Frege's New Science

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. 1. Frege, Hilbert, and Metatheory A little over a century ago a correspondence commenced between one of the most well-known mathematicians of the day and a lightly regarded colleague in Jena. This correspondence, between Hilbert and Frege, while of some but apparently not great interest to Hilbert, was thought by Frege to broach issues of central importance, so much so that he suggested that it warranted publication.1 Since Hilbert was unresponsive to this suggestion, Frege proceeded to write up his side of the discussion in a pair of articles under the title Die Grundlagen der Geometrie [13] in 1903. In response to Received March 29, 2001; printed September 30, 2002 2001 Mathematics Subject Classification: Primary, 03A05; Secondary, 00A30