# Hilbert's problems

Hilbert presented ten of his 23 problems at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne 1900.

In which of the 23 or 24 problems was rating (as he was speaking about actual mathematical problems he made estimations about their solvableness etc.) totally wrong and if it is possible to explain, why?

• Well he rated some of the problems as solvable in a certain time, easy and unsolvable – Medi1Saif Apr 13 '16 at 10:20

Contrary to what you say, Hilbert very rarely says anything which can be interpreted as a "rating".

There is no objective criterion to "rate" the problems except by the time it took to solve them.

Some problems were solved even before Hilbert stated them, for example, problem 3. Problem 22 (uniformization) was solved (in dimension 1) in 1906.

Problem 7 was rated by Hilbert as very difficult. It was solved in 1935. On the other hand, Hilbert was optimistic about problem 8 (Riemann hypothesis), which is understandable because there was a recent breakthrough at that time.

In some cases Hilbert was wrong on the answers: For example, problem 2 was "to prove that axioms of arithmetic are consistent". This is impossible to prove, as Godel showed in 1936.

Many of the problems were not specific questions but wide research programs. So one cannot really say whether they are solved or not.

EDIT. I asked Gerry Myerson about his answer mentioned in KCd's comment. The information comes from the book of C. Reid, Hilbert. In it Hilbert's talk in 1920s is mentioned where he gave this assessment of the three problems. His opinion about Fermat's problem was right. Only the comparison of the Riemann problem and transcendence problem turned out to be "wrong". I suspect that majority of mathematicians would agree with Hilbert's rating of these two problems at that time.

• I just remembered a professor who said that Hilbert's estimations on 3 problems in number theory were completely wrong, the easy to solve, was impossible to prove, the impossible was easy and so on (I'm not exactly sure about that, i hated number theory with this guy!) – Medi1Saif Apr 13 '16 at 14:26
• @Medi1Saif, I believe you are thinking of Hilbert's comparison of The Riemann Hypothesis, Fermat's Last Theorem, and transcendence of $2^{\sqrt{2}}$. See the top answer by Gerry Myerson to math.stackexchange.com/questions/88709/…. – KCd Apr 14 '16 at 16:20
• Gerry Myerson refers to some 1920 talk of Hilbert, which is different from his main 1900 talk where the problems were stated. For example, "Fermat theorem" is not one of the "Hilbert problems". – Alexandre Eremenko Apr 14 '16 at 17:53
• @AlexandreEremenko I think KCd has found the problems. according to your remark I might have mixed Hilbert's talk in Paris with the one of 1920. Because I'm quit sure that Fermat was one point which was quoted and which I really missed among the "Hilbert problems". – Medi1Saif Apr 15 '16 at 5:07