Talk:Mathematical coincidence
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A few comments. Maybe mathematical coincidence might yet be a better page name. I'm not sure what a 'theoretical explanation' would be, in all cases: but certainly in the case of exp(π√137), if I have that right, there is a very good if hidden reason. Perhaps the pigeonhole principle could be invoked. After all, it is because of Dirichlet's theorem on diophantine approximation, based on it, that we know that good rational approximations exist. And so on, for other kinds of relations. By the way, I added the law of small numbers link, but that really needs disambiguation since the Poisson distribution meaning is a separate thing.
Charles Matthews 16:10, 21 Jun 2004 (UTC)
Hi Charles
you're probably right about the page name. I chose the name originally because I reckoned that there would be hundreds and hundreds. In the end, the best I could do wasn't very extensive! There must be more out there! (exactly the same thing happened with List of scientific howlers in literature)
- The scientific howlers list is likely to grow, slowly but surely. Dpbsmith 22:58, 21 Jun 2004 (UTC)
Anyway, the exact definition of a mathematical coincidence is problematical. I spent some considerable time pondering the best definition, only to get myself tangled up in philosophy, and at one point defining everything to be a coincidence. Of course <math>\pi^2\simeq 10<math>! How could it possibly be otherwise? Nevertheless, there is definitely _something_ that all the examples have in common. Except maybe the exp(pi*sqrt(137)) one, which as you point out is a result of some algebraic number theory (all of which I forgot immediately after my finals). And possibly the continued fraction ones, although it _is_ coincidental that the continued fraction for pi has a large coefficient early on (isn't it?)
Best wishes
Robinh 20:51, 21 Jun 2004 (UTC)
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From a concerned mathematician
I am a mathematician who studies (among other things) topological (Nielsen) coincidence theory. The word "coincidence" has a widely accepted and recognized technical meaning: given two functions f and g, the Coincidence Set Coin(f,g) is the set of points x such that f(x) = g(x).
This is important as it is (perhaps) the most natural generalization of fixed point (mathematics) theory (where g is taken to be the identity map) and is the original setting for many results (e.g. the Lefschetz fixed-point theorem) which are now commonly misidentified as being fixed point results. (Lefschetz proved first the coincidence version of his result, and noted the fixed point version as a specific case.)
Coincidence theory also has an important reduction when the function g is taken to be a constant map- this setting is called root theory (finding points x with f(x) = c for some constant c).
As a mathematician, I feel like a wikipedia page on coincidences should refer to the above concept. The content now described as "mathematical coincidence" would perhaps be more accurately described as "mathematical curiosity".
My $0.02. Chris Staecker, UCLA Math Dept
Two levels of "coincidence"
It seems to me that there's an important distinction between "coincidences" that arise out of a general theory, like the one about exp(pi sqrt 163), and ones for which no explanation of any kind is known. To take a few examples from the page as of 2005-04-10:
- e^pi ~= pi^e is true because pi is close to e and x / log x is stationary at e. Not much of a coincidence, really; the same would be just as true if 3 or sqrt(10) were used in place of pi.
- pi ~= 355/113 is a "real" coincidence, so far as I know: that is, no reason is known why pi should have so large a coefficient so early in its continued fraction.
- sqrt(2) ~= 17/12 is not much of a coincidence; there's a general theorem that says that roots of quadratic equations always have rational approximations that are at least about that good.
- exp(pi) ~= pi+20 seems to be a "real" coincidence.
- 1 mile / 1 km ~= phi is an absolute coincidence, if anything is. (Incidentally, it might be worth mentioning the Zeckendorff representation in connection with this...)
Is it worth making this distinction on the page? Gareth McCaughan 13:48, 2005 Apr 10 (UTC)
The title of this page
How about "numerical coincidences"? Gareth McCaughan 13:48, 2005 Apr 10 (UTC)
these are not coincidences
None of these can be classified as coincidences, except maybe
- <math>e^\pi\simeq\pi^e<math>. -Hmib 02:48, 20 Jun 2005 (UTC)