I am looking for examples of the following situation in mathematics:
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every object of type $X$ encountered in the mathematical literature, except when specifically attempting to construct counterexamples to this, satisfies a certain property $P$ (and, furthermore, this is not a vacuous statement: examples of objects of type $X$ abound);
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it is known that not every object of type $X$ satisfies $P$, or even better, that “most” do not;
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no clear explanation for this phenomenon exists (such as “constructing a counterexample to $P$ requires the axiom of choice”).
This is often presented in a succinct way by saying that “natural”, or “naturally occurring” objects of type $X$ appear to satisfy $P$, and there is disagreement as to whether “natural” has any meaning or whether there is any mystery to be explained.
Here are some examples or example candidates which come to my mind (perhaps not matching exactly what I described, but close enough to be interesting and, I hope, illustrate what I mean), I am hoping that more can be provided:
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The Turing degree of any “natural” undecidable but semi-decidable (i.e., recursively enumerable but not recursive) decision problem appears to be $\mathbf{0}'$ (the degree of the Halting problem): it is known (by the Friedberg–Muchnik theorem) that there are many other possibilities, but somehow they never seem to appear “naturally”.
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The linearity phenomenon of consistency strength of “natural” logical theories, which J. D. Hamkins recently gave a talk about (Naturality in mathematics and the hierarchy of consistency strength), challenging whether this is correct or even whether “naturality” makes any sense.
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Are there "natural" sequences with "exotic" growth rates? What metatheorems are there guaranteeing "elementary" growth rates? concerning the growth rate of “natural” sequences, which inspired the present question.
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The fact that the digits of irrational numbers that we encounter when not trying to construct a counterexample to this (e.g., $e$, $\pi$, $\sqrt{2}$…) experimentally appear to be equidistributed, a property which is indeed true of “most” real numbers in the sense of Lebesgue measure (i.e., a random real is normal in every base: those which are are a set of full measure) but not of “most” real numbers in the category sense (i.e., a generic real is not normal in any base: those which are are a meager set).
What other examples can you give of the “most $X$ do not satisfy $P$, but those that we actually encounter in real life always do (and the reason is unclear)” phenomenon?
Best Answer
Most finite groups empirically are 2-groups (in the sense of being a p-group with $p=2$ not in the other sense of the word). There are a lot of them. Conjecturally almost all finite groups are 2-groups. That is it is conjectured that if you count all groups up to isomorphism with at most $n$ elements, then the fraction of those which are 2-groups goes to 1 as n goes to infinity. In practice, while we often encounter small 2-groups and a few specific 2-groups like $(Z/(2Z))^k$, when dealing with "largish" finite groups all these weird 2-groups don't seem to often show up.