# Has a discrete/quantum theory of probability based on the Cournot-Borel principle ever been developed as a step towards Hilbert sixth problem?

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In 1930, Émile Borel, the father of measure theory together with H. Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement négligeables (On universally negligible probabilities) in Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 190, pp. 537-40. Here it is:

http://gallica.bnf.fr/ark:/12148/bpt6k3143v.f539

According to this question

https://mathoverflow.net/questions/236758/have-some-works-by-%C3%89mile-borel-ever-been-translated-from-french-to-english-or-an

and to the best of my knowledge, this note has never been translated in any foreign language. I would be happy to translate it entirely upon request despite my poor English.

Borel is concerned by Cournot principle. As the bridge, the connection between the mathematical theory of probability and the real world of experience, Borel considers Cournot principle to be the most important and fundamental principle of probability theory: he used to call it the fundamental law of randomness or the unique law of randomness. Hence, Borel seeks for a quantitative version of Cournot principle. He starts like this:

We know that, in the applications of the calculus of probability, when the probability becomes extremely close to unity, it can and must be practically confounded with certainty. Carnot principle, the irreversibility of many phenomena, are well-known examples in which the theoretical probability equals practical certainty. However, we may not have, at least to the best of my knowledge, sufficiently specified from which limits a probability becomes universally negligible, that is negligible in the widest limits of time and space that we can humanly conceive, negligible in our whole universe.

and concludes:

The conclusion that must be drawn is that the probabilities that can be expressed by a number smaller than ${10^{ - 1000}}$ are not only negligible in the common practice of life, but universally negligible, that is they must be treated as rigorously equal to zero [emphasized by Borel] in every questions regarding our Universe. The fact that they are not effectively null may be of interest for the metaphysicist; for the scientist they are [emphasized by Borel] null and the phenomena to which they relate are absolutely impossible [emphasized by Borel].

This Cournot-Borel principle

$\left\{ \begin{array}{l} p \in \left[ {{{0,10}^{ - 1000}}} \right]\;\;\;\;\,\; \Rightarrow p = 0\quad \quad {\rm{Borel - supracosmic}}\;{\rm{probabilities}}\\ p \in \left[ {1 - {{10}^{ - 1000}},1} \right] \Rightarrow p = 1\quad \quad \,{\rm{Borel - supercosmic}}\;{\rm{probabilities}} \end{array} \right.$

implies that there are only discrete probability measures/distributions in every probabilistic questions regarding our universe.

Indeed, consider for instance a cumulative distribution function $F\left( x \right):\mathbb{R} \to \left[ {0,1} \right]$. Suppose $F\left( x \right)$ is left-continuous at some point ${x_0}$, that is $F\left( x \right)$ is continuous at ${x_0}$ since it is right-continuous by definition:

$\forall \varepsilon > 0\;\exists \eta > 0,\forall x,{x_0} - \eta < x < {x_0} \Rightarrow \left| {F\left( x \right) - F\left( {{x_0}} \right)} \right| = F\left( {{x_0}} \right) - F\left( x \right) = {\text{Prob}}\left( {y \in \left[ {x,{x_0}} \right]} \right) = \mu \left( {\left[ {x,{x_0}} \right]} \right) < \varepsilon$

In particular, by the Cournot-Borel principle

$\forall {10^{ - 1000}} > \varepsilon > 0\;\exists \eta > 0,\forall x,{x_0} - \eta < x < {x_0} \Rightarrow \mu \left( {\left[ {x,{x_0}} \right]} \right) = 0$

Hence, either $F\left( x \right)$ is constant or it is discontinuous: $F\left( x \right)$ is nothing but a discrete cumulative distribution function or cumulative mass function.

Hence, following Borel, at least two different mathematical theories of probability would coexist: the mathematical, metaphysical, continuous one that relies heavily on measure theory, and the scientific, physical, discrete one where measure theory is almost irrelevant.

This Borel-Cournot discrete theory of probability is not necessarily inconsistent nor trivial because continuous r.v.s have discrete probability measures. By construction and definition, it constitutes another potential answer or proposal to Hilbert sixth problem or program. We can also talk about a quantum theory of (classical and quantum?) probability (not the theory of quantum probability) with Borel probabilistic quanta $b = {10^{ - 1000}}$, analogous to the energy quanta in QM.

Has something like this theory ever been developed? If not, why?
Should it be developed? If not, why?

edited Oct 8, 2017

In practice, one can very often use distributions, in particular Dirac $\delta$-functions, with the usual need to take care when using products of distributions. A distribution $\frac{1}{2}(\delta(x+1)+\delta(x-1))$ corresponds, for example, to a probability of $\frac{1}{2}$ of obtaining the result $-1$ and likewise for $+1$. The theory of distributions is fairly well established, although there are variants that take care of products in different ways.

@PeterMorgan

Discrete probability distributions can be seen as linear combinations of Dirac $\delta$ distributions/functions (Schwartz). In your exemple, we have a Bernoulli $B\left( {\frac{1}{2}} \right)$ discrete probability distribution.

In standard probability theory, discrete r.v.s. have discrete probability distributions and continuous r.v.s. have continuous probability distributions.

In this hypothetical Cournot-Borel discrete probability theory, even continuous r.v.s. would have discrete probability distributions and this is the reason why it is not necessarily trivial! We could also call it another kind  of quantum probability theory: we would have probabilistic quanta (${10^{ - 1000}}$).

Hoilbert's 6th problem as regards probability theory was solved to everyone's satisfaction by Kolmogorov.

As regards the axiomatization of ohysics, the problem is still open, as physics has expanded so much since Hilbert posed his problems. Today it would require to have axioms generalizing Wightman's  or Haag & Kastler's axioms for quantum field theory so that they would apply to the standard model.

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