"The principles of mathematics have always had an important relation to philosophy. Mathematics apparently contains
a priori knowledge of a high degree of certainty, and most philosophy aspires to
a priori knowledge. Ever since Zeno the Eleatic, philosophers of an idealistic cast have sought to throw discredit on mathematics by manufacturing contradictions which were designed to show that mathematicians had not arrived at real metaphysical truth, and that the philosophers were able to supply a better brand. There is a great deal of this in Kant, and still more in Hegel. During the nineteenth century, the mathematicians destroyed this part of Kant’s philosophy. Lobatchevski, by inventing non-Euclidean geometry, undermined the mathematical argument of Kant’s transcendental aesthetic. Weierstrass proved that continuity does not involve infinitesimals; Georg Cantor invented a theory of continuity and a theory of infinity which did away with all the old paradoxes upon which philosophers had battened. Frege showed that arithmetic follows from logic, which Kant had denied. All these results were obtained by ordinary mathematical methods, and were as indubitable as the multiplication table. Philosophers met the situation by not reading the authors concerned." --Bertrand Russell
