It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.

... the student skit at Christmas contained a plaintive line: "Give us Master's exams that our faculty can pass, or give us a faculty that can pass our Master's exams."

The beginner should not be discouraged if he finds he does not have the prerequisites for reading the prerequisites.

The library is the mathematician's laboratory.

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing - one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.

You are allowed to lie a little, but you must never mislead.

The only way to learn mathematics is to do mathematics.