The New York Review of Books

“Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one?”

—Lewis Carroll
What the Tortoise Said to Achilles

During the nineteenth century, mathematics, the acknowledged foundation of all the sciences, began to turn its attention to its own foundations. With the work of Boole, De Morgan, Frege, Peano, Russell, and others, a great project began to take shape: mathematics would examine its own structure with all the rigor it had brought to other explorations. The goal was a system that would be unquestionable and secure; a mathematical language would be developed whose simplicity and clarity would dispel all doubts, reveal all mathematical truths.

The goal of the project was similar to the dream of many seventeenth-century thinkers who wished to create or discover a “universal language” in which a set of symbols would be used to describe all of man’s knowledge. Its structure would be so tied to the structure of the universe that the language itself could be used to discover truths. Its syntax would prevent falsity. There would be no ambiguous meanings. Leibniz, with his ideas of a precise symbolic language and a calculus of reasoning, was, in fact, a major influence on the nineteenth-century systemization of logic which was to provide the syntax of the new mathematical language.

In 1884, for example, Gottlob Frege claimed that arithmetic is “simply a development of logic,” and in the Principia Mathematica Russell and Whitehead attempted to show that all of “pure mathematics” might be derived from the propositions of logic. The full extent of the project was articulated in 1904 by David Hilbert: mathematics would be based upon a system of axioms similar to the one later elaborated in Principia Mathematica. New mathematical statements would be produced according to a set of rules; no two statements produced would contradict each other, making mathematics consistent. Moreover, the system would “fill” its chosen universe; it would be complete, producing every truth. Mathematical knowledge would then be secure. Just by manipulating signs of mathematics according to the rules of logic—its “grammar”—one could eventually discover all necessarily true statements, and one would never produce a falsehood.

The dream of producing such a language was shattered in 1931 by a twenty-five-year-old Austrian mathematician, Kurt Gödel. In a paper entitled (in English translation) “On Formally Undecidable Propositions of Principia Mathematica and Related Systems I,” he showed that in sufficiently powerful mathematical systems, consistency could not be proven in the way Hilbert wished, and more importantly, that such systems were incomplete: there are true statements in mathematics that the system will not produce, true statements, that is, that cannot be proven. This result has since taken its place among the other metaphorically powerful scientific results of our century—the Theory of Relativity and the Uncertainty Principle. Gödel’s result is known as the Incompleteness Theorem.

The proof of the theorem is so shrouded in abstraction that it is impenetrable to an outsider; one can hardly read Gödel’s original paper without previous preparation.^* Yet in the contemporary intellectual climate it has become almost essential to gain some understanding of Gödel’s theorem, it is not only called upon in discussions of philosophy of mathematics, but also in essays on science, music, and literature. There is, fortunately, an excellent book about the theorem by Ernest Nagel and James R. Newman that gives the historical background and a simplified exposition that is as direct and brief as its title: Gödel’s Proof. There is also a playful presentation of the theorem in Raymond Smullyan’s recent What Is the Name of This Book? Now Douglas Hofstadter has also written a welcome guide for the layman through the mathematical thickets, but his intentions are more grandiose.

“I began,” he writes, “intending to write an essay at the core of which would be Gödel’s Theorem. I imagined it would be a mere pamphlet.” What emerged instead is a large, expensive, eclectic, beautifully designed and lucidly written book that attempts to link Gödel’s theorem to the art of M.C. Escher and to the music of J.S. Bach. The text is punctuated with over 150 illustrations, ranging from Escher’s prints to a drawing of an arch constructed by termites. Hofstadter discusses cellular reproduction and Zen koans. Each chapter is preceded by playful dialogues between Achilles and the Tortoise from Zeno’s paradox, and assorted other characters, including the author himself. The dialogues are modeled on Bach’s musical compositions and bear titles like “Three-Part Invention,” “Chromatic Fantasy, and Feud,” and “The Magnificrab, Indeed.” The book, Hofstadter assures us, has “many levels of meaning”; it is meant to symbolize “at once Bach’s music, Escher’s drawings, and Gödel’s Theorem.” Hofstadter, an Assistant Professor of Computer Science at Indiana University, is also interested in arguing in this “metaphorical fugue on minds and machines” that someday it will be possible to create “Artificial Intelligence” in a computer.

If Gödel’s theorem shattered a sort of dream, it seems as if Hofstadter is intent on creating another one, a dream filled with wordplay and association in which Gödel plays a metaphorical role. Such a dream could have dissipated into chaos; but instead the book is exhilarating, challenging, valuable, and frustrating. Hofstadter writes directly and playfully for the lay reader, explaining the most abstract and wideranging arguments in short sections of great virtuosity. He is sophisticated in his understanding of the systems he explores and is adventurously speculative about their limits. But the book resists simple evaluation; it is at once surprisingly subtle and annoyingly naïve, exuberantly clever and embarrassingly silly. In order to understand Hofstadter’s project, we shall first have to understand the one that preceded Gödel.

