Who was Kurt Godel the mathematician

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Kurt Godel
Mathematician of the century and great explorer

Rebecca Goldstein
Piper, 2nd edition 2006, 312 pages, € 19.90
Piper TB, 2007, 312 pages, € 9.95

ISBN: 3-492-04884-6
ISBN: 3-492-24960-4

Two men stroll through Princeton, New Jersey. The scene comes either from the 1940s or early 1950s and forms the introduction to the biography of one of these two men. Together with Heisenberg's uncertainty principle, these two scientists, because they are the ones involved, have made probably the most spectacular and most revolutionary scientific contributions of the 20th century. One of the two is Albert Einstein, whose theory of relativity is well known even to non-experts (at least by name, much less often by the exact content) and whose person has achieved one of the highest degrees of familiarity in the world. His interlocutor and frequent companion on walks during this time is Kurt Gödel. A mathematician who for the most part, in contrast to Einstein, is himself known only to experts as a person (regardless of his results). His incompleteness theorems, however, are certainly equivalent to Einstein's results in their scientific meaning and effect even beyond mathematics on logic and philosophy. It is this person Kurt Gödel whose life and scientific work are dealt with in this book by Rebecca Goldstein.
Put simply, Gödel's two theorems of incompleteness say that in every formal system that has at least one theory of natural numbers there are undecidable formulas and that the consistency of such a system cannot be demonstrated within the system itself. In plain language this means: In mathematics there are formulas that can neither be identified as true nor as false and it is impossible to prove the consistency of mathematics with mathematical methods. The effects are even greater. Even if we modify our mathematical system of axioms, for example by axiomatically deciding such an undecidable formula, i.e. its answer as an axiom in ours Set of rules add, there are further undecidable formulas in the new system and the freedom from contradictions can be proven just as little as before. So it is not up to our system, but the problem is logically unavoidable.
Understandably, this does not sound as spectacular to the non-mathematician or non-logician (non-philosopher) like Einstein's curved space-time, in which time even slows down at extremely high speeds, but the scientific importance cannot be overestimated . Note, for example, that the great mathematician David Hilbert gave his famous speech at the International Congress of Mathematicians in Paris in 1900, at which he presented his 23 most important mathematical problems, the solution of which represented the most important task of the mathematicians of the 20th century should deliver the following as the second problem:
"But above all, among the numerous questions that can be asked with regard to the axioms, I would like to designate this as the most important problem, to prove that they are mutually free from contradictions, that is, that one can never reach them on the basis of a finite number of logical inferences Can achieve results that are in contradiction to one another. "
So Godel has this Second Hilbert problem solved by proving that it cannot be solved.
Another reason for the different effects that the two walkers Gödel and Einstein exerted on the public is of course the completely different personalities of the two. Godel withdrew as far as possible from his environment throughout his life. He also always remained reserved and almost shy in meeting others. Even when his results, which Gödel himself was well aware of, were initially hardly recognized and appreciated by his colleagues, he made no great effort to convince the mathematician community of them. In addition, as Rebecca Goldstein points out in several places, he was characterized by an almost self-denial obedience to the authorities and a certain cosmopolitanism, which on the one hand hardly perceive or correctly recognize the political developments around him (e.g. the takeover of power by the National Socialists in Germany) was assessed. On the other hand, it led to the fact that, even after his achievements and results were universally recognized and appreciated, he was constantly worried about his employment at the institute.
As early as 1934 he spent a few weeks in the sanatorium as a result of a nervous breakdown which can be traced back to overwork. When he was in the United States for a second time to study in the summer semester of 1939, Czechoslovakia was annexed by the German Reich. To the horror of his friends and acquaintances in America, however, Godel insisted on returning to Vienna for fear that the new rulers might deprive him of his lectureship. Rebecca Goldstein illustrates his cosmopolitanism and at the same time problematic psychological condition very well with the statement: "A man who was frightened by a refrigerator because he believed that it emitted toxic gases, returned to a Vienna that was run by the Nazis Had been "taken over" in order to enforce "his rights."
Only after he was caught and almost beaten by a gang of young thugs because of his "Jewish appearance" (Godel was not a Jew) (his wife came to his aid) and after he was unexpectedly classified as "fully fit" for military service (despite his alleged heart defect, in which he believed since he was eight years old), he made the decision to leave Vienna for good and to accept the invitation from Princeton. In an effort to get him an exit permit, the director of the Institute for Advanced Studies argued, among other things, that Gödel's departure could not set a precedent because there were "only a few people of similar scientific importance". It was there in Princeton where Gödel met Einstein and where he found his most important caregiver until his death.
Both the young years in Vienna, the membership in Vienna CircleRebecca Goldstein describes the influences of Wittgenstein on the intellectual life there and the mutual rejection of the views of Gödel and Wittgenstein, as well as the increasing withdrawal from the world after Einstein's death, which ultimately led to her own death through malnutrition and exhaustion, describes Rebecca Goldstein in a worthy and interesting way Wise. She also manages to present Gödel's mathematical and logical-philosophical results in an understandable way and to show their meanings and effects on science.
The book offers a balanced presentation of scientific interpretation, biographical biography and interesting and entertaining anecdotes, such as the now very famous story that Gödel informed Einstein before his appointment to obtain American citizenship that he had discovered a logical contradiction in the constitution, which prompts him not to leave Gödel on the said day and to distract him from the topic with all kinds of conversations on the way.
It is an extremely readable biography for mathematicians as well as for people who have not dealt with the subject matter, and one can assume that it brings Kurt Gödel a little closer to the position he actually deserved , as perhaps the greatest mathematician of the 20th century.

(Review: Jörg Beyer)

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