While some problems that Hilbert listed were solved, perhaps what has been overlooked is the intellectual malleability needed for their formulation as: questions, tentative answers? Do we need more David Hilbert?
I'm sure the answer to the last question is "yes" (we can always use additional geniuses, particularly in mathematics, the tool of the sciences for some and the queen of the sciences for others). As to the "intellectual malleability needed for their formulation", I'm not sure that really has been over-looked. Perhaps it has mostly been examined in the context of the nature of whether or not (and to what extent) certain answers are or should be considered solved (and who solved them), but even so this is impossible to study without getting into the nature of the questions and answers in all manner of respects. Most of the problems considered solved or partially solved were not solved by one individual or in a straight-forward manner, and there is disagreement over many answers.
Consider especially question 6: "For the general statements of the 6th Problem it seems unclear now how to formulate criteria of solutions" (see attached link). Granted, question 6 was unique among the total 23 (or 10, or 24, depending upon whether one goes with the problems as presented, as first published, or as including the additional question). However, it isn't the only one that posed problems due to being vague, even if the vagueness comes mainly from retrospection. Problem 1, for example, is as "solved" as it will ever be, in that Gödel and Cohen's proofs, taken together, show that it cannot be solved. Gödel actually contributed to two such problems, the other being question 2 (and the other contributor being Gentzen and possibly Novikov). Gödel's seems to have shown that the question, as intended/meant by Hilbert, is again unsolvable, although Gödel himself did not consider is incompleteness proof to have shown question 2 to be unanswerable. Part of the problem (the clearest part) revolves around the issue of a formal treatment of finitary proof. But part of it rests upon ambiguity in the question, especially as it relates to Hilbert's program which it is quite connected to:"ob etwa gewisse Aussagen einzelner Axiome sich untereinander bedingen und ob nicht somit die Axiome noch gemeinsame Bestandteile enthalten, die man beseitigen muß, wenn man zu einem System von Axiomen gelangen will, die völlig von einander unabhängig sind....Vor Allem aber möchte ich unter den zahlreichen Fragen, welche hinsichtlich der Axiome gestellt werden können, dies als das wichtigste Problem bezeichnen, zu beweisen, daß dieselben untereinander widerspruchslos sind, d.h. daß man auf Grund derselben mittelst einer endlichen Anzahl von logischen Schlüssen niemals zu Resultaten gelangen kann, die miteinander in Widerspruch stehen. " ["whether in some way there [are] particular statements about individual axioms' interdependence and whether therefore the axioms might not share common elements, which must isolate, if one is to arrive at a system of axioms that are from one another wholly independent...Above all I would like among the many questions, which can be asked of the axioms, this be designated the most important problem, to prove that these are not contradictory, i.e., that, based on these through a finite number of logical steps, one can never reach conclusions that are contradictory."]
I'm not happy with the translation but it's not worth spending the time to determine how best to render it into idiomatic English. The main point is that, while problem two is often summarized in terms of the proof that the axioms of arithmetic are consistent & complete, What Hilbert actually says of the axioms of arithmetic is "Zum Nachweise für die Widerspruchslosigkeit der arithmetischen Axiome bedarf es dagegen eines direkten Weges." ["For proof of the non-contradictoriness of the arithmetic axioms, though, is a direct method required."]
Although here Hilbert is more direct, he then continues onward on properties of proofs and mathematical concepts in a way that makes even this direct statement muddier. Hence to this day there remains widespread disagreement over whether it was or wasn't solved.
The resolution of the 13th problem hangs upon the requirement of analyticity. Several other problems have specific positive solutions but general proofs that the no solution is possible.
A two-volume work was released by the AMS on the progress in mathematics as a result of Hilbert's questions.
I'd say Hilbert's influence here was profound in many ways. A few years ago I bought the edited volume 23 Problems in Systems Neuroscience (Computational Neuroscience), which was deliberately styled after Hilbert's decision to center his address around the most important problems in the field (in his view). A host of similar examples of the influence of his address could easily be marshalled.
http://www.ams.org/journals/bull/2014-51-02/S0273-0979-2013-01439-3/S0273-0979-2013-01439-3.pdf
Hi Andrew,
Thanks for sharing, in particular of how, even in neuroscience, a crystalized list of problems worth investigating in the spirit of Hilbert, can help us find direction in the dizzying developments of our field. Albeit who will take up the challenge, as a singular individual, or as an "editorial team" remains to be debated. Some would say the days of the lone polymath (e.g. Leonardo da Vinci) are behind us.
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