Boole Between the Equations

Two questions are investigated. Why is it that, despite his general expla- nations, Boole’s logical operations can be difficult to pin down when he solves specific problems? Why did his late manuscript attempt to get rid of division by zero fall short of its goal? It is suggested that the former difficulty arises from his segmentation of solution-routines into stages, with the identity of the operations differing from stage to stage; while the latter failure stems from his continuing confinement to strictly equational reasoning. 1 Two questions Why is it that despite Boole’s general explanations, it is so difficult to pin down even his basic operations of multiplication, addition and subtraction when reading his solutions to logical problems? Why is it that when, in a late manuscript only now being published, he finally got down to the task of trying to eliminate the controverted operation of division from his procedures, he did not fully succeed? These are the central questions of the present paper. In outline, the conclusions are as follows: • For Boole, solving a logical problem consisted essentially in running a routine. The routines have stages, and the identities of even the basic operations of multiplication, addition and subtraction are not fixed, but vary according to the stage of the routine in which they appear. These stages and their opera- tions are analysed in section 3, revealing also several different ways of reading Boole’s celebrated {0, 1} principle. The author thanks David Waszek for generously making available the edited text of Boole’s manuscript ‘The nature of thought’ before its publication and general discussions of Boole’s work. Thanks also to Alex Citkin, Antonielly Garcia Rodrigues and again David Waszek for helpful information, comments and corrections on drafts.

Authors

• David Makinson

Keywords

No keywords are indexed for this paper.