Formalism, Incompleteness, and the Implicit Commitment Thesis

(Logic/Philosophy of Mathematics, English)

urt Gödel, in his famous article (Gödel 1953/1995), claimed to refute the formalism or at least one of its emanations: the Syntactic Interpretation of Mathematics (SIM). According to SIM, the sentences of mathematics have no extra-linguistic content and can be justified by means of syntactical rules of the mathematical system alone. Gödel claimed to refute this interpretation of mathematics on the basis of his celebrated Incompleteness Theorem. He argued that SIM entails the consistency of the accepted formal system which, according to his formal result, cannot be justified by formal means.

In my talk I will defend the formalist position against Gödel's argument. I will argue that although the argument is formally valid, its conclusion depends on the assumption that the formalists need to justify the consistency statement within the formal system itself. I will show that this assumption contradicts the Implicit Commitment Thesis (ICT) which is widely discussed in the recent literature on philosophy of mathematics (cf. Dean 2015, Cie&_347;li&_324;ski 2017, Nicolai & Piazza 2019, Fischer & Horsten & Nicolai 2019, Fischer 2021, &_321;e&_322;yk & Nicolai ms). The ICT states that in accepting a formal theory S, one is also committed to some statements that are not provable in S, but whose acceptance is implicit in the acceptance of S. Examples of such commitments include the consistency statement and various reflection principles. Moreover, some authors (e.g. Horsten 2019) claim that these statements need no further justification than the one given for acceptance of the formal theory S.

In the talk, I characterize the formalism as SIM and contrast it with Carnap's formalist position, and discuss Gödel's argument against SIM. I introduce the ICT and show how this thesis can be used to refute Gödel's argumentation. The talk ends with a discussion of some controversies concerning the ICT and arguments for its plausibility.

Chair: Leon Commandeur

Time: 10:00-10:30, 09. September 2022 (Friday)

Location: SR 1.006

Remark: (Online Talk)

Maciej Glowacki

(University of Warsaw, Poland)

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