathematical biology? category theory? Also: stochastic processes: about 2 years ago, roughly one out of 2 fresh Phd holder that I met was doing his/her thesis on something related to stochastic processes. I don’t know if that still holds.
Consider this: nearly 30 years ago, numerical anaylsis was considered a fad. I don’t know how common it is nowadays to get a PhD in this subject, but it lasted long enough.
Another interesting question would be: what fields are “dead” (very loosely) nowadays? Can someone get a PhD in pure linear algebra these days?
December 12th, 2005 at 7:55 am
Wavelet theory might be new, but I doubt it will be a fad. It simply has too many useful applications in compression technologies nowadays. All the latest sound and video encoders are using the technique.
December 13th, 2005 at 12:58 pm
I agree, ObsessiveMathsFreak. But it seems that at least some of the time, the reason wavelets are used for some particular application is not because of a reasoned time-frequency analysis, but instead because they’re the latest greatest.