Mathematical Physics and Elementary Logic

PSA Proceedings of the Biennial Meeting of the Philosophy of Science Association 1990 (1):288-301 (1990)
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Modern mathematical physics uses real-number variables, and therefore presupposes set theory. (A real number is defined as a certain kind of set or sequence of natural or rational numbers.) Set theory is also used to define the operations of differential calculus, needed to state physical laws as differential equations constraining the numerical variables representing physical quantities. The derivative f' = df(t)/dt is defined as the limit of an infinite sequence of terms [f(t+e)-f(t)]/e as e → 0, and this definition can only be expressed in a language allowing reference to infinite sequences. Moreover, closure under these limit operations again requires the underlying field of numbers to be Dedekind complete, hence to include all of the reals.The possibility that mathematical physics depends essentially upon set theory is disturbing, for two distinct reasons. From a logical viewpoint, it is disturbing that an important branch of natural science should depend upon a part of logic or mathematics whose status remains uncertain and controversial.



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