In response to:
The Scientist as Rebel from the May 25, 1995 issue
To the Editors:
In an eloquent article on “The Scientist as Rebel” [NYR, May 25], Freeman Dyson writes of “the great mathematician David Hilbert, (who) after thirty years of high creative achievement … walked into a blind alley of reductionism…. (He) proposed to solve the problems of mathematics by finding a general process that could decide … (a) statement…. He called the process the Entscheidungsproblem. He dreamed of solving the Entscheidungsproblem and thereby solving as corollaries all the famous unsolved problems of mathematics…. Finally, when Hilbert was seventy years old, Kurt Gödel proved by a brilliant analysis that the Entscheidungsproblem as Hilbert formulated it cannot be solved.”
These statements are each and all false or misleading. It so happens that I was a student of Mathematical logic in Göttingen (Hilbert’s University) in 1931–33, just after the publication of the famous 1931 paper by Gödel. Hence I venture to reply.
Hilbert had long (at least since 1900) been interested in the famous problems of mathematics. He also sharply disagreed with the insightful Dutch mathematician, L. E. J. Brouwer, an intuitionist. Brouwer rejected one of the classical rules of logic, (“the tertium non datur”) which claims that any specific statement is either true or false. This principle was (and is) vital to many standard proofs in mathematics. Hilbert opposed Brouwer’s views (on this and other matters) and considered them an attack on the basic aim of mathematics. He held that the problems of mathematics can all ultimately be solved: this he formulated in a famous speech in the words “Wir müssen wissen; wir werden wissen.” (We must know, we will know). Also in 1928, in an influential small book with W. Ackerman, he formulates (as Dyson states) an explicit version of the Entscheidungsproblem for a certain specific formal system (the first order predicate calculus, which covers essentially the logic of “and,” “or,” “not,” “implies,” and “there exists.”) Dyson states that “he dreamed of solving the Entscheidungsproblem and thereby solving as corollaries all the famous unsolved problems of mathematics.”
I was not and am not privy to Hilbert’s dreams. I know no evidence that Hilbert thought that the first solution would automatically give a solution to all other problems (e.g. Fermat’s last theorem). He hoped chiefly that somehow they all could someday be solved. This was the tone of his general lecture course on the philosophy of science which I heard him give in late 1931. In his massive 1934 book with his talented assistant Paul Bernays, he first discusses special cases, then writes (my translation) “we are still very far from a general solution to this problem.” With a footnote “the impossibility of such a general situation has since been proved by Alonzo Church in 1936.” I talked extensively in Göttingen with Professor Bernays: I am confident that the first sentence here was written before Gödel.
In 1931, just as Hilbert asserted that we must know, Kurt Gödel showed that the Entscheidungsproblem in fact could not be solved—more explicitly that it could not be solved in the formal system of Whitehead and Russell’s “Principia Mathematica” and the methods derived in that system. The 1936 paper by Church did the same for the specific formal system (the predicate calculus) used by Hilbert.
It remains possible that the problem for a given system can be solved by means of some other more powerful system. Such results have in fact been obtained by Gerhard Gentzen, a fellow student of mine 1931–33 in Göttingen, and an early leader in the deep branch of logic called (Hilbert’s) “proof theory.”
Dyson has used his inaccurate descriptions of Hilbert to attack “reductionism.” In this case he has missed the target. At the age about 65, Hilbert extracted the formulation of the first order predicate calculus from the pedantic morass of “Principia Mathematica” (1910). Hilbert’s work was essential background for Gödel. As Dyson stated, Hilbert was “reducing mathematics to a set of marks written on paper, and deliberately ignoring the context of ideas and applications that give meaning to the marks.”
(Hilbert himself called this “metamathematics.” He used this for a specific limited purpose, to show mathematics consistent.) Without this reduction, no Gödel’s theorem, no definition of computability, no Turing machine, and hence no computers.
Dyson simply does not understand reductionism and the deep purposes it can serve. Hilbert was not “sterile.”
Saunders Mac Lane
Department of Mathematics
The University of Chicago
Freeman Dyson replies:
I am delighted that Saunders Mac Lane, a legendary figure in the world of mathematics, takes the trouble to read my article and to disagree with it. He certainly knows more about Hilbert than I do. He is one of the few mathematicians still alive who heard Hilbert lecture and knew Hilbert’s Göttingen in its days of glory before 1933. I am glad to be corrected by him on questions of historical fact. But I find that the differences between us lie in questions of interpretation rather than in questions of fact. I agree with all the facts stated in his letter. I too was a student of mathematical logic before I turned to physics. I too read Hilbert and Bernays and Gödel and Church and Gentzen. I too grieved when I heard the news of Gentzen’s untimely death. I too was exhilarated and inspired by the enormous deepening of mathematical understanding that grew in the 1930’s out of the ruins of Hilbert’s program of formalization. Only, Mac Lane would use the words “upon the foundations” where I say “out of the ruins.” Solid foundations and ruined hopes are not incompatible. Both were essential parts of the legacy that Hilbert left to his successors. The main difference between Mac Lane and me, as I see it, is a matter of temperament. He is by temperament a reductionist, and I am not. He believes that the reduction of mathematical concepts to their abstract logical components is the main road of progress in understanding. I prefer to deepen my understanding of abstract components by building them up into concrete structures. I do not deny the power and the beauty of reductionist science, as exemplified in the axioms and theorems of abstract algebra or algebraic topology. But I assert the equal power and beauty of constructive science, as exemplified in Gödel’s construction of an undecidable proposition or in Gentzen’s construction of an enlarged domain for mathematical logic.
Hilbert himself was, of course, a master of both kinds of mathematics.