Let me begin by setting the scene, embroidering only a little on Weir’s opening pages.
Consider then the following claims, ordinarily regarded as mathematical truths:
- 3 is prime.
- The Klein four-group is the smallest non-cyclic group.
- There is an uncountably infinite set of nested subsets of Q, the set of rationals.
On the surface, (1) looks structurally very similar to ‘Alan is clever’. The latter is surely about some entity, namely Alan, and is true because that entity has the property attributed. Likewise, we might initially be pretty tempted to say, (1) also is about something, namely the number three, and is true because that thing has the property attributed. Similarly (2) is about something else, the Klein four-group. And (3) is about the set of rationals, and claims there is a further thing, an uncountably infinite family of sets of rationals nested inside each other.
So what kind of things are three, the Klein group, the set of rationals? Not things we can see or kick, but non-concrete things, surely — i.e. abstract objects. And given the standard view that (1), (2) and (3) don’t just happen to be true but are necessarily true, it would seem that these abstract objects must be necessary existents.
But how can we possibly know about such things? Once upon a time (Weir might have reminded us), the thought seemed attractive that we are made in God’s image, and — albeit to a limited extent — partake in his rational nature (for Spinoza, indeed, ‘the human mind is part of the infinite intellect of God’). And God, the story went, can just rationally see all the truths of mathematics: sharing something of his nature, in a small way we can come to do that too. Thus Salviato, speaking for Galileo in his Dialogue, says that in grasping some parts of arithmetic and geometry, the human intellect ‘equals the divine in objective certainty, for here it succeeds in understanding necessity’. And Leibniz writes `minds are … the closest likenesses of the first Being, for they distinctly perceive necessary truths’. (For more on this theme, see Ch. 1 of Edward Craig’s wonderful The Mind of God and the Works of Man (OUP, 1987), from which the quotations are taken.)
However, this conception of ourselves as approximating to the God-like has lost its grip on us. So, getting back to Weir, given the sort of beings we now think we are, with the sorts of limited and ramshackle cognitive powers with which evolution has provided us to enable us to survive in our small corner of the universe, the question becomes pressing: how come that we can possibly get to know anything about supposedly necessarily existent abstract entities (entities in Plato’s heaven, as they say)? How are we supposed to cognitively ‘lock on’ to such things? Indeed, we might wonder, how do we ever manage even to frame concepts of things apparently so remote from quotidian experience?
Now, note that to find this sort of question pressing it isn’t that we already have to bought into the idea that epistemology should be ‘naturalized’ in some strong sense, or that a Quinean ‘naturalized epistemology’ exhausts the legitimate parts of what used to be epistemology. And though Weir does talk about mathematical platonism being “put to the test by naturalized epistemology”, he officially means no more than that our conception of ourselves as natural agents without God-like powers “imposes a non-trivial test of internal stability” (p. 3) when combined with views like platonism. The problem-setting issue, then, is an entirely familiar one: as Benacerraf frames it in his classic paper, ‘a satisfactory account of mathematical truth … must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have’.
So far, then, so good. We have a familiar but still pressing question, and in the next post, I’ll say something about various lines we might take in response and indicate how, in rest of his Introduction, Weir situates on the map the position he wants to defend.
But first just a word or two more about Weir’s enthusiasm for naturalism. He goes on to write “I accept, as a general methodological maxim, the prescription that one should push a naturalistic approach as far as it will go.” (p. 5) And what does that involve? “The methodological naturalist … prescribes that one ought to follow scientific method, at a level of sophistication appropriate to the problem at hand, whenever attempting to find out the truth about anything.” Really? If using the methods of science is construed more narrowly as involving the careful rational weighing of evidence to test specific, antecedently formulated, empirical conjectures, etc., then of course this isn’t the only way to discover truths. Evolution has thankfully provided us with other quick-and-dirty ways of fast-tracking to the truth reliably enough to avoid fleet-footed predators often enough! While if the methods of science are understood in a more relaxed and embracing way, as whatever goes into the mix as we develop our best overall theory of nature, then (a familiar old-Quinean point) these methods would seem to subsume the methods of (much) mathematics which seem so entangled, and ‘methodological naturalism’ in itself has no special bite again platonism (over and above Benacerraf’s problem, which doesn’t depend on the naturalism).
But we really don’t want to start off on that debate again, about the appropriate formulation of a possibly-defensible ‘naturalism’. So let’s re-emphasize the key point that I think Weir would make, which his rhetoric hereabouts could possibly obscure: we don’t need to endorse any strong form of naturalism, or have any commitment to naturalized epistemology as the one legitimate residuary legatee of the epistemological tradition, to find troubling the combination of platonism with our conception of ourselves as limited creatures epistemically geared to the sublunary world. That‘s enough of a problem to get Weir’s project going.