Is God a Mathematician_ - Mario Livio [92]
Kurt Gödel (Figure 53) was born on April 28, 1906, in the Moravian city later known by the Czech name of Brno. At the time, the city was part of the Austro-Hungarian Empire, and Gödel grew up in a German-speaking family. His father, Rudolf Gödel, managed a textile factory and his mother, Marianne Gödel, took care that the young Kurt got a broad education in mathematics, history, languages, and religion. During his teen years, Gödel developed an interest in mathematics and philosophy, and at age eighteen he entered the University of Vienna, where his attention turned primarily to mathematical logic. He was particularly fascinated by Russell and Whitehead’s Principia Mathematica and by Hilbert’s program, and chose for the topic of his dissertation the problem of completeness. The goal of this investigation was basically to determine whether the formal approach advocated by Hilbert was sufficient to produce all the true statements of mathematics. Gödel was awarded his doctorate in 1930, and just one year later he published his incompleteness theorems, which sent shock waves through the mathematical and philosophical worlds.
Figure 53
In pure mathematical language, the two theorems sounded rather technical, and not particularly exciting:
1. Any consistent formal system S within which a certain amount of elementary arithmetic can be carried out is incomplete with regard to statements of elementary arithmetic: there are such statements which can neither be proved nor disproved in S.
2. For any consistent formal system S within which a certain amount of elementary arithmetic can be carried out, the consistency of S cannot be proved in S itself.
The words may appear to be benign, but the implications for the formalists’ program were far-reaching. Put somewhat simplistically, the incompleteness theorems proved that Hilbert’s formalist program was essentially doomed from the start. Gödel showed that any formal system that is powerful enough to be of any interest is inherently either incomplete or inconsistent. That is, in the best case, there will always be assertions that the formal system can neither prove nor disprove. In the worst, the system would yield contradictions. Since it is always the case that for any statement T, either T or not-T has to be true, the fact that a finite formal system can neither prove nor disprove certain assertions means that true statements will always exist that are not provable within the system. In other words, Gödel demonstrated that no formal system composed of a finite set of axioms and rules of inference can ever capture the entire body of truths of mathematics. The most one can hope for is that the commonly accepted axiomatizations are only incomplete and not inconsistent.
Gödel himself believed that an independent, Platonic notion of mathematical truth did exist. In an article published in 1947 he wrote:
But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.
By an ironic twist of fate, just as the formalists were getting ready for their victory march, Kurt Gödel—an avowed Platonist—came and rained on the parade of the formalist program.
The famous mathematician John von Neumann (1903–57), who was lecturing on Hilbert’s work at the time, canceled the rest of his planned course and devoted the remaining time to Gödel’s findings.
Gödel the man was every bit as complex as his theorems. In 1940, he and his wife Adele fled Nazi Austria so he could take up a position at the Institute for Advanced Study in Princeton, New Jersey. There he