Age of Stochasticity? 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!

 A: 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.
A: 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.) 
