# Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?

In 1930, Émile Borel, the father of measure theory together with his student 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

Have some works by Émile Borel ever been translated from French to English or another foreign language?

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, by a purely physical reasoning (emphases by Borel):

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 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 null and the phenomena to which they relate are absolutely impossible.

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. Of course, this value should be updated according to our modern knowledge about the Universe.

Has something like this theory ever been developed? I would be happy with non-probabilistic answers too, i.e. mathematical theories formalizing such a concept of finite resolution and indistinguishability. I found such a theory many years ago but I don't remember!

• C'est Carnot, non? Oct 4, 2017 at 15:13
• @YemonChoi. No, Cournot principle in probability theory and Carnot principle in thermodynamics. Oct 4, 2017 at 15:15
• I'm not an expert in history or measure theory, but I don't see why we should take it seriously. In my view, modern measure theory perfectly resolves the issue raised by Borel in these writings, by allowing sets with probability measure zero that are nonetheless "possible" (in the support). If one is concerned with practical physics rather than theoretical mathematics, there are many phenomena which are actually discrete but are modeled with continuous mathematics (e.g. a volume of liquid is actually a finite collection of molecules); why not probability as well?
– usul
Oct 4, 2017 at 16:01
• Why do you call a published paper of Borel a "confidential note"? Because it is published in French? Oct 4, 2017 at 17:26
• with respect to an English text by Borel on this topic, see chapter 3 "Negligible probabilities and the probabilities of everyday life" in the Dover edition "Probabilities and Life" (1962). Oct 4, 2017 at 18:19

This question belongs more to philosophy rather than mathematics, so it might be out of scope of this site. But the general answer is that mathematical models are only an approximation to reality, and we choose those approximations which are convenient. For example, we can discover sometimes that the space/time is not really continuous but consists of some discrete objects. But this will not make calculus based on the concept of real number useless or obsolete. One can argue without end whether real numbers really correspond to something "real". Nevertheless they are useful in physics and engineering. (See, for example, N. J. Wildberger, Real fish, real numbers, real jobs, The Mathematical Intelligencer, Volume 21, Issue 2, pp 4–7.)

Same with probability. Continuous distributions historically arise as approximations to discrete distributions (normal distribution is an approximation of the binomial distribution via the de Moivre-Laplace theorem). But normal distribution is much nicer from the mathematical point of view and therefore it must be used, even if when the "real" distribution under consideration is binomial.

In other words, the accepted axioms of Probability are chosen (from the several proposed systems) because of their mathematical convenience, rather than because they better approximate the "real world" from the philosophical point of view.

(Of the systems of mathematical foundations of probability which were competing with Kolmogorov's axioms, I can mention those proposed by S. Bernstein, R. von Mises and H. Steinhaus. And philosophic considerations played a secondary role in the choice).

Remark: Here is Wildberger's reading of his paper on Youtube, for those who have no subscription for Intelligencer.

• We apply probability theory to mathematical phenomena. In this case small probabilities occur. For example R. Brent, van de Lune, and I compute the probability of $|\arg(\zeta(\sigma+it)|>\pi/2$ for $\sigma=1.165$ as $1.279\dots\times10^{-283}$ and this really happens for some $t$. We can obtain smaller probabilities for other $\sigma>1$.
– juan
Oct 4, 2017 at 17:52
• @alexandre eremenko . Sorry if you don't find this thread mathematical enough. But there are tags for historical and philosophical aspects of maths in MO, so that I hope discussing "philosophical" works... by well-known mathematicians is not too much out of scope. Oct 4, 2017 at 18:32
• @alexandre eremenko. The problem is precisely that Borel philosophical considerations seemsto have some drastic mathematical consequences! Do you agree that this "Cournot-Borel principle" implies that there are only discrete probability measures in every questions regarding our universe??? Oct 4, 2017 at 18:35
• @alexandre eremenko. My question nevertheless pertains to Hilbert sixth problem. Discrete or not discrete, that is the question. Is quantifying Cournot principle not a kind of mathematical physics? I propose to cross-post the question on PO if MO doesn't mind. Oct 4, 2017 at 21:25
• @alexandre eremenko Some years ago, I bought an original copy of Borel Le hasard (Randomness), second edition, 1947. Believe it or not, the book was brand new, I cut the sheets by myself! Tried to find his complete works in 4 volumes but no way. Not exactly best sellers! Oct 4, 2017 at 22:48