The universe is an enormous direct product of representations of symmetry groups.
We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.
Nobody can say what a variable is.
My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.
Besides language and music, [mathematics] is one of the primary manifestations of the free creative powers of the human mind, and it is the universal organ for world-understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation.
‘Mathematizing’ may well be a creative activity of man, like language or music, of primary originality…
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.