Analytical geometry has never existed. There are only people who do linear geometry badly, by taking coordinates, and they call this analytical geometry. Out with them!

The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying.

Analytical geometry has never existed. There are only people who do linear geometry badly, by taking coordinates, and they call this analytical geometry. Out with them!

Now ... the basic principle of modern mathematics is to achieve a complete fusion [of] 'geometric' and 'analytic' ideas.

We have not begun to understand the relationship between combinatorics and conceptual mathematics.

On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,'... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.

It is indubitable that a 50-year-old mathematician knows the mathematics he learned at 20 or 30, but has only notions, often rather vague, of the mathematics of his epoch, i.e. the period of time when he is 50.