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Update post on the dawning of the age of stochasticity

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--- ---
title: "The Dawning of the Age of Stochasticity" title: "The Dawning of the Age of Stochasticity"
date: 2022-03-12 date: 2022-03-24
tags: maths, foundations, paper, statistics, probability tags: maths, foundations, paper, statistics, probability
toc: false 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 for a new set of foundations of mathematics on probability and
statistics. It argues that logic has had its time, and now we should 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 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 [fn::{-} This is probably the best definition of mathematics I have
seen. Before that, the most satisfying definition was "mathematics is seen. Before that, the most satisfying definition was "mathematics is
what mathematicians do". It also raises an interesting question: what 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 #+begin_quote
The study of mental objects with reproducible properties is called mathematics. The study of mental objects with reproducible properties is called mathematics.
[@davis2012_mathem_exper_study_edition] [cite:@davis2012_mathem_exper_study_edition]
#+end_quote #+end_quote
What are the categories of reproducible mental objects? Mumford 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 plausibility, and sampling from posterior distributions to refine
estimates. estimates.
He then makes a quick historical overview of the principal notions of As such, random variables are rooted in our inspection of our own
probability, which mostly mirror the detailed historical perspective mental processes, in the self-conscious analysis of our minds. Compare
in @hacking2006_emerg_probab. There is also a short summary of the to areas of mathematics arising from our experience with the physical
work into the foundations of mathematics. 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 Mumford also claims that although there were many advances in the
foundation of probability (e.g. Galton, Gibbs for statistical physics, 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"? * 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 Here, Mumford discusses the various definitions we can apply to the
notion of random variable, from an intuitive and a formal point of notion of random variable, from an intuitive and a formal point of
view. The conclusion is essentially that a random variable is a view. The conclusion is essentially that a random variable is a
complex entity that do not easily accept a satisfying definition, complex entity that do not easily accept a satisfying definition,
except from a purely formal and axiomatic point of view. (Similar to except from a purely formal and axiomatic point of view.
the notion of "set", "collection", or "category".)
* 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 * Putting random variables into the foundations
→ natural numbers → real numbers → measures → random
variables. Instead, we could put random variables at the foundations, The usual way of defining random variables is:
and define everything else in terms of that. 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 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 that it is possible. However, to make his case, Mumford presents two
developments. One is from Jaynes, who has a complete formalism of developments. One is from [[https://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E. T. Jaynes]], who has a complete formalism
Bayesian probability from a notion of "plausibility". With a few of Bayesian probability from a notion of "plausibility". With a few
axioms, we can obtain an isomorphism between a vague notion of axioms, we can obtain an isomorphism between an intuitive notion of
plausibility and a true probability function. plausibility and a true probability function.
The other example is a proof that the continuum hypothesis is false, 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 * Stochastic methods have invaded classical mathematics
I think this is by far the most convincing argument to give a greater This is probably the most convincing argument to give a greater
importance to probability and statistics methods in the foundations of importance to probability and statistical methods in the foundations
mathematics: there tend to be everywhere, and extremely productive. A of mathematics: there tend to be everywhere, and extremely
prime example is obviously graph theory, where the "probabilistic productive. A prime example is obviously graph theory, where the
method" has had a deep impact, thanks notably to Erdős. (See "probabilistic method" has had a deep impact, thanks notably to
@alon2016_probab_method and [[https://www.college-de-france.fr/site/timothy-gowers/index.htm][Timothy Gowers' lessons at the Collège de Erdős. (See [cite:@alon2016_probab_method] and [[https://www.college-de-france.fr/site/timothy-gowers/index.htm][Timothy Gowers' lessons
France]] on the probabilistic method for combinatorics and number 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 theory.) Probabilistic methods also have a huge importance in the
analysis of Partial differential equations, chaos theory, and analysis of differential equations, chaos theory, and mathematical
mathematical physics in general. physics in general.
* Thinking as Bayesian inference * 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 link with formal definitions of induction, such as PAC learning, which
is very present in machine learning. is very present in machine learning.
I'll conclude this post by quoting directly the last paragraph of the
article:
#+begin_quote #+begin_quote
My overall conclusion is that I believe stochastic methods will My overall conclusion is that I believe stochastic methods will
transform pure and applied mathematics in the beginning of the third transform pure and applied mathematics in the beginning of the third