The Universal Computer: The Road from Leibniz to Turing.(Review)

Martin Davis The Universal Computer: The Road from Leibniz to Turing. W. W. Norton & Co., 237 pages $26.95

"Every morning I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty.... [T]he two summers of 1903 and 1904 remain in my mind as a period of complete intellectual deadlock.... [I]t seemed quite likely that the whole of the rest of my life might be consumed in looking at that blank sheet of paper."

That is from Bertrand Russell's autobiography. What was stumping him was the attempt to find a definition of "number" in terms of pure logic. What does "three," for example, actually mean? The German logician Gottlob Frege had come up with an answer: "three" is merely the set of all threesomes, the set of all those sets whose members can be exhaustively paired off with Larry, Curly, and Moe.

However, if the concept "the set of all sets with a certain property" can be used indiscriminately, as Frege used it, then we can construct the set [Omega] of all sets that are not members of themselves. The set of all turtles is not a member of itself, since it is a set, not a turtle. It is therefore a member of [Omega]. But the set of all things that can be defined in fewer than a hundred words is a member of itself, and therefore not a member of [Omega]. Now pose the question: is [Omega] a member of [Omega]? If it is, it isn't, by definition; and if it isn't, it is. This contradiction is named "Russell's antinomy" and, until a way round it could be found, the enterprise that both Frege and Russell were embarked upon--the derivation of mathematics from logic--was dead in the water.

If you had asked Russell, during those summers of frustration, whether his perplexities were likely to lead to any practical application, he would have hooted with laughter. This was the purest of pure intellection, to the degree that even Russell, a pure mathematician by training, found himself wondering what the point was: "It seemed unworthy of a grown man to spend his time on such trivialities." So little can we tell where disinterested inquiry will lead! In fact, Russell's work brought forth Principia Mathematica, a key advance in one of the strangest and most unexpected enterprises of the modern age. Among the fruits of that enterprise have been, so far, victory in World War Two (or at any rate, victory at a lower cost than would otherwise have been possible) and machines like the one on which I am writing this review.

The Universal Computer tells this story in eight chapters, each concentrating on a key figure in the story: Leibniz, Boole, Frege, Cantor, Hilbert, Godel, Turing, and von Neumann. Some of those names will be familiar to any educated person; some have even escaped into the larger culture. Turing was the subject of a rather good play by Hugh Whitmore, Breaking the Code (1986). He and Godel both turn up as characters in Apostolos Doxiadis's 1992 novel Uncle Petros and Goldbach's Conjecture, of which an English translation was published last year to considerable success.

The strength of this book is in its tracing the continuous chain of events from Leibniz's early attempt at a calculus of propositions all the way through to the stored-program computers of our own time. To follow the stored-program concept backwards: developed by John von Neumann, it rests on the idea that code (the instructions that tell a computer how to act) and data (the sniff that is to be acted upon) can be represented in just the same way in a computer's memory.