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Is there any reason why Hilbert's problems are ordered in the way they are?
Also, from whom does this ordering come?
I know that Hilbert talked about several problems on his list, namely 1, 2, 6, 7, 8, 13, 16, 19, 21 and 22, so I don't think that his talk in 1900 affected the order.
Yes, they are ordered by the broad areas of mathematics. If you read his paper, you will see that he hints this.
1,2 - foundations, set theory.
3,4, and 5 to some extent - foundations of geometry.
6 - foundations of physics.
7-12 - number theory,
13,14 -algebra,
15,16,17 algebraic geometry,
18 - metric geometry,
19-23 - analysis and differential equations.
This seems to be a natural order of any exposition of the whole mathematics (compare with French Courses d'Analyse, or early 20th century Encyclopedia of mathematics, or even Bourbaki).
