Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

Mathematical knowledge adds vigour to the mind, frees it from prejudice, credulity, and superstition.

The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions and Logarithms.

A marveilous newtrality have these things mathematicall, and also a strange participation between things supernaturall and things naturall.

A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.

To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.

Go to the roots of these calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe, is the mission of future mathematicians. This is the road on which I am embarking in this work.

Responding to Olbers’ attempt in 1768 to entice him to work on Fermat’s Last Theorem:

I confess that Fermat’s Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.

The Theory of Groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing.

Mathematics is no more computation than typing is literature.