The Limits of Abstraction
This paper has been written more from a sense of curiosity than commitment. I was fortunate enough to attend the Munich Conference on the Philosophy of Mathematics in the Summer of 1993 and to overhear a discussion of recent work on Frege's approach to the foundations of mathematics. This led me to investigate certain technical problems connected with the approach; and these led me, in their turn, to reflect on certain philosophical aspects of the subject. I was concerned to see to what extent a Fregean theory of abstraction could be developed and used as a foundation for mathematics and to place the development of such a theory within a general framework for dealing with questions of abstraction. My conclusions were somewhat mixed: a theory of abstraction could be developed somewhat along the lines that Frege has envisaged; and it could indeed be used as a basis for both arithmetic and analysis. When wedded to a suitable version of the context principle, the theory was capable of accounting for our reference to numbers and other abstract objects. But without the support of the principle, it was not clear that the theory had any great advantage over its rivals. Thus my results would be congenial to someone already committed to the Fregean approach though unconvincing to someone who was not. I therefore present them in somewhat the same spirit as someone who sends off a message in a bottle. I have no desire to announce my communication to the world; but if someone stumbles across it and finds it to be of interest, I shall be pleased.
The paper is in three parts. The first is devoted to philosophical matters, which help explain the motivation for the subsequent technical work and also its significance. It is centred on three main questions: What are the correct principles of abstraction? In what sense do they serve to define the
I am much indebted to the participants in that discussion--principally, Boolos, Clark, Hale, Heck, and Wright. Preliminary versions of the paper were given at the third Austrian philosophy conference in Salzburg, at a talk at the City University of New York, at a philosophy of mathematics workshop at UCLA, and at a workshop on abstraction in St Andrews; and I am grateful for the comments that I received on those different occasions. I have been greatly influenced by the writings of Michael Dummett and Crispin Wright and have greatly benefited from the comments of Tony Martin.