At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate. (Terence Tao)
Someone had told Linnik the news beginning with one of the corollaries: Matiyasevich can construct a polynomial with integer coefficients such that the set of all natural number values assumed by this polynomial for natural number values of the variables is exactly the set of all primes.” “That’s wonderful,” Linnik replied. “Most likely we soon shall learn a lot of new things about primes.” Then it was explained to him that the main result is in fact much more general: Such a polynomial can be constructed for every recursively enumerable set, i.e., a set the elements of which can be listed in some order by an algorithm. “It’s a pity,” Linnik said. “Most likely, we shall not learn anything new about the primes.” (Yuri Matiyasevich)
The very possibility of mathematical science seems an
insoluble contradiction. If this science is only deductive
in appearance, from whence is derived that perfect rigour
which is challenged by none? If, on the contrary, all the
propositions which it enunciates may be derived in order
by the rules of formal logic, how is it that mathematics
is not reduced to a gigantic tautology? (Henri Poincaré)
Questions of this kind show just how incomplete a picture of mathematics was provided by some philosophers early in the twentieth century, who held that it was merely a giant collection of tautologies. The way that tautologies relate to each other is fascinating and important, so even if one believes that mathematics consists of tautologies, the “merely” is unacceptable. (Timothy Gowers)