The Gödel Letter
Princeton, 20 March 1956
Dear Mr. von Neumann:
With the greatest sorrow I have learned of your illness. The news came to me as quite unexpected. Morgenstern already last summer told me of a bout of weakness you once had, but at that time he thought that this was not of any greater significance. As I hear, in the last months you have undergone a radical treatment and I am happy that this treatment was successful as desired, and that you are now doing better. I hope and wish for you that your condition will soon improve even more and that the newest medical discoveries, if possible, will lead to a complete recovery.
Since you now, as I hear, are feeling stronger, I would like to allow myself to write you about a mathematical problem, of which your opinion would very much interest me: One can obviously easily construct a Turing machine, which for every formula F in first order predicate logic and every natural number n, allows one to decide if there is a proof of F of length n (length = number of symbols). Let ψ(F,n) be the number of steps the machine requires for this and let φ(n) = maxF ψ(F,n). The question is how fast φ(n) grows for an optimal machine. One can show that φ(n) ≥ k ⋅ n. If there really were a machine with φ(n) ∼ k ⋅ n (or even ∼ k ⋅ n2), this would have consequences of the greatest importance. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem, the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine. After all, one would simply have to choose the natural number n so large that when the machine does not deliver a result, it makes no sense to think more about the problem. Now it seems to me, however, to be completely within the realm of possibility that φ(n) grows that slowly. Since it seems that φ(n) ≥ k ⋅ n is the only estimation which one can obtain by a generalization of the proof of the undecidability of the Entscheidungsproblem and after all φ(n) ∼ k ⋅ n (or ∼ k ⋅ n2) only means that the number of steps as opposed to trial and error can be reduced from N to log N (or (log N)2). However, such strong reductions appear in other finite problems, for example in the computation of the quadratic residue symbol using repeated application of the law of reciprocity. It would be interesting to know, for instance, the situation concerning the determination of primality of a number and how strongly in general the number of steps in finite combinatorial problems can be reduced with respect to simple exhaustive search.
I do not know if you have heard that “Post’s problem”, whether there are degrees of unsolvability among problems of the form (∃ y) φ(y,x), where φ is recursive, has been solved in the positive sense by a very young man by the name of Richard Friedberg. The solution is very elegant. Unfortunately, Friedberg does not intend to study mathematics, but rather medicine (apparently under the influence of his father). By the way, what do you think of the attempts to build the foundations of analysis on ramified type theory, which have recently gained momentum? You are probably aware that Paul Lorenzen has pushed ahead with this approach to the theory of Lebesgue measure. However, I believe that in important parts of analysis non-eliminable impredicative proof methods do appear.
I would be very happy to hear something from you personally. Please let me know if there is something that I can do for you. With my best greetings and wishes, as well to your wife,
Sincerely yours,
Kurt Gödel
P.S. I heartily congratulate you on the award that the American government has given to you.
I am curious about the lower bound result mentioned in the letter:
“it seems that φ(n) ≥ k ⋅ n is the only estimation which one can obtain by a generalization of the proof of the undecidability of the Entscheidungsproblem”.
Is it an easy result ? Has a proof sketch been found in the margin of Gödel’s notebook?
Was it rediscovered independently; does it follow from a result published thereafter ?
All right, I think I can half-answer my own question (but further comments are still very welcome) : there is a lower bound on the complexity of predicate calculus in a little-known letter by Steve Cook to the STOC’71 program committee.
A proof of the claim φ(n) ≥ kn is given in my paper “On Gödel’s theorems on lengths of proofs II: Lower bounds for recognizing k symbol provability”, in Feasible Mathematics II, P. Clote and J. Remmel (eds), Birkhauser, 1995, pp. 57-90. The paper is also on my web page.
The proof uses a diagonalization argument, so perhaps it is similar to the proof that Godel had in mind.
As I remember, this letter was originally written in German and a rather difficult translation was arranged by Mike Sipser. If you happen to know more details about this, it would be nice to mention them in the post.
The letter was indeed originally written in German. Mike Sipser’s translation can be found in “The History and the Status of the P versus NP Question”, in the 24th STOC proceedings, 1992, pp. 603-618. This also contains a number of other historical quotes.
Peter Clote also made a translation to English, which can be found in the book “Arithmetic, Proof Theory and Computational Complexity”, P. Clote and J. Krajicek, eds., Oxford Univ. Press, 1993.
Sam, thanks. That reference let me find Mike Sipser’s paper:
http://www.seas.harvard.edu/courses/cs121/handouts/sipser-pvsnp.pdf
According to the paper, the translation in it was done by Soren Istrail and Arthur S. Wensinger.
Footnote 14 of the translation contains a mistake. It sais:
“Gödel uses the English word ‘Analysis’ here, not the German term ‘Analytik’.”
Istrail and Wensinger (or whoever did the translation) overlook that the German technical term for “analysis” is “Analyis”, not “Analytik”. “Analytik” doesn’t mean the mathematical field of analyis, but the philosophical field of analytics. (Also, it may be used in chemistry, meaning “chemical analysis”.)
Unfortunately, I misspelled the word: “Analysis” is right (instead of “Analyis”). Sometimes, some words don’t want to be typed