Talk:List of probabilistic proofs of non-probabilistic theorems

Maybe this article is somewhat biased, just because I am acquainted with all my works but only a part of works of others. Hopefully, other people will add other examples, and the bias will disappear. That is wiki... Boris Tsirelson (talk) 21:42, 4 December 2008 (UTC)

Question about the normal number theorem
I don't think the normal number theorem is appropriate since the definition of normal numbers is inherently probabilistic. GeometryGirl (talk) 14:04, 7 December 2008 (UTC)


 * Yes, I feel this problem. However: (1) the definition is only inspired by probability theory; the question whether or not a single such number exists, is formally not probabilistic. (2) One could ask less, say, only that every digit appears infinitely often (not necessarily with right frequency); then the probabilistic flavor disappears (or almost disappears?) but I guess we still have no other proof. Boris Tsirelson (talk) 14:49, 7 December 2008 (UTC)

More items?
Should Loewner's torus inequality and Weierstrass approximation theorem be added? Michael Hardy (talk) 17:54, 22 December 2008 (UTC)


 * Indeed. Thank you! Weierstrass approximation theorem, I knew about it (and several other cases where probabilistic proofs exist but have little or no advantages over non-probabilistic proofs). In contrast, Loewner's torus inequality is new for me, and quite interesting. I'll be glad to add it (unless you like to do it yourself). But also I'll add a number of simpler things, including Weierstrass approximation theorem. (Again, feel free to act yourself, too.) Boris Tsirelson (talk) 19:29, 22 December 2008 (UTC)


 * Both are added (and some others, too). Boris Tsirelson (talk) 21:03, 25 December 2008 (UTC)

Only probabilistic proofs of non-probabilistic theorems, please
I revert the recent contribution by 81.241.200.112, since probability heuristics, interesting or not, do not fit into this article. Sorry. Boris Tsirelson (talk) 07:09, 14 January 2009 (UTC)

To User:Michael Hardy: do you really think that Probabilistic interpretation of Taylor series is a probabilistic proof of a non-probabilistic theorem? That is, of Taylor formula? I think that it is a simple and clear analytic proof, translated into probabilistic language by trivially converting integrals into expectations. What is it good for? Boris Tsirelson (talk) 19:13, 19 January 2009 (UTC)

I revert again; the theorem formulated recently by User:Michael Hardy cannot be treated as an example of non-probabilistic theorems (since its formulation involves probabilistic notions). Boris Tsirelson (talk) 16:22, 29 April 2009 (UTC)

Computable normal numbers
How does the existence of computable normal numbers follow from the fact that random numbers are normal? Algebraist 16:26, 18 April 2009 (UTC)


 * You are right; not quite from the fact that random numbers are normal, but from a bit stronger fact, namely, from some constructive estimations of probabilities. The fact that random numbers are normal follows immediately from these estimations. The existence of computable normal numbers follows from these estimations, but not immediately; additional arguments are involved. Boris Tsirelson (talk) 16:42, 18 April 2009 (UTC)
 * How should this be expressed in the article (if at all)? Algebraist 22:45, 18 April 2009 (UTC)
 * Please look now. I treat the items in this article as pointers rather than presentations. But of course you may disagree; and indeed, they should not be misleading. Boris Tsirelson (talk) 09:59, 19 April 2009 (UTC)
 * Looks good. Algebraist 13:07, 19 April 2009 (UTC)

More examples, and sections
There are more examples on the Math Overflow page, to which I just added a link (which also links back here). It would be good to add some of them, especially those for which we have articles. Also, this was a somewhat long list, so I have (very carelessly) split it into sections for various areas of mathematics. This is inaccurate, so feel free to revert, but feel freer to categorise them properly. Regards, Shreevatsa (talk) 17:46, 4 May 2010 (UTC)

Twice the normal numbers
Before, a doubt was expressed, whether normal numbers fit here. But now they appear twice: in Analysis (the very first example), and in Number theory (added recently). Why? Boris Tsirelson (talk) 17:47, 16 July 2011 (UTC)
 * My bad, I thought naively it belonged to number theory, and I did not see it in the Amalysis section, though it has the first place there. By the way, I don't feel normality is a probabilistic property by nature, as opposed to geometry girl, though I see her point : does the observation of frequencies characterize  probabilistic  properties ? Perhaps ...  But I think it is even more essential and preexistent to probability theory, or is it not ? . And, by the way, I like this page, and I would like to have a similar one in the French Wikipedia.  Am I wrong or is the Borel result about normal numbers the first occurrence of Borel Cantelli Lemma, the first occurrence of a strong law of large numbers, simultaneously, and perhaps even the first occurrence of the probabilistic method promoted later by Erdos ? Just asking, since I have an expert available, seemingly ... Pardon me if I am over-enthusiastic, but I discovered the role of Borel in this result and the date of it only recently ...  Chassain (talk) 08:26, 17 July 2011 (UTC)
 * Expert, well, in probability theory, but not in its history, sorry; try to ask someone else about "the first occurrence". To number theory? As far as I understand, number theory is the theory of arithmetical properties of natural numbers, not real numbers. Boris Tsirelson (talk) 08:54, 17 July 2011 (UTC)