FOM: Cantor and Hilbert knew what they were talking about

[Stewart Shapiro]

Martin Davis wrote:

>This is somewhat unfair to Hilbert. What he spoke of was "the conviction ...
>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". (emphasis mine) With this escape
>hatch, his call for a decision problem for diophantine equations was quite
>reasonable. Just a few years before Matiyasevich's work, Julia Robinson
>(having, as she said, "lost faith") was attempting to find such a decision
>procedure.
>

I second this.  In the 1900 Mathematical Problems lecture Hilbert says that
sometimes problems receive completely satisfactory solutions in unintended
ways.  He gives an example of the problem to construct a perpetual motion
machine, and its "solution" the development of the theory that shows that
such a machine is impossible.  It seems to me that this is a foreshadowing
of developments in mathematical logic.