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In My Collaboration with JULIA ROBINSON it is said

Hilbert foresaw the possibility of negative solutions to some mathematical problems

What evidence is there that Hilbert knew (around the time when he set the problems) or suspected that some decision problems (like resolution of Diophantine equations) can be impossible?

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Since Hilbert died several years after the publication of the first incompleteness and undecidability results, at some stage he knew. There is no reason to think that he knew or suspected at a much earlier stage. His famous pronouncements "In mathematics there is no ignorabimus" "We must know; we shall know" are in the opposite direction. Maybe the painful progress of his grad students in the late 1920s on a decision problem for the predicate calculus gave him cause to worry, but I do not know of any direct evidence. –  André Nicolas May 14 '11 at 16:22
Continuation of comment: Even after it had been proved that (if Peano Arithmetic is consistent) its consistency cannot be proved within PA, the Hilbert program of producing a consistency proof continued, culminating in the work of Hilbert's assistant Gentzen. This provides some tangential evidence that Hilbert held on to the "In mathematics there is no ignorabimus." Of course Hilbert always knew that some problems, like the question of equi-decomposability of tetrahedra of the same volume, are settled in the negative. –  André Nicolas May 14 '11 at 16:58
In 1902 an authorized translation of Hilbert's address was published in the Bulletin of the New York Mathematical Society (freely accessible, as far as I can tell). The passage on p.444f (Occasionally...) is especially relevant to your question. The famous "[...] in mathematics there is no ignorabimus." is on p. 445. –  t.b. May 14 '11 at 17:15

1 Answer 1

up vote 8 down vote accepted

See Hilbert's famous 1900 lecture:


Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts.

I don't think it gets more explicit than that.

That being said, I think he believed that his own problems (the Tenth problem, the Entscheidungsproblem...) can be solved - at least, I have never seen evidence to the contrary (not that I know much about these things...)

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So we referred to the exact same passage :) See my comment above. –  t.b. May 14 '11 at 17:31
Note also that even this statement is overly optimistic, since we now know that there are problems that are neither provable nor provably unprovable. –  Mario Carneiro Feb 26 at 18:09

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