“Mathematics is” quotes

Tuesday August 4, 2026

For at least six years, I've been collecting quotes about what mathematics is. Here are some of them.

“... mathematics is not an empirical science, ...” (Von Neumann, The Mathematician)

“Mathematics is not a deductive science—that's a cliché.” (Halmos, I want to be a mathematician)

“Pure mathematics is, in its way, the poetry of logical ideas.” (Einstein, epitaph for Noether)

“... mathematics is the science of skillful operations with concepts and rules invented just for this purpose.” (Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences)

“Mathematics is a tool for reasoning.” (Feynman, The Relation of Mathematics to Physics)

“Mathematics is no longer knowledge but a game of formulae, ruled by certain conventions, which is very well comparable to the game of chess.” (Weyl, Current Epistemological Situation in Mathematics)

“Mathematics is the art of giving the same name to different things.” (Poincaré, Science et Methode)

“... mathematics cannot amount to anything more than an immense tautology.” (Poincaré, Science and Hypothesis)

“Mathematics is the extension of common sense by other means.” (Ellenberg, How Not to Be Wrong)

“Mathematics is the art of explanation.” (Lockhart, A Mathematician's Lament)

“Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.” (Ellenberg, How Not to Be Wrong)

“[Mathematics is] about how things fit together.” (Kra, Inside Dynamical Systems and the Mathematics of Change)

“[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture.” (Halmos, quoted in his MAA obituary)

“Mathematics is prestidigitation.” (Linderholm, Mathematics Made Difficult)

“Mathematics is the science of the infinite.” (Weyl)

“Mathematics is the study of anything that obeys the rules of logic, using the rules of logic.” (Cheng, How to bake π)

“Mathematics is easy, as long as you have the right definition of ‘easy’” (Cheng, How to bake π)

“The pursuit of mathematics is the process of working out exactly what is easy, and the process of making as many things easy as possible.” (Cheng, How to bake π)

“Mathematics is interesting in so far as it occupies our reasoning and inventive powers.” (Pólya, How to Solve It)

“Mathematics is about problems, and problems must be made the focus of a student’s mathematical life.” (Lockhart, A Mathematician's Lament)

“Mathematics is not life” (Cheng, How to bake π)

“In the end mathematics is simply about things that are true.” (Cheng, How to bake π)

“Mathematics is not settled.” (Ellenberg, How Not to Be Wrong)