## Summary

Mathematician John L. Casti is simultaneously professor of operations research and system theory at the Technical University of Vienna and a member of the external faculty of the Santa Fe Institute in Santa Fe, New Mexico. A prolific author of both technical and popular works, Casti has produced some of the recent years’ most inventive and imaginative books on science and mathematics. His *Five Golden Rules: Great Theories of Twentieth Century Mathematics and Why They Matter* (1996) and *Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of Twentieth Century Mathematics* (2000) explore the intricacies of contemporary mathematics, making difficult concepts comprehensible and enjoyable for general readers. His work of “scientific fiction,” *The Cambridge Quintet: A Work of Scientific Speculation* (1998), dramatizes the debate about the possibility of artificial intelligence by staging a dinner party discussion of the topic by philosopher Ludwig Wittgenstein (1889-1951), quantum physicist Erwin Schrödinger (1887-1961), computing pioneer Alan Turing (1912-1954), geneticist J. B. S. Haldane (1892-1964), and novelist, scientist, and civil servant C. P. Snow (1905-1980). For *Gödel*, Casti has teamed up with Werner DePauli, a statistician and computer scientist at the University of Vienna who has previously published a number of books in German about the life and work of Kurt Gödel.

The title *Gödel: A Life of Logic* is somewhat misleading, since it suggests that the book is an intellectual biography of the great Austrian mathematician. Instead, it consists of a set of essays on Gödel’s personal life, his work, and the implications of his work for problems in science and logic. Although these essays serve as helpful introductions to Gödel and as fascinating meditations on his work, the connections among them are often weak.

As the authors explain in their opening chapter, Kurt Gödel is best known for his 1931 proof of the incompleteness of mathematics, known as “Gödel’s Theorem.” This theorem established that there are mathematical facts, or relations among numbers, that can be seen to be true but that cannot be proven by mathematical induction. Mathematics, then, is an incomplete system, since one cannot account for all mathematical statements using the techniques of mathematics. The discoverer of this apparently paradoxical theorem, Casti and DePauli explain, lived in an unusually rich intellectual environment. He grew up in a region that had given birth to the mysticism of Jakob Böhme (1575-1624), the physics and philosophy of Ernst Mach (1838-1916), and the eerie literature of Franz Kafka (1883-1924). In 1924 Gödel settled in Vienna to study at the city’s university. There he came into contact with the influential group of physicists, mathematicians, and philosophers known as the Vienna Circle. He also became familiar with the work of the philosopher Ludwig Wittgenstein, who would argue that truth cannot be completely described by language. Casti and DePauli see Gödel’s Theorem as essentially a mathematical version of Wittgenstein’s view.

In their efforts to explain Gödel’s work, Casti and DePauli occasionally indulge in elaborate metaphors that contribute little to clarity or simplicity. Gödel’s Theorem was a response to a challenge posed by German mathematician David Hilbert at the 1928 International Congress of Mathematicians in Bologna, Italy. Hilbert had challenged his colleagues to develop a system of proving the truth or falsity of all mathematical statements. Casti and DePauli choose to introduce Hilbert’s Bologna address with the example of a chocolate cake machine capable of producing all possible chocolate cakes. The authors suggest that this kind of machine is analogous to a method for producing mathematical theorems. Many readers may find the digression on chocolate cakes a heavy-handed rhetorical ploy that simply obscures Hilbert’s problem.

Avoiding the difficult mathematical details of Gödel’s Theorem, Casti and DePauli do manage to give some idea of the logic behind it. Basically, Gödel showed that there must be an arithmetical counterpart of the self-referential sentence “This statement is not provable” in every formalization of arithmetic. That self-referential statement must be true if the formal system is consistent. If additional axioms from another system are added to prove the unprovable sentence, the new system will itself necessarily have an unprovable Gödel sentence.

After their efforts to explain Gödel’s difficult but important work, the authors turn back to their subject’s biography. The mathematician...

(The entire section is 1905 words.)