In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.
Mathemagical mathematics combines the beauty of mathematical structure with the entertainment value of a trick.
The sudden hunch, the creative leap of mind that “sees” in a flash how to solve a problem in a simple way, is something quite different from general intelligence.
In no other branch of mathematics is it so easy for experts to blunder as in probability theory.
Indeed, there is something to be said for the old math when taught by a poorly trained teacher. He can, at least, get across the fundamental rules of calculation without too much confusion. The same teacher trying to teach new math is apt to get across nothing at all...
There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry.
A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?
One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery – and more totally useless – than the perfect numbers.
Debunking bad science should be constant obligation of the science community, even if it takes time away from serious research or seems to be a losing battle. One takes comfort from the fact there is no Gresham’s laws in science. In the long run, good science drives out bad.
Modern science should indeed arouse in all of us a humility before the immensity of the unexplored and a tolerance for crazy hypotheses.
If all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime.
As Bertrand Russell once wrote, two plus two is four even in the interior of the sun.
The violence and double-talk in the Alice books probably does no harm to children, but the novels should not be allowed to circulate indiscriminately among adults who are undergoing analysis.
The computers are not replacing mathematicians; they are breeding them.
Although Lewis Carroll thought of The Hunting of the Snark as a nonsense ballad for children, it is hard to imagine – in fact one shudders to imagine – a child of today reading and enjoying it.
As I have often said, electrons and gerbils don’t cheat. People do.
The Rail Fence Cipher Suppose.
Let us hope that Lysenko’s success in Russia will serve for many generations to come as another reminder to the world of how quickly and easily a science can be corrupted when ignorant political leaders deem themselves competent to arbitrate scientific disputes.
We cannot tell that we are constantly splitting into duplicate selves because our consciousness rides smoothly along only one path in the endlessly forking chains.
My wife and I own a cat we call Eureka, after Dorothy’s cat in the fourth Oz book. In Eureka’s dim mind she must be a kind of polytheist, fed as she is by the two of us, and by neighbors when we take a trip; surrounded on all sides by giant creatures who move about on their hind legs to do things utterly beyond her ken. But we who are her gods have a power of speculation far greater than that of her tiny feline brain.