It might seem, at first, that the mathematical project that Gödel shattered was a totally unrealistic one, particularly if it is seen as part of a continuing search for a “universal language.” But some such project was needed: the certainty of mathematical knowledge after the nineteenth century really was in doubt. Difficulties had arisen even in that most concrete field, geometry. The ancient “axiomatic method,” in which certain self-evident fundamental propositions are used as basic axioms from which the theorems may be derived, itself began to pose problems.

There had always been a problem with Euclid’s system, despite its longevity. The first four of Euclid’s five axioms from which he derived plane geometry had seemed indubitable (e.g., a straight line segment can be drawn joining any two points). But the fifth axiom (whose modern equivalent is: through a point outside a line only one line can be drawn parallel to the given line) had always seemed less than self-evident, and over the centuries there were numerous attempts to demonstrate its superfluity by deriving it from the first four. Finally, it was recognized that it could not be proven; it was, in fact, independent of the other axioms. Yet this independence is puzzling, for if the fifth axiom were contradicted, if, for example, one were to assert that through a point outside a line no line can be drawn parallel to a given line, and if this new axiom were added to the first four, the results seem absurd. One would find, for example, that the sum of the angles of a triangle would be more than 180 degrees.

Several mathematicians simultaneously realized that by reinterpreting certain undefined terms in Euclidean geometry, like “line,” sense could be made out of the absurdities of the system with a different fifth axiom. The system above, for example, could be modeled by a sphere in which “lines” are interpreted to be great circles (like longitudinal or equatorial circles). Not only do the first four axioms still hold, but so does the new fifth axiom: through a point outside a line (a great circle) no line (great circle) can be drawn parallel to it (since all great circles intersect). And on a sphere the sum of the angles of a triangle is indeed greater than 180 degrees. This new interpretation yielded a non-Euclidean geometry.

The discovery of non-Euclidean geometries raised serious questions about how mathematics was to be understood. Mathematicians tended to assume that the terms of mathematics, whether “point,” or “line,” or “numeral,” referred clearly to objects in our experience. This was no longer certain. In fact, by reinterpreting those terms, one could generate quite different systems, some of which might have nothing to do with out experience at all. As Nagel and Newman put it: “It came to be acknowledged that the validity of a mathematical inference in no sense depends upon any special meaning that may be associated with the terms.” The business of mathematics was not interpretation, or truth, but logical derivation.

But if it was difficult in a field as concrete as geometry to determine which statements could be proven and what the contradictions were, how much more difficult would it be in more abstract systems? All of mathematics was becoming more abstract in the nineteenth century; it was no longer tied to a description of our universe, its space, or its numbers. It was also, then, cut off from the confirmation experience might give to mathematics. How could one be certain that one was not producing contradictions in these unearthly realms?

David Hilbert, who did more than any other contemporary mathematician to refine and clarify the axiomatic foundations of geometry, believed that only the axiomatic method could guarantee the certainty of mathematical results. His dream, mentioned earlier, was nurtured by this belief. He proposed a radical “formalization” of mathematics; all mathematical signs would be drained of meaning in order to avoid such confusions as arose in geometry. Statements in mathematics would be no more than “strings” of such arbitrarily chosen signs as “strokes,” hyphens, and letters. Axioms would be strings that were given with the system. Then by using specified mechanical rules, these strings could be manipulated to produce other strings to be known as “theorems.” A “derivation” of a theorem would just be an “array” of strings, arranged according to the rules. Mathematics would become a mechanical, syntactical game.

The study of such mathematical systems, known as “meta-mathematics,” was also to be systematized. The questions of meta-mathematics were substantial. For example, how can we determine whether a particular string is a theorem, i.e., whether it can be proven? Given a particular interpretation of the system, are there any contradictions within it?

Hilbert’s formalist project has had enormous influence. By treating systems as if they were purely syntactic he focused attention upon the minute nuances in mathematical thought and helped make abstract rigor the measure of mathematical presentation. And by defining the realm of meta-mathematics, Hilbert explicitly raised questions about truth and provability in mathematics.

The most thorough attempt to ground mathematics in a system was made by Russell and Whitehead in Principia Mathematica: all of mathematics was to be developed out of the rules of a logical calculus. Given such a system, Hilbert asked, can we prove that it is both consistent and complete, that it contains no contradictions and yields every true statement? If we could, if just a formalized arithmetic would have these properties, we would be on our way toward the construction of a powerful mathematical language. Hilbert recognized that in accomplishing this dual task of proving both consistency and completeness, one could not appeal to ordinary mathematical methods, for it was just those methods whose legitimacy was in question. Nor could proofs in meta-mathematics make reference to infinite numbers of strings or operations since such infinite sets had created paradoxes and confusion in the mathematics of the time. Such proofs would have to construct any mathematical object they wished to make use of, demonstrating its existence clearly and concretely.