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.
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.
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,