Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries Series) Book

I've always been fascinated by Kurt Gödel and his incompleteness theorems. While Douglas Hofstadter did a fine job in explaining the latter in his book Gödel, Escher, Bach , and also in a video lecture, there's hardly any biographical/personal information about the human behind the mathematician here to be found. That's where Rebecca Goldstein jumps in. Her book focuses on the life of the "greatest logician since Aristotle". About his time at the Vienna Circle (a.k.a. the Schlick-Group) in the late 1920s, the emigration to Princeton (the Institute for Advanced Study), where Gödel became friends with Albert Einstein, and finally his personal incompleteness and his tragic death, apparently brought on by self starvation in 1978.

There is, of course, a chapter devoted to Gödel's proof of the incompleteness of formal systems, which is not entirely without mathematics and formulas. The proof itself is rather simplified and no great mathematical knowledge is required. Logical understanding is very helpful, however!

As opposed to most of the participants in the Viennese circle, Gödel was not a positivist, but rather held it with Plato: He had an axiom by which he looked at the world: nothing that happens in it is due to accident or stupidity. Gödel believed in an abstract reality, and "that the truths of mathematics are independent of any human activities, such as the construction of formal systems ' with their axioms, definitions, rules of inference, and proofs." The positivists, on the other hand, believed in the creation of the (meta-) mathematical reality by man alone. Everything outside this reality, all meta-mathematics, was meaningless and of no importance to them. Since Gödel also acted in this circle of positivists, there was reason for many to believe that his proof of incompleteness was a success story of positivism, but, according to Goldstein, "could not be further from the truth."

For me the most exciting topic was the confrontation of Gödel and Ludwig Wittgenstein, who was venerated by the Viennese circle in an almost mystical way.

Wittgenstein's Tractatus Logico-Philosophicus was apparently read and read again in the circle for months, as if it were a holy book. Gödel never acknowledged Wittgenstein's work, and Wittgenstein was "adamant in denying the possibility of a proof such as Gödel's." The characters of the two could not be more different. Ironically, however, one can conceive the final statement of the Tractatus as a kind of incompleteness theorem: "Whereof one cannot speak, thereof one must be silent." Unfortunately, Gödel was anything but ironically inclined, and perhaps he did not see this, so that one can almost speak of a meta-irony here.

Gödel's other important relationship was the warm and close friendship with Albert Einstein. The two of them were very dissimilar (Einstein was 27 years older than Gödel), but they understood each other well and appreciated each other extremely. Einstein confessed that "he only went to his office to have the privilege of walking home each day with the logician, the two great minds of the twentieth century able to share, at least for a while, their intellectual exile with one another." After Einstein's death in 1955 Gödel became very lonely indeed. The final chapter shows the incessant descent of this brilliant thinker quite impressively, as I find.

Apart from the formal parts of the third chapter, this is a very easy to grasp and recommendable reading for all those who want to make themselves more familiar epistemology, history of science, meta-mathematics and logic.


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