One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here.

The question is this:

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!

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    $\begingroup$ I think the ultimate crux is whether doing so (putting stochasticity at the foundation) will make any difference to the real issue: the connection with computability and computational complexity---because in the "dawn of stochasticity" the real gains that we care about are often of a computational nature (expected value or high probability answers rather than exact, expensive ones, etc.)...but really nice question worth pondering about. $\endgroup$ – Suvrit Apr 18 '13 at 15:19
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    $\begingroup$ In the last few years there has been a fascinating trend in theoretical computer science called "probabilistic programming." It is a huge step towards stochastic foundations with the computational complexity aspects in mind. If people are interested they should look at this wonderful survey article: danroy.org/papers/FreRoyTen-Turing.pdf $\endgroup$ – Jeremy Hahn Apr 20 '13 at 0:43

Here's an example of something that I think Mumford might advocate in the foundations of mathematics: Solovay's model.

The axiom of choice is generally accepted by mathematicians, but it has always suffered from the nagging problem that it violates certain intuitions we have. Almost all these counterintuitive consequences of the axiom of choice are related in one way or another to the existence of non-measurable sets. (See this related MO question for more information, in particular Ron Maimon's answer.) Solovay's model shows that we can come close to having our cake and eating it too: We can simultaneously have the axioms "all Lebesgue sets are measurable" and the axiom of dependent choice. The former pretty much eliminates all the probabilistic paradoxes while the latter gives us almost all of the "desirable" consequences of the axiom of choice.

The reason that I think this is the sort of thing Mumford might advocate is that Mumford's discussion of Freiling's theorem shows that he really wants to preserve probabilistic intuition even at the expense of jettisoning a well-accepted axiom. In the paper he suggests getting rid of the power-set axiom, but my guess is he was probably not familiar with Solovay's model at the time, and if he were, he would have been favorably disposed towards it.

EDIT: In particular, in Solovay's model, all the following hold: (1) the axioms of ZF, including powerset; (2) all sets are Lebesgue measurable (which is most of what we need to capture probabilistic intuitions); (3) Freiling's axiom of symmetry; (4) the continuum hypothesis in the form "every uncountable subset of $\mathbb R$ can be put into 1-1 correspondence with $\mathbb R$." The only price one pays is that the axiom of choice has to be weakened to dependent choice. (Thanks to Ali Enayat for pointing this out.) My view is that Freiling's argument shows only that probabilistic intuition is incompatible with full-blown AC (which is something we knew already); the continuum hypothesis is a red herring.
For more information about the practical impact of adopting Solovay's model and some speculation on why it hasn't already been adopted widely, see this MO question and Andreas Blass's answer to this MO question.

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Here is a result that gives the flavor of the kind of thing along these lines I hope to see in the future. Recall Tarski's undefinability of truth: under suitable assumptions, a formal system can't be equipped with a truth predicate $\text{True}$ such that $\text{True}(G)$ if and only if $G$ is true. The reason is that under suitable assumptions, we can write down a sentence $G$ which is equivalent to $\text{True}(\neg G)$ (the liar paradox), and then we obtain a contradiction.

Christiano, Yudkowsky, Herreshoff, and Barasz recently showed, however, that a formal system can be equipped with a probability predicate $\mathbb{P}(G)$ satisfying a weaker reflection principle, namely that

$$\mathbb{P}(G) \in (a, b) \Leftrightarrow \mathbb{P}(\mathbb{P}(G) \in (a, b)) = 1.$$

The corresponding probability assignments to sentences may be thought of as probability distributions over models of some theory. See the draft for more details. (Disclaimer: I was involved in a small way with a workshop one of whose goals was to see how far this result could be pushed.)

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