Remembering Kurt Gödel
Tea with Kurt and Adele
Glimpses of Gödel from Freeman Dyson
Looking through his collection of letters written to his parents in England, Freeman Dyson, Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study, discovered a report of his visit to the home of Kurt and Adele Gödel, written less than a month after he first arrived at the Institute in 1948: “It is remarkable that Gödel invited me to his home, comments Professor Dyson, “I was a fresh post-doc and he had never seen me before. He was much more friendly and much more hospitable than I had expected. Here is what I wrote."
Before I stop I must tell you about my meeting with Gödel. We talked mainly about mathematics and physics. He is an amusing talker, and not so pathologically shy in his home as he is at the Institute. He is a little man of about forty, with a fat little Austrian wife, and they live together seeing very little of anybody, and will no doubt continue to do so for many years to come since he is a permanent member of the Institute. The interesting thing to me was to learn what Gödel is doing and proposes to do in the way of research. He produced during his youth two epoch-making discoveries in pure mathematics, one in 1932 and one in 1939, and since then nothing has been heard from him. With the whole of mathematics to choose from and his unrivalled talents, I was very curious to know what such a man would choose to do. The answer, when it came, was completely baffling. It turns out that he has spent the last few years working in physics, in collaboration with Einstein, on problems of general relativity.
I will try to explain why this is so baffling. In the first place, there is no question of Gödel suffering from deterioration of intellect. He understands general relativity and its position in physics as well as anybody, and knows quite well what he is about. He has found some results which will certainly be of interest to specialists in relativity. On the other hand, it is fairly clear to most people that general relativity is one of the least promising fields that one can think of for research at the present time. The theory is from a physical point of view completely definite and completely in agreement with all experiments. It is the general view of physicists that the theory will remain much as it is until there are either some new experiments to upset it or a development from the direction of quantum theory to include it. And in spite of all this, there is Gödel.
Professor Dyson comments: “Unfortunately there is only one more glimpse of Gödel in the letters. Two months later [letter of November 25, 1948] he comes with his wife to the Institute dance, which was held in the common room. His wife dances with me but he does not dance. He was standing around by himself miserably while all this went on. He did not seem to talk to anybody all the evening. My memories of those early years at the Institute are faded and fragmentary. In later years I interacted very little with Gödel. He sometimes called me by telephone to ask whether the astronomers had any new observations, which could either confirm or contradict his model of a rotating universe. That was the model that he had been working on when he talked to me in 1948. I had to tell him each time that the observations of galaxy distributions and of the cosmic microwave background radiation were not yet accurate enough to set interesting limits on the speed of rotation of the universe. He always remained intensely interested in finding out whether his model might give a true picture of the universe we live in. Now, thirty years later, the observations are far more precise, and we still see no evidence that we live in Gödel’s rotating universe.
Gödel's Influence on the Development of Theoretical Computing
Gödel’s work in the 1930s had a strong influence on the development of theoretical models of computation (and thereby, to the creation of real computers and the computer revolution we live today). Firstly, his definition of recursive functions gives one of the first models of computation, influencing Turing and Church in their study of the universality of this notion. Next, Alan Turing’s seminal work on defining computation (via Turing machines) has strong analogies with Gödel’s work on Incompleteness, and was certainly strongly influenced by it. In particular, the use of the diagonal argument in Turing’s work for proving Incomputability of basic computational tasks strongly resembles Gödel’s argument on Incompleteness, especially the dual role played by integers as simultaneously representing data as well as programs/formulae.
Gödel’s insight into computing has gone far beyond computability. In the 1950s he was interested in efficient computation the main focus of current research in Theoretical Computer Science. In a (recently discovered) letter to von Neumann, Gödel foreshadowed some of the insights of the 1970s. In it, he almost precisely defines a major problem of this field (and more generally of Mathematics), the P versus NP problem, with which we are struggling to this very day.
--Avi Wigderson, Herbert H. Maass Professor in the School of Mathematics, Institute for Advanced Study
Kurt Gödel was logician, mathematician, and philosopher, all rolled into one. He was perhaps the foremost logician of the twentieth century, with major ground-breaking results in the theory of formal deductive systems, in automata theory, in intuitionistic mathematics, in set theory, and in relativistic physics, most within the span of a scant 20 years. His most celebrated result is the pair of 1931 theorems on the incompleteness of formal systems of arithmetic, in which he showed that it is impossible to devise a system of axioms for even the elementary arithmetic of whole numbers that are sufficient for answering every question that can be framed in their terms—including the question of their own consistency. If devising axioms and constructing proofs that consist in the derivation of consequences from those axioms is a central way that mathematics proceeds, these theorems show its task to be essentially incompletable, even within areas already under investigation.
The questions Gödel chose to investigate and the results he achieved were, by his own admission, guided by his philosophical views on the nature of mathematical activity as a mental activity that cannot be modeled as any sort of mechanical process.
He wrote as follows in 1963 by way of suggested correction to a proposed article that was to appear in TIME magazine:
Before Gödel’s work it had been widely conjectured that any precisely formulated mathematical yes or no question can be decided by the mechanical rules of logical inference on the basis of a few mathematical axioms. In 1931, Gödel showed that, no matter what axioms are chosen, there exist number-theoretical yes or no questions not decidable from the axioms. Combining this proof with A.M. Turing’s theory of computing machines, one arrives at the following conclusion: either there are infinitely many number-theoretical questions which human reason is unable to answer or human reason contains an element which, in its action, is totally different from any finite combinatorial mechanism and its parts. Gödel hopes it will be possible to prove that the second alternative holds.
--Paul Benacerraf, James S. McDonnell Distinguished University Professor of Philosophy, Princeton University