Today I came across D. Mumford's 1999 article The Dawning of the Age of Stochasticity, which is quite remarkable even after more than a decade. The title already indicates the theme, but I copy the abstract for the convenience of the reader:
For over two millennia, Aristotle's logic has ruled over the thinking of western intellectuals. All precise theories, all scientific models, even models of process of thinking itself, have in principle conformed to the straight-jacket of logic. But from its shady beginnings devising gambling strategies and counting corpses in medieval London, probability theory and statistical interference now emerges as better foundations for scientific models, especially those of the process of thinking and as essential ingredients of theoretical mathematics, even foundation of mathematics itself. We propose that this sea change in our perspective will affect virtually all of mathematics in the next century.
In the article he proposes a new approach to mathematical science, putting random variables and stochasticity into foundations of mathematics (rather than building them upon measure theory), especially in theory of differential equations and artificial intelligence.
I am wondering how is this program going? I know something about stochastic differential equations from finance, and I know probability theory is fundamental to machine learning and artificial intelligence.
However, it seems to me stochasticity is still far from the foundations of mathematics, and much mathematics is still ruled by logic. Of course as an undergraduate maybe I am just too far from the frontier.
So can someone tell me how is this program going? Is it really some advantage in this new approach Mumford proposed?
Thanks very much!