Kurt Gödel

Even in High School my brother was somewhat more one-sided than me and to the astonishment of his teachers and fellow pupils had mastered university mathematics by his final Gymnasium years. ... Mathematics and languages ranked well above literature and history. At the time it was rumoured that in the whole of his time at High School not only was his work in Latin always given the top marks but that he had made not a single grammatical error.

Kurt entered the University of Vienna in 1923. He was taught by Furtwängler, Hahn, Wirtinger, Menger, Helly and others. As an undergraduate he took part in a seminar run by Schlick which studied Russell's book Introduction to mathematical philosophy. Olga Tausky-Todd, a fellow student of Gödel's, wrote:-

It became slowly obvious that he would stick with logic, that he was to be Hahn's student and not Schlick's, that he was incredibly talented. His help was much in demand.

He completed his doctoral dissertation under Hahn's supervision in 1929 and became a member of the faculty of the University of Vienna in 1930, where he belonged to the school of logical positivism until 1938.

He is best known for his proof of Gödel's Incompleteness Theorems. In 1931 he published these results in Uber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme . He proved fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved.

This ended a hundred years of attempts to establish axioms to put the whole of mathematics on an axiomatic basis. One major attempt had been by Bertrand Russell with Principia Mathematica (1910-13). Another was Hilbert's formalism which was dealt a severe blow by Gödel's results. The theorem did not destroy the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than that envisaged by Hilbert's.

Gödel's results were a landmark in 20th-century mathematics, showing that mathematics is not a finished object, as had been believed. It also implies that a computer can never be programmed to answer all mathematical questions.

Gödel met Zermelo in Bad Elster in 1931. Olga Taussky-Todd, who was at the same meeting, wrote:-

The trouble with Zermelo was that he felt he had already achieved Gödel's most admired result himself. Scholz seemed to think that this was in fact the case, but he had not announced it and perhaps would never have done so. ... The peaceful meeting between Zermelo and Gödel at Bad Elster was not the start of a scientific friendship between two logicians.

In 1933 Hitler came to power. At first this had no effect on Gödel's life in Vienna. He had little interest in politics. However after Schlick, whose seminar had aroused Gödel's interest in logic, was murdered by a National Socialist student, Gödel was much affected and had his first breakdown. His brother Rudolf wrote

This event was surely the reason why my brother went through a severe nervous crisis for some time, which was of course of great concern, above all for my mother. Soon afer his recovery he received the first call to a Guest Professorship in the USA.

In 1934 Gödel gave a series of lectures at Princeton entitled On undecidable propositions of formal mathematical systems. At Veblen's suggestion Kleene, who had just completed his Ph.D. this at Princeton, took notes of these lectures which have been subsequently published.

He returned to Vienna, married Adele Porkert in 1938, but when the war started he was fortunate to be able to return to the USA although he had to travel via Russia and Japan to do so.

In 1940 Gödel emigrated to the United States and held a chair at the Institute for Advanced Study in Princeton, from 1953 to his death. He received the National Medal of Science in 1974.

His work Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1940) is a classic of modern mathematics.

His brother Rudolf, himself a medical doctor, wrote:-

My brother had a very individual and fixed opinion about everything and could hardly be convinced otherwise. Unfortunately he believed all his life that he was always right not only in mathematics but also in medicine, so he was a very difficult patient for doctors. After severe bleeding from a duodenal ulcer ... for the rest of his life he kept to an extremely strict (over strict?) diet which caused him slowly to lose weight.

Towards the end of his life Gödel became convinced that he was being poisoned and, refusing to eat to avoid being poisoned, starved himself to death.

In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules an axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths ... It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.

Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions ... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced.

He proved it impossible to establish the internal logical consistency of a very large class of deductive systems - elementary arithmetic, for example - unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves ... Second main conclusion is ... Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set... Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set.

The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:

1. Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
2. Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
3. Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
4. Now Gödel laughs his high laugh and asks UTM whether G is true or not.
5. If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
6. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
7. "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."

With his great mathematical and logical genius, Gödel was able to find a way (for any given P(UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics.

Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth ... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.

All consistent axiomatic formulations of number theory include undecidable propositions ...

Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved ...

How can you figure out if you are sane? ... Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is "peculiar' or not, given that you have only your own logic to judge itself? I don't see any answer. I am reminded of Gödel's second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent.

The other metaphorical analogue to Gödel's Theorem which I find provocative suggests that ultimately, we cannot understand our own mind/brains ... Just as we cannot see our faces with our own eyes, is it not inconceivable to expect that we cannot mirror our complete mental structures in the symbols which carry them out? All the limitative theorems of mathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally.

References:

Jones and Wilson, An Incomplete Education
Boyer, History of Mathematics
Nagel and Newman, Gödel's Proof
Rucker, Infinity and the Mind
Hofstadter, Gödel, Escher, Bach