It is impossible, and it has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form. - William Rowan Hamilton

There is no religious denomination in which so much sin has been committed through the misuse of metaphorical expressions as in mathematics.

A set is a unity of which its elements are the constituents. It is a fundamental property of the mind to comprehend multitudes into unities. Sets are multitudes which are also unities. A multitude is the opposite of a unity. How can anything be both a multitude and a unity? Yet a set is just that. It is a seemingly contradictory fact that sets exist. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics. Thinking a plurality together seems like a triviality: and this appears to explain why we have no contradiction. But “many things for one” is far from trivial.

More often than not, a piece of mathematics worked out years before—and believed to be totally without practical value—finds a role in the “real” world.