Add post on the dawning of the age of stochasticity

bib/bibliography.bib

@@ -615,3 +615,56 @@
   url =		 {https://doi.org/10.1080/09537289208919407},
   DATE_ADDED =	 {Thu Mar 18 13:53:35 2021},
 }
+
+@article{mumford2000_dawnin_age_stoch,
+  author =	 {Mumford, David},
+  title =	 {The Dawning of the Age of Stochasticity},
+  journal =	 {Atti della Accademia Nazionale dei Lincei. Classe di
+                  Scienze Fisiche, Matematiche e Naturali. Rendiconti
+                  Lincei. Matematica e Applicazioni},
+  volume =	 11,
+  pages =	 {107--125},
+  year =	 2000,
+  url =		 {http://eudml.org/doc/289648},
+  ISSN =	 {1120-6330; 1720-0768/e},
+  MSC2010 =	 {00A69 01A99 60A05 62A01},
+  Zbl =		 {1149.00309},
+  month =	 12,
+  publisher =	 {Accademia Nazionale dei Lincei},
+}
+
+@book{davis2012_mathem_exper_study_edition,
+  author =	 {Philip J. Davis and Reuben Hersh and Elena Anne
+                  Marchisotto},
+  title =	 {The Mathematical Experience, Study Edition},
+  year =	 2012,
+  publisher =	 {Birkh{\"a}user Boston},
+  url =		 {https://doi.org/10.1007/978-0-8176-8295-8},
+  doi =		 {10.1007/978-0-8176-8295-8},
+  isbn =	 978-0-8176-8294-1,
+  pages =	 500,
+  series =	 {Modern Birkh{\"a}user Classics},
+}
+
+@book{hacking2006_emerg_probab,
+  author =	 {Hacking, Ian},
+  title =	 {The Emergence of Probability: A Philosophical Study
+                  of Early Ideas about Probability, Induction and
+                  Statistical Inference},
+  year =	 2006,
+  publisher =	 {Cambridge University Press},
+  url =		 {https://doi.org/10.1017/CBO9780511817557},
+  doi =		 {10.1017/CBO9780511817557},
+  edition =	 2,
+  place =	 {Cambridge},
+}
+
+@book{alon2016_probab_method,
+  author =	 {Alon, Noga and Spencer, Joel H.},
+  title =	 {The Probabilistic Method},
+  year =	 2016,
+  publisher =	 {Wiley},
+  edition =	 {4th},
+  isbn =	 9781119061953,
+}
+

posts/dawning-of-the-age-of-stochasticity.org

@@ -0,0 +1,117 @@
+---
+title: "The Dawning of the Age of Stochasticity"
+date: 2022-03-12
+tags: maths, foundations, paper, statistics, probability
+toc: false
+---
+
+
+This article [@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
+better foundations.
+
+* The taxonomy of mathematics
+
+[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?]
+
+#+begin_quote
+The study of mental objects with reproducible properties is called mathematics.
+[@davis2012_mathem_exper_study_edition]
+#+end_quote
+
+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.
+
+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.
+
+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"?
+
+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".)
+
+* Putting random variables in the foundations
+
+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.
+
+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
+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.
+
+* 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
+theory.) Probabilistic methods also have a huge importance in the
+analysis of Partial 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.
+
+#+begin_quote
+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.
+#+end_quote
+
+* References