Mathematics as culture and knowledge

Mathematics is an intellectual activity, arguably one of the most sophisticated ever produced by human civilization. Hermann Hesse sketched a portrait of the activities of mathematicians through the metaphor of the Glass Bead Game. That was perhaps the best literary attempt at catching a glimpse of some inner workings of the society of mathematicians. One does not blame a fictional work for inaccuracy, but it is indeed a difficult task to say something meaningful about what it is like to do mathematics. There are quite a few mathematicians who entertain a Platonist view of mathematics. By this I mean essentially the belief that mathematical objects and constructs enjoy some kind of existence in a “world of ideas”, independent of the human mind. As in the case of the mythical heaven, the proposers of such beliefs tend to be quite vague on the location and consistency of this extra-mental Platonic world. One reason that is often produced in support of the Platonic standpoint is the effectiveness of mathematics in modeling the physical world. No doubt that Kepler’s laws would eventually be observed and understood by any technological intelligence living on a planet bound by gravity to revolve around a star (but would such a discovery follow the course we know had the planet been revolving around two stars?) but one can hardly make an equally strong case in support of other very beautiful but far more abstract branches of mathematics. If no one would perhaps doubt that any sufficiently evolved extraterrestrial intelligence would understand the notion of prime number, there is far less compelling evidence that they would have our same notions of derived categories or shtukas. Recent years have gotten us used to more and more sophisticated mathematics being recruited to account for increasingly complicated models of high-energy physics. This sort of evidence notwithstanding, I personally remain extremely skeptical about the Platonist hypothesis.