Editing Hilbert's problems (section)

==Knowability==
Following [[Gottlob Frege]] and [[Bertrand Russell]], Hilbert sought to define mathematics logically using the method of [[formal system]]s, i.e., [[finitism|finitistic]] [[Mathematical proof|proofs]] from an agreed-upon set of [[axiom]]s.<ref name="Frege_Gödel">{{cite book |pages=464ff |editor-first=Jean |editor-last=van Heijenoort |year=1976 |orig-year=1966 |title=From Frege to Gödel: A source book in mathematical logic, 1879–1931 |publisher=Harvard University Press |location=Cambridge MA |isbn=978-0-674-32449-7 |edition=(pbk.) |quote=A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational 'crisis' that was on-going at the time (translated into English), appears as Hilbert's 'The Foundations of Mathematics' (1927).}}</ref> One of the main goals of [[Hilbert's program]] was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.{{refn|group=lower-alpha|See Nagel and Newman revised by Hofstadter (2001, p.&nbsp;107),<ref name="Hofstadter_2001">{{Cite book |last1=Nagel |first1=Ernest |title=Gödel's proof |last2=Newman |first2=James R. |last3=Hofstadter |first3=Douglas R. |date=2001 |publisher=New York University Press |isbn=978-0-8147-5816-8 |editor-last=Hofstadter |editor-first=Douglas R. |editor-link=Douglas Hofstadter |edition=Rev. |location=New York}}</ref> footnote&nbsp;37: "Moreover, although most specialists in mathematical logic do not question the cogency of [Gentzen's] proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency." Also see next page: "But these proofs [Gentzen's et al.] cannot be mirrored inside the systems that they concern, and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program." Hofstadter rewrote the original (1958) footnote slightly, changing the word "students" to "specialists in mathematical logic". And this point is discussed again on page&nbsp;109<ref name=Hofstadter_2001/> and was not modified there by Hofstadter (p.&nbsp;108).<ref name=Hofstadter_2001/>}}

However, [[Gödel's second incompleteness theorem]] gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible.  Hilbert lived for 12&nbsp;years after [[Kurt Gödel]] published his theorem, but does not seem to have written any formal response to Gödel's work.{{refn|group=lower-alpha|Reid reports that upon hearing about "Gödel's work from Bernays, he was 'somewhat angry'. ... At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. ... It was not yet clear just what influence Gödel's work would ultimately have" (p.&nbsp;198–199).<ref name="Reid_1996">{{cite book |first=Constance |last=Reid |year=1996 |title=Hilbert |publisher=Springer-Verlag |location=New York, NY |isbn=978-0387946740 |url-access=registration |url=https://archive.org/details/hilbert0000reid }}</ref> Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p.&nbsp;199).<ref name="Reid_1996"/>}}{{refn|group=lower-alpha|Reid's biography of Hilbert, written during the 1960s from interviews and letters, reports that "Godel (who never had any correspondence with Hilbert) feels that Hilbert's scheme for the foundations of mathematics 'remains highly interesting and important in spite of my negative results' (p.&nbsp;217). Observe the use of present tense – she reports that Gödel and Bernays among others "answered my questions about Hilbert's work in logic and foundations" (p.&nbsp;vii).<ref name=Reid_1996/>}}

Hilbert's tenth problem does not ask whether there exists an [[algorithm]] for deciding the solvability of [[Diophantine equations]], but rather asks for the ''construction'' of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in [[rational integer]]s". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.

In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.<ref group=lower-alpha>This issue that finds its beginnings in the "foundational crisis" of the early 20th&nbsp;century, in particular the controversy about under what circumstances could the [[Law of Excluded Middle]] be employed in proofs. See much more at [[Brouwer–Hilbert controversy]].</ref> He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "[[Ignoramus et ignorabimus|ignorabimus]]" (statement whose truth can never be known).<ref group=lower-alpha>"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ''ignorabimus''." (Hilbert, 1902, p.&nbsp;445)</ref>  It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus.

On the other hand, the status of the first and second problems is even more complicated: there is no clear mathematical consensus as to whether the results of Gödel (in the case of the second problem), or Gödel and Cohen (in the case of the first problem) give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.{{refn|group=lower-alpha|Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as ''Principia Mathematica'' is not excluded by Gödel's results. ... His argument does not eliminate the possibility ... But no one today appears to have a clear idea of what a finitistic proof would be like that is ''not'' capable of being mirrored inside ''Principia Mathematica'' (footnote&nbsp;39, page&nbsp;109). The authors conclude that the prospect "is most unlikely".<ref name=Hofstadter_2001/>}}