Earlier this term I had a chance to speak with Robert Friedman, a distinguished mathematician here at Columbia. During our conversation I suggested to Professor Friedman that it is important for philosophy to maintain contact with other disciplines and that surely he was familiar with analogous issues within mathematics -- lest you think that distinguished mathematicians at Columbia walk over to Philosophy Hall to seek the opinions of junior faculty, let me assure you that this was not the case on this occasion. Anyway, it was in the context of this exchange that I referred Professor Friedman to the following passage by von Neumann:
As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I'art pour I'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. (from "The Mathematician")
I'm not sure if Professor Friedman shares concerns of the sort that are suggested in this quote -- concerned that I had said something inappropriate, I reminded him of the following joke:
Q: What is the difference between a mathematician and a philosopher?A: The mathematician only needs paper, pencil, and a trash bin for his work - the philosopher can do without the trash bin...
No, I don't think that this is the difference between mathematicians and philosophers, but it is a nice joke to have in your pocket if you are a junior philosopher in need of a little self-deprecation while in the company of a distinguished mathematician at your university.