He is still all potential. The potential to be great, the potential to be mad. He will achieve both magnificently.
Everyone gathered on this Thursday, the rotating numbers accounting for some three dozen, believe in their very hearts that mathematics is unassailable. Gödel has come tonight to shatter their belief until all that is left are convincing pieces that when assembled erect a powerful monument to mathematics, but not an unassailable one — or at least not a complete one. Gödel will prove that some truths live outside of logic and that we can't get there from here. Some people — people who probably distrust mathematics — are quick to claim that they knew all along that some truths are beyond mathematics. But they just didn't. They didn't know it. They didn't prove it.
Gödel didn't believe that truth would elude us. He proved that it would. He didn't invent a myth to conform to his prejudice of the world — at least not when it came to mathematics. He discovered his theorem as surely as if it was a rock he had dug up from the ground. He could pass it around the table and it would be as real as that rock. If anyone cared to, they could dig it up where he buried it and find it just the same. Look for it and you'll find it where he said it is, just off center from where you're staring. There are faint stars in the night sky that you can see, but only if you look to the side of where they shine. They burn too weakly or are too far away to be seen directly, even if you stare. But you can see them out of the corner of your eye because the cells on the periphery of your retina are more sensitive to light. Maybe truth is just like that. You can see it, but only out of the corner of your eye.
And Alan Turing:
Where is God in 1 + 1 = 2? Nowhere.
Over the coming days and weeks and on into months, he feels the last of his spiritualism evaporate like the cooling remains of a fever. He is left without remorse and wonders how he ever clung to his awkward faith with such emotion. We're just machines. At the age of twenty-three and for the rest of his life he embraces, without reservation, a mathematics that exists independently of it. We are biological machines. Nothing more. We have no souls, no spirits. But we are bound to mathematics and mathematics is flawless. This has to be true.
Where is God in 1 + 1 = 2? There is no God.
[. . .]
To his great surprise his materialism is not so sad. He moves with greater ease and resilience and less fear. It is as though his eyes have suddenly come into focus from a nauseating blur and the whole world looks brilliant. All of his senses have sharpened so that colors and sounds and smells and textures are splendid and vibrant, his experience of them a heart-soaring joy. Every blade of grass glistens. The hard Cambridge wind batters respect out of him. The barren twist of every branch of every tree, even the weak fog of light, the whole of the world sings out to him as though he has never seen or heard anything before. With the sheer pleasure of this tactile awakening, his love of nature intensifies as though he has finally given over to her, wholly and without inhibition. Within days his spiritualism is no more than a mildly embarassing, childish memory. In its place is a calm, impervious materialism — nothing like the sad bleak emptiness he feared. He would have a bad time trying to put it into words. No single word could mean this thing. He would have to write something lengthy with many caveats and tangents and even then he knows he would not successfully express the immediacy or the splendor of the visceral experience. Maybe in another's mind better words would come, but not in his. And so his mind offers him something simple. Only one word comes to him over and again, and it could be only this — a word he doesn't often think to use: "beautiful." It is beautiful.
I won't pretend to know what this book actually means or what it aspires to do, beyond speculating as to the character and thought of these two eccentric minds, but it's quite beautiful. This is a fictional account of the lives of Gödel and Turing. There is no hard math here. The philosophical concepts Gödel and Turing played with are presented quite simply and often tangentially. There is no direct relationship between Gödel and Turing. Turing's work drew on Gödel's ideas, but they never met. But parallels are drawn, perhaps no more complicated than the fact that these are two geniuses firmly detached from reality. The story is told with a cool detachment, which proves to be highly poetic.
Review by M John Harrison.
Interview (Quirks & Quarks, CBC) (mp3).
Jonathan Lethem and Janna Levin discuss truth and beauty (transcript):
Levin: Something I find particularly interesting is that science, I think, is the last realm in which people talk to each other seriously, with a straight face, about beauty. Visual artists would never say that's a beautiful piece of work, not in really contemporary, cutting-edge art.
Lethem: That's a very difficult phrase. After modernism, beauty is terrifically suspect.
Levin: Right, absolutely. And it's considered kind of provincial to aim for something beautiful. We're not doing pretty pictures here; we're doing something else. But in science, we really hold on to beauty and elegance as the goal because, for reasons that I think nobody fully understands, it's a good criterion for distinguishing what's right from what's wrong. And if something is beautiful and elegant, it's probably right. Occasionally, you'll see something that's so beautiful and so elegant, and it's not right, and you can't believe it's not right.