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Hilbert uses the expression «das inhaltliche logische Schliessen »¹ which I translate by "internal logic", rather than logic of content. Brouwer and H. Weyl² use also the expression to designate an inner logic different from formal (external) logic which mirrors only the superficial structure of mathematics. For Hilbert, internal logic is not ordinary or formal logic, the rôle of which is only ancillary, that is the demonstration of theorems in a given mathematical theory. But internal logic, often identified with metamathematics³, should be considered as an "intramathematics" in the sense that the inner consistency of axioms is more important than the deduction of particular theorems. In other words, proof theory <Beweistheorie> or <Metamathematik> is an internal logic to the extent it describes the inner workings of a mathematical theory. Proof theory has been seen as the theory of formal systems and, by extension, as the very embodiment of formalism. The hypothesis that I want to defend goes the other way: internal logic is the opposite of formalism and Hilbert's endeavour or programme could be formulated in the following terms: internal finitary logic reduces infinitary formal logic in the same manner that a finitary mathematical theory (like arithmetic) reduces the infinite problems of the theory of forms or the theory of invariants to a finite calculus.

DOI: 10.1007/978-94-017-0083-2_2

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