A non-trivial congruence - or the real beauty of Mathematics

Something I often find myself struggling to explain is the concept of mathematical beauty. It's something that mathematicians and physicists instinctively recognise, but it's not easy to articulate to the lay person!

But I was most heartened to find the following excerpt on John Polkinghorne's website (a wierd site set-up means it's impossible to link straight to the article). He doesn't answer the question of beauty per se but he makes a most appealing case for the amazingness of mathematics generally!

He's discussing aspects of metaphysics... 'aspects of the laws of physics which raise questions beyond physics' competence to answer':

"The first is a property of the physical world that is so familiar to us that we take it for granted. It is, in fact, the necessary basis of the whole scientific endeavor. It is this: that we can understand the world, that it is intelligible to us, that it is rationally transparent. Not only do we understand the world, but it is mathematics which is the key to the understanding of the physical universe. In fundamental physics one looks for theories which in their mathematical expression are economic and elegant, which are mathematically beautiful. Mathematical beauty is a very recognizable characteristic. There is an expectation -- an expectation that has been justified time and again in the history of physics -- that it is just those theories which have the character of mathematical economy and elegance which will prove to be the ones that explain what is going on in the physical world. If you have a friend who is a theoretical physicist, and you wish to upset them, you simply say to them, "That new theory of yours looks rather ugly and contrived to me." They will be truly upset, because you are saying that it does not have the character which successful theory always has had.

When we use mathematics in that way as a heuristic tool, a device for finding out what's going on in the world, something very odd is happening. After all, what is mathematics? Mathematics is the free exploration of the finite human mind. Our mathematical friends sit in their studies and out of their heads they spin the beautiful patterns of mathematics. Mathematics can be thought of as a pattern creating, pattern analyzing, subject. Yet some of the most beautiful patterns that are dreamt up by the pure mathematicians in their studies are found actually to occur in the structures of the physical world around us. In other words, there is a deep-seated congruence between the reason that we experience within (in our minds) and the reason that we experience without (in the physical world around us). They fit together like a glove. That seems a fact about the physical world that is what the mathematicians in their modest way would call non-trivial. 'Non-trivial' is a mathematical word meaning 'highly significant.'"

Wow wow wow (or is it just me?!) I love this guy!