From d4c1f12c25f786e29483115faf7c9b20a5f12175 Mon Sep 17 00:00:00 2001 From: Dimitri Lozeve Date: Thu, 24 Mar 2022 22:07:24 +0100 Subject: [PATCH] Update post on the dawning of the age of stochasticity --- posts/dawning-of-the-age-of-stochasticity.org | 95 +++++++++++++------ 1 file changed, 67 insertions(+), 28 deletions(-) diff --git a/posts/dawning-of-the-age-of-stochasticity.org b/posts/dawning-of-the-age-of-stochasticity.org index 915e719..1e3491d 100644 --- a/posts/dawning-of-the-age-of-stochasticity.org +++ b/posts/dawning-of-the-age-of-stochasticity.org @@ -1,12 +1,18 @@ --- title: "The Dawning of the Age of Stochasticity" -date: 2022-03-12 +date: 2022-03-24 tags: maths, foundations, paper, statistics, probability toc: false --- +#+begin_quote +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. +#+end_quote -This article [@mumford2000_dawnin_age_stoch] is an interesting call +This article [cite:@mumford2000_dawnin_age_stoch] 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 @@ -17,11 +23,11 @@ better foundations. [fn::{-} 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?] +would the study of /non-reproducible/ mental objects be?] #+begin_quote The study of mental objects with reproducible properties is called mathematics. -[@davis2012_mathem_exper_study_edition] +[cite:@davis2012_mathem_exper_study_edition] #+end_quote What are the categories of reproducible mental objects? Mumford @@ -37,10 +43,16 @@ be based. People are constantly weighing likelihoods, evaluating plausibility, and sampling from posterior distributions to refine estimates. -He then makes a quick historical overview of the principal notions of -probability, which mostly mirror the detailed historical perspective -in @hacking2006_emerg_probab. There is also a short summary of the -work into the foundations of mathematics. +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 [cite:@hacking2006_emerg_probab]. 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, @@ -57,42 +69,66 @@ 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. (Similar to -the notion of "set", "collection", or "category".) +except from a purely formal and axiomatic point of view. -* Putting random variables in the foundations +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. -The usual way of defining random variables is : predicate logic → sets -→ natural numbers → real numbers → measures → random -variables. Instead, we could put random variables at the foundations, -and define everything else in terms of that. +* Putting random variables into the foundations + +The usual way of defining random variables is: +1. predicate logic, +2. sets, +3. natural numbers, +4. real numbers, +5. measures, +6. 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 Jaynes, who has a complete formalism of -Bayesian probability from a notion of "plausibility". With a few -axioms, we can obtain an isomorphism between a vague notion of +developments. One is from [[https://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][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. +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 -I think this is by far the most convincing argument to give a greater -importance to probability and statistics 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 -@alon2016_probab_method and [[https://www.college-de-france.fr/site/timothy-gowers/index.htm][Timothy Gowers' lessons at the Collège de -France]] on the probabilistic method for combinatorics and number +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 [cite:@alon2016_probab_method] and [[https://www.college-de-france.fr/site/timothy-gowers/index.htm][Timothy Gowers' lessons +at the Collège de France]][fn::In French, but see also [[https://www.youtube.com/c/TimothyGowers0][his YouTube +channel]].] on the probabilistic method for combinatorics and number theory.) Probabilistic methods also have a huge importance in the -analysis of Partial differential equations, chaos theory, and -mathematical physics in general. +analysis of differential equations, chaos theory, and mathematical +physics in general. * Thinking as Bayesian inference @@ -104,6 +140,9 @@ 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: + #+begin_quote My overall conclusion is that I believe stochastic methods will transform pure and applied mathematics in the beginning of the third