A question that has always fascinated me about mathematics is that of how the field manages to stay healthy and not degenerate in the way I’ve seen theoretical physics do as it lost new input from experiment. On Twitter, Ash Joglekar gave a wonderful quote from von Neumann that addresses this question. The quote was from a 1947 essay “The Mathematician” (available here and here). von Neumann argues that:
…mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science.
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 l’art pour l’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.
which describes all too well what has happened to string theory. What saves a field from this? “Men with an exceptionally well-developed taste”? He poses the general question this way:
What is the mathematician’s normal relationship to his subject? What are his criteria of success, of desirability? What influences, what considerations, control and direct his effort?
Normally mathematicians are loath to debate this kind of “soft” topic, but the rise of computer software capable of producing proofs has recently led several first-rate mathematicians to take an interest. Each year the Fields Institute in Toronto organizes a Fields Medal Symposium, structured around the interests of a recent Fields Medalist. This year it’s Akshay Venkatesh, and the symposium will be devoted to questions about the changing nature of mathematical research, specifically the implications of this kind of computer software. Last year Venkatesh wrote an essay exploring the possible significance of the development of what he called “Alephzero” (denoted $\aleph(0)$):
Our starting point is to imagine that $\aleph(0)$ teaches itself high school and college mathematics and works its way through all of the exercises in the Springer-Verlag Graduate Texts in Mathematics series. The next morning, it is let loose upon the world – mathematicians download its children and run them with our own computing resources. What happens next – in the subsequent decade, say?
Among the organizers of the conference is Michael Harris, who has written extensively about mathematical research and issues of value in mathematics. Recently he has been writing about the computer program question at his substack Silicon Reckoner, with the most recent entry focusing on Venkatesh’s essay and the upcoming symposium.
One of the speakers at the symposium will be Fields medalist Tim Gowers, who will be addressing the “taste” issue with Is mathematical interest just a matter of taste?. Gowers is now at the Collège de France, where he is running a seminar on La philosophie de la pratique des mathématiques.
I’ve tried asking some of my colleagues what they think of all this activity, most common response so far is “why aren’t they proving theorems instead of spending their time talking about this?”
Update: For yet more about this happening at the same time, there’s a talk this afternoon by Michael Douglas on “How will we do mathematics in 2030?”.
Update: The talks from the Fields Institute program are now available online.
Terry Tao is one of the organizers of a planned February workshop at UCLA involving many of the same people, much the same topic.