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According to Peter Machamer's "A Brief Historical Introduction to the Philosophy of Science" (The Blackwell Guide to the Philosophy of Science, p. 7)

... classical mechanics was not axiomatized until 1949 ...

Machamer does not include a citation here, but nor does this fact seem to belong to "general knowledge".

Who first axiomatized classical mechanics (in 1949), and where can I read more about historical attempts to axiomatize classical mechanics?

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Not sure who Machamer and Silberstein had in mind specifically, but, in any case, they are mistaken. Simon published The axioms of Newtonian mechanics in 1947, Hermes Eine Axiomatisierung der allgemeine Mechanik in 1938, and Hamel Die Axiome der Mechanik in 1927. Von Neumann even axiomatized quantum mechanics back in 1932. In a follow-up note (1954) Simon mentions an alternative axiomatization of Kinsey-Sugar-Suppes from 1953, but nothing from 1949. Kinsey-Sugar-Suppes cite Hamel, Hermes and Simon, but the only citation from 1949 is Hamel's monograph Theoretische Mechanik. In a later survey paper, Suppes also mentions Khinchin's Statistical Mechanics from 1949, but there is no axiomatization there. So perhaps Machamer and Silberstein refer to Hamel's book or misremembered 7 as 9.

See Tiemeyer's monograph Axiome der Klassischen Mechanik for a more comprehensive outlook.

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