Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, "This is from the book!"
When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached. It is this way that in mathematics speculative theorems and practical canons are reduced by analysis to definitions, axioms and postulates.
Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems.
It is not so much whether a theorem is useful that matters, but how elegant it is.
How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!
One cannot really argue with a mathematical theorem.
We often hear that mathematics consists mainly of "proving theorems. " Is a writer's job mainly that of "writing sentences?"
Now, one of my beliefs, one of my theorems that I have evolved over the years is that when it comes to Democrats and the media they will always tell us who they fear. And all we have to do to learn that is look at who they're trying to damage andor destroy.
The product of mathematics is clarity and understanding. Not theorems, by themselves. . . . In short, mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new.
Phyllis explained to him, trying to give of her deeper self, 'Don't you find it so beautiful, math? Like an endless sheet of gold chains, each link locked into the one before it, the theorems and functions, one thing making the next inevitable. It's music, hanging there in the middle of space, meaning nothing but itself, and so moving. . . '
For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.
I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths.
