Notre Dame Journal of Formal Logic

Frege's New Science

Aldo Antonelli and Robert May

Abstract

In this paper, we explore Fregean metatheory, what Frege called the New Science. The New Science arises in the context of Frege's debate with Hilbert over independence proofs in geometry and we begin by considering their dispute. We propose that Frege's critique rests on his view that language is a set of propositions, each immutably equipped with a truth value (as determined by the thought it expresses), so to Frege it was inconceivable that axioms could even be considered to be other than true. Because of his adherence to this view, Frege was precluded from the sort of metatheoretical considerations that were available to Hilbert; but from this, we shall argue, it does not follow that Frege was blocked from metatheory in toto. Indeed, Frege suggests in Die Grundlagen der Geometrie a metatheoretical method for establishing independence proofs in the context of the New Science. Frege had reservations about the method, however, primarily because of the apparent need to stipulate the logical terms, those terms that must be held invariant to obtain such proofs. We argue that Frege's skepticism on this score is not warranted, by showing that within the New Science a characterization of logical truth and logical constant can be obtained by a suitable adaptation of the permutation argument Frege employs in indicating how to prove independence. This establishes a foundation for Frege's metatheoretical method of which he himself was unsure, and allows us to obtain a clearer understanding of Frege's conception of logic, especially in relation to contemporary conceptions.

Article information

Source
Notre Dame J. Formal Logic Volume 41, Number 3 (2000), 242-270.

Dates
First available in Project Euclid: 26 November 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1038336844

Digital Object Identifier
doi:10.1305/ndjfl/1038336844

Mathematical Reviews number (MathSciNet)
MR1943495

Zentralblatt MATH identifier
1009.03004

Subjects
Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 00A30: Philosophy of mathematics [See also 03A05]

Keywords
Frege Hilbert logicism logical truth logical constants independence proofs

Citation

Antonelli, Aldo; May, Robert. Frege's New Science. Notre Dame J. Formal Logic 41 (2000), no. 3, 242--270. doi:10.1305/ndjfl/1038336844. http://projecteuclid.org/euclid.ndjfl/1038336844.


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