The mathematician predominates at the expense of the philosopher

Cartan developed a general scheme of infinitesimal geometry in which Klein's notions were applied to the tangent plane and not to the n-dimensional manifold M itself.

This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.

We cannot hope to give here a final clarification of the essence of fact, judgement, object, property; this task leads into metaphysical abysses; about these one has to seek advice from men whose name cannot be stated without earning a …

It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought …

In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.

In my work, I have always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.