revised edition, D. R. Hofstadter (ed)
155 pages, ISBN 0-8147-5816-9, New York University Press, New York, 2001.www.nyupress.org
Reviewed by J. M. Haile, Macatea Productions, http://www.macatea.com/
In an axiomatic study we stipulate a (few) axioms and from those axioms deduce theorems that describe features of the objects under study. The classic example is Euclid's use of five axioms as the basis for deriving all the theorems of plane geometry. Motivated by Euclid's success, we may reasonably expect that the method could be applied to other domains; can we, for example, develop arithmetic on a purely axiomatic basis?
In 1931 Kurt Gödel proved two important theorems about such attempts. (1) A consistency theorem (consistency here means freedom from contradiction): Not all axiomatic systems can be proven to be consistent by appealing only to the theorems and axioms of the axiomatic system itself. (2) A completeness theorem: Within any consistent axiomatic system there are statements that can be neither proved nor disproved although they are in fact true. Gödel's results are profound, for they not only showed that a complex domain, such as arithmetic, cannot be fully constructed by an axiomatic approach, but they also stimulated new modes of thinking. For example, the completeness theorem draws a clear line between truth and provability.
Gödel's 1931 paper is highly technical but of such importance that in 1958 Nagel and Newman published a brief description of Gödel's reasoning, aimed at those interested in mathematics, logic, and philosophy. That original edition has now been lightly edited and embellished with a personal foreword by Douglas Hofstadter.
Besides traditional views of Gödel's work, Hofstadter emphasizes that Gödel was the first to demonstrate that any pattern can be reduced to numbers: his proofs depend on such mappings. Today we not only take such mappings for granted, but we are hardly aware that we continuously invoke them. We word process, we create and view images, we compose and play music, we calculate, we simulate, we communicate, we design—and all those many patterns are ultimately mapped into numbers—ones and zeroes deep in some computer's hardware. So not only are Gödel's proofs important, but his method of proof was seminal and prophetic.
Before getting to the proofs themselves, Nagel and Newman use six short chapters to place Gödel's work within its historical and mathematical context. These clear and carefully written chapters are a great help not only in understanding Gödel's work but also in appreciating the history of mathematics. They touch on many important aspects of mathematical thought: models, formal systems, the nature of proof, mapping, metamathematical statements, logical paradoxes, tautologies. The presentation, in some measure, traces the transition of mathematics from the science of counting to the science of abstract deductive proof. If the book contained only those six chapters, it would be a valued contribution to the literature. As it is, it is a small gem (How do you prove that a statement cannot be proved?).
Depending on your background, you may be able to read through this book only once and grasp the ideas, but most of us need several readings—not because the language is obscure (it is not), but because the presentation is rich and you must follow the arguments closely to finally visualize the logical structure that is being assembled. No matter—this little book will repay the effort.
(jmh 22 June 06) © 2006 by J. M. Haile. All rights reserved.