From mathematics in logic to logic in mathematics : Boole and Frege

This project proceeds from the premise that the historical and logical value of Boole's
logical calculus and its connection with Frege's logic remain to be recognised. It begins by
discussing Gillies' application of Kuhn's concepts to the history oflogic and proposing the
use of the concept of research programme as a methodological tool in the historiography
oflogic. Then it analyses'the development of mathematical logic from Boole to Frege in
terms of overlapping research programmes whilst discussing especially Boole's logical
calculus.
Two streams of development run through the project: 1. A discussion and appraisal of
Boole's research programme in the context of logical debates and the emergence of
symbolical algebra in Britain in the nineteenth century, including the improvements which
Venn brings to logic as algebra, and the axiomatisation of 'Boolean algebras', which is due
to Huntington and Sheffer. 2. An investigation of the particularity of the Fregean research
programme, including an analysis ofthe extent to which certain elements of Begriffsschrift
are new; and an account of Frege's discussion of Boole which focuses on the domain
common to the two formal languages and shows the logical connection between Boole's
logical calculus and Frege's.
As a result, it is shown that the progress made in mathematical logic stemmed from two
continuous and overlapping research programmes: Boole's introduction ofmathematics in
logic and Frege's introduction oflogic in mathematics. In particular, Boole is regarded as
the grandfather of metamathematics, and Lowenheim's theorem ofl915 is seen as a revival
of his research programme.