mathematics

Increasingly, the mathematics will demand the courage to face its implications.

Eternal truths are ultimately invisible, and you won't find them in material things or natural phenomena, or even in human emotions. Mathematics, however, can illuminate them, can give them expression - in fact, nothing can prevent it from doing so.

The play is independent of the pages on which it is printed, and __ure geometries_ are independent of lecture rooms, or of any other detail of the physical world.

Man had been given a brain that could think in numbers, and it could not be coincidence that the world was unlocked by that very tool. To understand any aspect of the cosmos was to look on the face of God: not directly, but by a species of triangulation, because to think mathematically was to feel the action of God in oneself.

Everything can be summed up into an equation.

Mathematics ... is indispensable as an intellectual technique. In many subjects, to think at all is to think like a mathematician.

...no thought, if it be non-mathematical in spirit, can be trusted, and, although mathematicians sometimes make mistakes, the spirit of mathematics is always right and always sound.

...317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way.

Mathematics is the art of explanation.

[The golden proportion] is a scale of proportions which makes the bad difficult [to produce] and the good easy.

On Mars, the joke went, a man__ hole was his castle where values of castle approached dorm room.

It now becomes clear that consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system.

A mathematician tells you that the wall of warped space prevents the Moon from flying out of its orbit yet can't tell you why an astronaut can go back and forth across that same space.

...it would not be quite right to say that the problem is unsolvable in principle; only so complicated that it is not worth anybody__ time to think about it. So what do we do?In probability theory there is a very clever trick for handling a problem that becomes too difficult. We just solve it anyway by:(1)__making it still harder;(2)__redefining what we mean by __olving_ it, so that it becomes something we can do;(3)__inventing a dignified and technical-sounding word to describe this procedure, which has the psychological effect of concealing the real nature of what we have done, and making it appear respectable.

In the Principia Mathematica, Bertrand Russell and Alfred Whitehead attempted to give a rigorous foundation to mathematics using formal logic as their basis. They began with what they considered to be axioms, and used those to derive theorems of increasing complexity. By page 362, they had established enough to prove "1 + 1 = 2.

Last period of the day. Charles had decided that morning that he would talk about tessellations. Last period, they should have been covering sines and cosines. They should have been starting to graph, but he just didn't have it in him. Tessellations were his favorite.

I think we need more math majors who don't become mathematicians. More math major doctors, more math major high school teachers, more math major CEOs, more math major senators. But we won't get there unless we dump the stereotype that math is only worthwhile for kid geniuses.

Instead of concentrating just on finding good answers to questions, it's more important to learn how to find good questions!