**Mathematics**is the study of quantity

, space

, structure

, and change

. Mathematician

s seek out patterns and formulate new conjecture

s. Mathematicians resolve the truth or falsity of conjectures by mathematical proof

s, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic

.

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.

If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.

Great fleas have little fleas upon their backs to bite 'em,And little fleas have lesser fleas, and so ad infinitum,And the great fleas themselves, in turn, have greater fleas to go on,While these again have greater still, and greater still, and so on.

Numbers exist only in our minds. There is no physical entity that is number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe.

Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!