"A mathematician is a device for turning coffee into theorems"
-Alfréd Rényi
"Let bigons be bygone"
-Willam Menasco
"The tangent bundle is the true beginning of the study of differentiable manifolds, and you should not read further until you grok it."
-Michael Spivak
"If numbers aren't beautiful, I don't know what is."
-Paul Erdős
"I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude."
-Gottfried Wilhelm Leibniz
(geometria situs - See page 5 of Lectures in Knot Theory for some interesting history about this quote and the dawn of knot theory)
"Matter tells spacetime how to curve. Spacetime tells matter how to move."
-John A. Wheeler