... there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified as "pure mathematicians" (though "purist" would be more adequate) while the second is, with equal inadequacy, referred to as "applied".
Quoted in R W Ritchie, New directions in mathematics
Steinhaus, with his predilection for metaphors, used to quote a Polish proverb, 'Forturny kolem sie tocza' [Luck runs in circles], to explain why π, so intimately connected with circles, keeps cropping up in probability theory and statistics, the two disciplines which deal with randomness and luck.
Enigmas of Chance
...to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.
Statistical Independence in Probability Analysis and Number Theory
To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.
Mark Kac and Stanislaw Ulam, Mathematics and Logic (1968)