Mumford, David. 2000. “The Dawning of the Age of Stochasticity.” Atti Della Accademia Nazionale Dei Lincei. Classe Di Scienze Fisiche, Matematiche E Naturali. Rendiconti Lincei. Matematica E Applicazioni 11: 107–25. http://eudml.org/doc/289648.
This article (Mumford 2000) is an interesting call for a new set of foundations of mathematics on probability and statistics. It argues that logic has had its time, and now we should make random variables a first-class concept, as they would make for better foundations.
The taxonomy of mathematics
This is probably the best definition of mathematics I have
seen. Before that, the most satisfying definition was “mathematics is
what mathematicians do”. It also raises an interesting question: what
would the study of non-reproducible mental objects be?
The study of mental objects with reproducible properties is called mathematics. (Davis, Hersh, and Marchisotto 2012)
What are the categories of reproducible mental objects? Mumford considers the principal sub-fields of mathematics (geometry, analysis, algebra, logic) and argues that they are indeed rooted in common mental phenomena.
Of these, logic, and the notion of proposition, with an absolute truth value attached to it, was made the foundation of all the others. Mumford’s argument is that instead, the random variable is (or should be) the “paradigmatic mental object”, on which all others can be based. People are constantly weighing likelihoods, evaluating plausibility, and sampling from posterior distributions to refine estimates.
As such, random variables are rooted in our inspection of our own mental processes, in the self-conscious analysis of our minds. Compare to areas of mathematics arising from our experience with the physical world, through our perception of space (geometry), of forces and accelerations (analysis), or through composition of actions (algebra).
The paper then proceeds to do a quick historical overview of the principal notions of probability, which mostly mirror the detailed historical perspective in (Hacking 2006). There is also a short summary of the work into the foundations of mathematics.
Mumford also claims that although there were many advances in the foundation of probability (e.g. Galton, Gibbs for statistical physics, Keynes in economics, Wiener for control theory, Shannon for information theory), most important statisticians (R. A. Fisher) insisted on keeping the scope of statistics fairly limited to empirical data: the so-called “frequentist” school. (This is a vision of the whole frequentist vs Bayesian debate that I hadn’t seen before. The Bayesian school can be seen as the one who claims that statistical inference can be applied more widely, even to real-life complex situations and thought processes. In this point of view, the emergence of the probabilistic method in various areas of science would be the strongest argument in favour of bayesianism.)
What is a “random variable”?
Random variables are difficult to define. They are the core concept of any course in probability of statistics, but their full, rigorous definition relies on advanced measure theory, often unapproachable to beginners. Nevertheless, practitioners tend to be productive with basic introductions to probability and statistics, even without being able to formulate the explicit definition.
Here, Mumford discusses the various definitions we can apply to the notion of random variable, from an intuitive and a formal point of view. The conclusion is essentially that a random variable is a complex entity that do not easily accept a satisfying definition, except from a purely formal and axiomatic point of view.
This situation is very similar to the one for the notion of “set”. Everybody can manipulate them on an intuitive level and grasp the basic properties, but the specific axioms are hard to grasp, and no definition is fully satisfying, as the debates on the foundations of mathematics can attest.
Putting random variables into the foundations
The usual way of defining random variables is:
- predicate logic,
- natural numbers,
- real numbers,
- random variables.
Instead, we could put random variables at the foundations, and define everything else in terms of that.
There is no complete formulation of such a foundation, nor is it clear that it is possible. However, to make his case, Mumford presents two developments. One is from E. T. Jaynes, who has a complete formalism of Bayesian probability from a notion of “plausibility”. With a few axioms, we can obtain an isomorphism between an intuitive notion of plausibility and a true probability function.
The other example is a proof that the continuum hypothesis is false, using a probabilistic argument, due to Christopher Freiling. This proof starts from a notion of random variable that is incompatible with the usual definition in terms of measure theory. However, this leads Mumford to question whether a foundation of mathematics based on such a notion could get us rid of “one of the meaningless conundrums of set theory”.
Stochastic methods have invaded classical mathematics
This is probably the most convincing argument to give a greater
importance to probability and statistical methods in the foundations
of mathematics: there tend to be everywhere, and extremely
productive. A prime example is obviously graph theory, where the
“probabilistic method” has had a deep impact, thanks notably to
Erdős. (See (Alon and Spencer 2016) and Timothy Gowers’ lessons at the Collège de FranceIn French, but see also his YouTube channel.
on the probabilistic method for combinatorics and number theory.) Probabilistic methods also have a huge importance in the analysis of differential equations, chaos theory, and mathematical physics in general.
Thinking as Bayesian inference
I think this is not very controversial in cognitive science: we do not think by composing propositions into syllogisms, but rather by inferring probabilities of certain statements being true. Mumford illustrates this very well with an example from Judea Pearl, which uses graphical models to represent thought processes. There is also a link with formal definitions of induction, such as PAC learning, which is very present in machine learning.
I’ll conclude this post by quoting directly the last paragraph of the article:
My overall conclusion is that I believe stochastic methods will transform pure and applied mathematics in the beginning of the third millennium. Probability and statistics will come to be viewed as the natural tools to use in mathematical as well as scientific modeling. The intellectual world as a whole will come to view logic as a beautiful elegant idealization but to view statistics as the standard way in which we reason and think.