  To illustrate the difference between statements in mathematics proper and the philosophy of mathematics, consider as an example of the former the Green-Tao theorem, proved in 2004, which states that the sequence of prime numbers contains arbitrarily long arithmetic progressions (but of course no infinitely long one); as examples of the latter take the platonist claim that the number two exists and is abstract or the claim that because mathematical entities are created rather than discovered, impredicative definitions are not permissible. For any given philosopher, the Green-Tao theorem and its proof are to be assessed by the standards of the first column. For example, Quine accepts the theorem if and only if it is part of the best systematisation of science (which it is assuming that the principles from which it is deduced and the logic by which it is deduced from those principles are). Similarly, for any given philosopher, platonism and the stricture regarding impredicativity are to be assessed by the standards of the second column. For example, Quine accepts platonism because he takes it to be the scientifically sanctioned interpretation of mathematics: in his view, the mathematics contained in the best systematisation of science is platonic.