250 J.J. Haldane
1991); and The Seas of Language and Other Essays (Oxford: Oxford University
Press, 1993).
18 Berkeley, PrinciplesinThe Works, paragraph 80 (volume 2, p. 75).
19 Michael Dummett, ‘Reply to McGuiness’, in Brian McGuiness and Gianluigi
Oliveri (eds) The Philosophy of Michael Dummett (Dordrecht: Kluwer, 1994)
pp. 358 and 359.
20 Quaestiones Disputate, VIII, q. 1, a. 1.
21 For contemporary versions of ‘traditional’ mathematical Platonism see John
Bigelow,The Reality of Numbers (Oxford: Clarendon Press, 1988), and James
Robert Brown, ‘π in the Sky’ in A.D. Irvine (ed.) Physicalism in Mathematics
(Dordrecht: Kluwer, 1990). Bigelow’s last chapter is entitled ‘Platonism and
Necessity’. It is five pages long and only engages the ontological question on the
last page, commenting on the necessity of mathematical entities in its last para-
graph. I quote: ‘Mathematical properties and relations instantiate one another;
and that makes truths about them largely independent of the world of changing
individuals. This independence is what underlies their air of necessity and
certainty ; it provides them with allthe necessity and certainty they are going to get.
And that will be enough’ (p. 178). The italics are mine. Brown characterizes math-
ematical Platonism as involving four ingredients, the first two of which are meta-
physical, the latter two being epistemological. The metaphysical ones are objective
existence and abstractness (transcendence of space and time). Necessary existence
does not even feature. Notwithstanding the decline of verificationism there is
evidently a residual reluctance to contemplate metaphysical non-contingency.
22 See Saul Kripke, Naming and Necessity (Oxford: Blackwell, 1980) pp. 34ff.
23 These reflections were written during a period of tenure as Royden Davis Chair
of Humanities at Georgetown University. I am grateful to Georgetown, especially
to the Department of Philosophy, for the benefits bestowed by this appointment.