May 2005

Anders Floor


Incompleteness: The Proof and Paradox of Kurt Godel by Rebecca Goldstein

Kurt Godel, the greatest logician of our era and the heir to Aristotle, is best known for having proved the incompleteness of arithmetic. Rebecca Goldstein (best known for such novels as The Mind-Body Problem) has attempted to unpack the exceedingly esoteric notion of incompleteness for the lay reader, and give some idea of how his proof runs. In this attempt she largely succeeds. She situates the theorem that arithmetic is incomplete both in the history of ideas and in Godel's own life, showing how it related to his philosophical views and preoccupations. The result is Incompleteness: The Proof and Paradox of Kurt Godel, an entry in the "Great Discoveries" series, which notably has recently let us hear from David Foster Wallace on the subject of the infinite.

The introduction lays out Godel's incompleteness result for those unfamiliar: for any consistent formal system (a set of basic statements together with rules letting you deduce further statements) rich enough for arithmetic (i.e. we can define addition, subtraction, etc. in the language of the system), there will be a true proposition of arithmetic which cannot be deduced in the system. Goldstein also provides the setup for one of her main goals in this book: to use Godel's own ideas, and his attitude toward his proof, to put a stop to the use (or at least the unthinking use) of Godel's result to support subjectivism. Like Einstein, who gets a strong supporting role in this book, Godel believed in objective reality. And he thought that his theorem lent aid and succor to this view. That no one else seemed to see it that way caused him great despair.

The rest of the book lays out Godel's early years and views, the mathematical and philosophical climate in which he was operating, a simplified version of the proof of Godel's theorem, and Godel's last, tragic years. One of two major problems with the book is that Goldstein appears to have an axe to grind with logical positivism, and this distorts her account of it. Logical positivism, as the term is used here, is the body of ideas developed from the view that only sentences which you could (in theory) check on empirically are meaningful, and that the checking procedure is the sentence's meaning. So the proposition "My pen is on the desk" is okay, because I can look to see if it's true. Also, "My pen is made of atoms," and "There is a pen somewhere beneath the surface of Pluto" are OK, because the checking could in principle be carried out. But propositions like "God exists" or "An invisible intangible inaudible odorless gnome is sitting under my desk" are not OK. (You might think of logical positivism as raising to the level of ideology the common-sensical observation that one way of explaining the meaning of a sentence is to say how you would verify it: "What do I mean by 'The pen is on the desk'? Well, I mean if you looked in the direction of the desk, you'd see what looks to be my pen." And when someone refuses to give us a means of verification, we often have trouble figuring out what's meant: "Can I see God?" "No." "Can I touch Him?" "No"...) This is not subjectivism, as Goldstein seems to imply. Indeed, it would seem that on a positivist view both subjectivism and objectivism are meaningless.

Identifying positivism with subjectivism seems pretty basic to her situating of Godel in relation to his contemporaries. She says, "'Everything,' [a positivist manifesto] proclaimed, 'is accessible to man. Man is the measure of all things.' The ancient Sophist's words were reiterated verbatim." Earlier she identified the Sophistic philosophy with a certain kind of subjectivism. And it is the positivists, among others, who are meant to play the foil to Godel and his Platonism, in Goldstein's telling. So I think it is fairly important to get this point right, and I think Goldstein flubs it, albeit subtly.

A second problem is the forced artsiness she engages in from time to time. Thus at one point in the proof she interjects, "Soft whispers of self-referentiality are hovering in the hushed air." And throughout she insists on referring to first-order logic as "limpid logic." I agree that what she is discussing is beautiful, but she should let the beauty speak for itself, and not keep talking about it.

She does, however, do a good job of presenting the proof and the related ideas in such a way as to be understandable by someone with little mathematical background. She ably describes the tension between Godel's ideas and those of Wittgenstein, as well as those of the formalists (though her discussion of the latter is to some extent marred by the same problems which plague her discussion of the positivists). The biographical information, and the picture she paints of certain facets of intellectual life in the first half of the twentieth century, are presented so vividly and charmingly that one wishes one could oneself have lived then, and known such people. And she does an excellent job of rescuing Godel from those (and they are many) who would indeed use him to advance relativism. This is a book which should be read by anyone wishing to know something about one of the great intellectual accomplishments of our time, indeed of any time, and about the penumbra of issues and ideas surrounding it.

Incompleteness: The Proof and Paradox of Kurt Godel by Rebecca Goldstein
W. W. Norton
ISBN: 0393051692
296 Pages