Engines of Logic: Mathematicians and the Origin of the Computer. By Martin Davis, W.W. Norton, New York, 2001 (paperback), 257+xii pages, $14.95.
The latest in Gregory Chaitin's series (four and counting) of nontechnical accounts of his work consists of 12 short chapters, of which three are (edited) lecture transcripts, three are TV interviews, and three are ordinary interviews. It also includes a short introduction, a few final thoughts, and some recommended further reading. Intended to be the most accessible in the series---all published by Springer---the new book is at times oracular (as in the chapter titled ``Undecidability & Randomness in Pure Mathematics'') and at others deeply personal.
In chapter 10, for example, when asked what mathematics is for him, Chaitin describes it as ``fun''---especially when you've got an idea that seems to be leading somewhere---and ``sensual,'' like music, albeit a form of music that most people are unable to hear. On the other hand, he observes, mathematics can resemble an adventure, like trying to climb a previously unscaled mountain. Sometimes you reach the summit, ``where no one has been before, where the air is very pure, where the sky is a beautiful dark blue, where you get those incredible views, where you think you're closer to God, where you understand!'' Yet things seldom turn out that way. More often than not, mathematical mountain climbers are frustrated by unforseen features of the landscape, which block the more obvious routes to the summit. This is a natural analogy for Chaitin, who climbs mountains himself ``on weekends and holidays, especially in winter with snowshoes and ski poles.''
Chaitin goes on to reflect that a career in mathematics can easily degrade one's performance as a husband, a father, and a member of the community:
``Having mathematical talent is wonderful, but it's also a bit of curse. It tends to take over your life. Now I didn't want it to take over my life. I think I tried to have a normal human life. But it did sort of take over my life, I guess.''
Perhaps to console himself for opportunities missed, he recalls the answer given late in life by Einstein---a confessed failure as husband and father---to the wife of a former colleague who wrote to ask why her beloved (and reputedly talented) spouse had ``never accomplished anything.'' ``But of course,'' Einstein replied, ``it's because he's a good man.''
Chaitin describes his thoughts on randomness, and some of the reactions to them. While studying physics as a teenager, he began to wonder what it meant for something to exhibit no pattern, to obey no law. His answer, when he found it, seemed entirely natural as applied to physics, where randomness is known and expected. It seemed somewhat unnatural, on the other hand, as applied to the foundations of mathematics, where randomness was not previously known or expected to occur. As a result, he says, he has become persona non grata in a portion of the logic community, where there is resistance to the conclusion that ``God threw dice'' even while constructing the natural numbers.
According to Chaitin, his most fruitful idea occurred to him in 1970, in Rio de Janeiro, just before Carnival time. During the Carnival itself, he hastens to inform his (Brazilian) interviewer, he was far too busy ``dancing in the street and looking at those luscious carioca women'' to think about mathematics. The idea was that, although almost every natural number does in fact satisfy his personal definition of randomness---much as almost every real number is transcendental---mathematical reasoning can prove only in isolated cases that this is so. He had discovered a new limitation, akin to those imposed by the incompleteness theorems of Gödel and Turing, on what mathematical reasoning can achieve!
The book's most controversial thesis, admittedly an exageration, is that ``the computer was invented to clarify a question about the foundations of mathematics.'' He is referring, of course, to Alan Turing's investigation of what he originally called ``a-machines''---the ``a'' standing for ``automatic.'' It is easily forgotten that Turing invented the machines that now bear his name to investigate certain fundamental issues raised by logicians (particularly Hilbert) at the beginning of the 20th century. Although Hilbert's attempt to encompass all of mathematics within a single formalization foundered on the reef of Gödel's incompleteness theorem, Chaitin asserts that his formalizations live on in the languages by which we communicate with modern electronic computers, forming an indispensable part of the century's most spectacularly successful technology. He describes these events as a largely forgotten chapter in intellectual history, and devotes his opening lecture to a summary thereof. After making more or less the same claim, Martin Davis devotes an entire book to the thesis.
Davis's book, which was originally published in 2000 under the title The Universal Computer: The Road from Leibniz to Turing, was recently reissued in paperback...
Both Davis and Chaitin remark, in discussing Gödel's proof, that a large part of the effort went into the construction of the Gödel numbering system for formulas, which closely resembles a computer language like Fortran, the very name of which arose as a contraction of FORmula TRANSlator. It seems unlikely that the similarity would have been apparent to anyone not deeply immersed in both logic and the machine-level programming of computers. However different the two books (and authors) may be, each in its (or his) own way is both thought-provoking and informative.
James Case writes from Baltimore, Maryland.
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