A new biography sheds new light on the work of Alan Turing ...
The first one I read was Rebecca Goldsmith's about Kurt Gödel, the logician who proved conclusively that we will never be able to prove everything we know, and that reality really does exist whether we can prove it or not.
Or, as Gödel put it when approached at a cocktail party and asked about his work, "I am attempting to prove that all knowledge is a priori." How's that for a conversation opener? Good luck with that, Kurt!
It's no accident that the other book that appealed to me was the one about Alan Turing, the man whose theoretical work on "provability" made history when he used it to break codes and help to win World War II.
In fact, Turing and Gödel were both working on the same problem from different directions. Gödel came from the background of pure logic, rooted in philosophy from his studies with Wittgenstein, so his solution was a construction of abstract symbols that can only be fully appreciated by another logician. (Maybe that's why he and Einstein were such good buddies.)
Turing, on the other hand, had more of a background in applied science. He loved to build things. So it is perhaps not surprising that he resorted to the use of an imaginary mechanical device (now known as the "Turing Machine") for his proof.
Using a hypothetical tape of infinite length, a sprocket to move it back and forth, and a sort of recording head to read and write symbols on it, he was able to show that any possible computation could be done mechanically. Building on this foundation he went on to demonstrate that the machine could analyze its own written instructions on the tape to determine if the problem to be solved was solvable. Of course, the instructions on how to do that could also be analyzed, and he was able to show that it was an example of a problem that could not be solved. Therefore, there will always be some things we can't prove even if they are true. Whew!
The main thing I took away from David Leavitt's new biography, The Man Who Knew Too Much, is an appreciation of why Turing's approach was so unique and far reaching. Gödel's proof may seem definitive to those who can do the math, but Turing invented a proof from which the answer emerges with the mechanical inevitability of a clockwork gear. It is compelling because it is physical, as irrefutable as the force of gravity.
My only complaint about Leavitt's presentation is that he spends so many of his pages meticulously explaining Turing's 1936 paper, "On Computable Numbers."* Having understood it himself, he takes great pains to inflict the same achievement on his readers. But this is also the book's strength. I confess I skimmed the last part of that section, but not before the import had become clear.
In the 1940's Turing found himself in the perfect time and place, and with a compelling reason, to actually turn his imaginary device into reality. The result was the breaking of the German Enigma code, the changing of history, and the invention, by Turing and others, of the programmable digital computer -- the spirit of mathematics made flesh. We are only now beginning to realize how great an achievement that was, as the machines continue to work their way into quite literally everything we do.
Turing went on to theorize about the future potential of what he was the first to call "thinking machines." Having helped to give birth to them, he naturally had an interest in how his progeny would turn out. He seems to have understood at once that there was no fundamental difference between the logical operations of computer equipment and the functioning of the human brain. The concept was almost unimaginable back then, but it is now considered only a matter of time before a computer can pass the Turing Test -- proving itself indistinguishable from a human being in written conversation.
[Trials have already begun, as explained in this recent BBC News story. However, many now believe this test is too easy and that a higher standard will have to be used to determine true intelligence.]
Missing from this version of Turing's saga is much of the exciting detail of the code breaking project, and the drama of looking over his shoulder as he invented one by one all the key elements of programming. For that and much more about his brilliant and tragic life, I recommend (as does Leavitt) Alan Turing: The Enigma by Andrew Hodges, still the definitive work on this remarkable man.
[*A copy of Turing's original paper, now available online in PDF form, was recently valued at $15,000 to $20,000 by Christies. See also the Alan Turing Scrapbook, and the Turing Digital Archive.]