Frege's Conception of Numbers As Objects

In the middle and later years of this century Frege’s ideas on a wide class of issues in the philosophy of language have assumed a deserved centrality in the thinking of philosophers interested in that area. Of his philosophy of mathematics, in contrast, it is fair to say that its felt importance to contemporary work remains largely historical. No-one who wished to understand current philosophical controversy about mathematics could reasonably omit study of Frege’s texts; but there are not many Fregeans actively engaged in contemporary issues in the philosophy of mathematics as there are, say, in contemporary philosophical semantics. Even Professor Dummett allows that Frege’s work in the philosophy of mathematics has now come to seem ‘indisputably archaic’! in a way in which that of Brouwer, or Hilbert, has not. It would be possible to accept that assessment and still recognise Frege’s 1884 ‘Logicist Manifesto’, Grundlagen der Arithmetik, as one of the most brilliant short treatises ever written in all of philosophy. It is therefore surprising that the text has so far attracted a relatively slight corpus of direct exegesis and commentary. Partly this may be a result of the very clarity and vigour of the writing: where Frege seems right, he seems obviously right; and the technical failings of the programme which he announces are thought to be well known. Yet, almost a hundred years after its publication, there is still, to my knowledge, in English no systematic introduction to or exposition of the central themes of the Grundlagen. The present essay is directed, in part, at this lacuna. It is my particular hope that it may help (second and subsequent year) undergraduate students of philosophy to grasp some of the depth behind the superficial clarity of the positive, that is, non-critical, parts of Grundlagen, and so may encourage a more widespread use of Frege’s classic as an introductory philosophy of mathematics text; a role for which it is still, in my