Timeline for Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?
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| Jan 13, 2023 at 14:54 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| Oct 9, 2017 at 18:41 | comment | added | Fabrice Pautot | @alexandre eremenko As one example of the kind of mathematics we may miss if we don't care about the real world/physics, please see my unanswered question about probability theory and dynamical system theory: mathoverflow.net/questions/232043/… . If we could ever compute those probability distributions, we could solve many practical signal processing problems quite easily. But unfortunately, it seems that nobody can. | |
| Oct 9, 2017 at 9:05 | comment | added | Fabrice Pautot | @alexandre eremenko Given that Borel reasoning is purely physical, there is nothing metaphysical or philosophical in this story. Philosophical tag removed. I would be happy with non-probabilistic answers too, i.e. mathematical theories formalizing concepts such as finite resolution and indistinguishability. | |
| Oct 7, 2017 at 18:39 | comment | added | Fabrice Pautot | @juan Conversely, a pure mathematician does not need to know the reasons why a mathematical axiom like this Cournot-Borel principle was introduced in order to explore and develop the corresponding theory, isn't it? That's the reason why my question may be of interest for some pure mathematicians as well. But clearly, it should sound more interesting for those who care about the applications of maths to the real world. | |
| Oct 7, 2017 at 18:35 | comment | added | Fabrice Pautot | @juan For what set does your "we" stand please? As you know, Borel also made good use of PT in pure maths, including number theory like you (Borel normal numbers...). No doubt he would appreciate your nice results on the zeroes of the zeta function... that really happen in the mathematical world. But, as many others, Borel had much interest in the applications of mathematics too, not only in the physical sciences but also in game theory (Théorie mathématique du bridge). And not only PT but also geometry for instance (Introduction géométrique à quelques théories physiques). | |
| Oct 7, 2017 at 13:51 | comment | added | Fabrice Pautot | @alexandre eremenko A priori, any system of probability can be Cournot-Borel quantified. Moreover, quantification may depend on the interpretation of probability, e.g. fequentist or Bayesian... | |
| Oct 7, 2017 at 12:48 | comment | added | Fabrice Pautot | @alexandre eremenko We can mention A. Renyi's axiomatic system too, that accomodates for improper probability distributions (that would certainly disappear in the Cournot-Borel theory). | |
| Oct 7, 2017 at 12:43 | comment | added | Fabrice Pautot | @alexandre eremenko My favorite axiomatic system of probability is R. T. Cox's. Bernstein's is nice too, as far as I can remember. | |
| Oct 7, 2017 at 12:41 | comment | added | Fabrice Pautot | @alexandre eremenko As you pointed it out, just like space quantification can yield interesting theories like quantum loop gravity, even the purest mathematician can wonder if we can ever get something interesting (mathematically but not only) by quantifying probability as well, as suggested by Borel. Of course, just like QM does not make classical physics obsolete or irrelevant, quite the opposite, both continuous and quantum theories of probability are not mutually exclusive (a priori). | |
| Oct 5, 2017 at 16:57 | comment | added | Fabrice Pautot | @alexandre eremenko I did my best to make the question more mathematical. On the philosophical side, it is worth observing that Borel considers the current, continuous, measure-theoretic theory as the metaphysical, philosophical one. I'm asking for the discrete, quantum, scientific and physical one. | |
| Oct 4, 2017 at 23:55 | comment | added | Fabrice Pautot | @alexandre eremenko My question is 100% mathematical: take probability theory and add a new non-trivial axiom, the Cournot-Borel principle. | |
| Oct 4, 2017 at 22:48 | comment | added | Fabrice Pautot | @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 21:25 | comment | added | Fabrice Pautot | @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 18:35 | comment | added | Fabrice Pautot | @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 18:32 | comment | added | Fabrice Pautot | @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 18:01 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
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| Oct 4, 2017 at 17:52 | comment | added | juan | 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 $\sigma>1$. | |
| Oct 4, 2017 at 17:23 | history | edited | Alexandre Eremenko | CC BY-SA 3.0 |
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| Oct 4, 2017 at 17:15 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |