In a recent exchange with Huw Price (Expressivism, Pragmatism and Representationalism) Paul Horwich makes an 11-point case against nominalism (he calls this a case against naturalism, but it seems to be nominalism in particular that is being targeted). The argument is quite condensed, but it provides a nice summary of why (I think) so many philosophers embrace some form of platonism (or 'anti-nominalism' for those who prefer the apophatic formulation) as well as why (I think) the case against nominalism goes awry.
Firstly, there is the purported motivation for nominalism (for 'naturalism' read 'nominalism'):
1. Naturalism rests on the impression that non-natural facts would be intolerably weird.
2. That impression has three sources: first, the singular practical and explanatory importance of naturalistic facts; second the very broad scope of the naturalistic order – the striking range and diversity of the facts that it demonstrably encompasses; and, third, the feeling that reality must 'surely' be fundamentally uniform – so all facts must be naturalistic. [EPR: 124-5]
This theme is continued:
6. The committed naturalist will not be greatly perturbed by the accusation that [his defence of naturalism is] ad hoc, contrived and intrinsically implausible. For he will reason that although such defects may indeed be present, and are indeed unwelcome in themselves, they are a price well worth paying for the wonderfully simple metaphysics that naturalism provides. [EPR: 125]
Now, perhaps some nominalists really are motivated by metaphysical simplicity, and for some reason take simple pictures of reality to be intrinsically more plausible than complicated ones; but nominalists don't, as a matter of habit, primarily motivate their view by appealing to any metaphysical claims of this kind. On the contrary, the most prominent defenders of nominalism, such as Field or Leng, object to platonism on epistemological grounds. Knowledge of abstract (and hence acausal) objects is problematic because the existence or non-existence of mind-independent abstract objects can make no difference to any grounds one might have for believing in them. Since the existence of abstract objects has no consequences for anything could possibly take place it seems impossible in principle to (i) provide justificatory grounds for belief in abstract objects, or even (ii) provide some hardcore externalist model of belief in abstract objects that could explain how these beliefs could count as knowledge, even in the absence of justification. (Note that this applies to indispensability arguments: the indispensability of quantification over mathematical objects in scientific theories does not depend on the existence of a domain of abstract mathematical objects—it is a function of more mundane things, such as the complexity of the concrete systems being modelled and the expressive resources available to the agents doing the modelling.)
The crux of the matter however seems to lie in the pro-case for platonism:
4. Note, to start with, that it's prima facie extremely implausible that amongst the facts we recognise, some are non-natural – for example, that there are numbers […] An unbiased consideration of such facts will indicate that they aren't naturalistic. For it's as plain as day (to anyone not 'in the grip of a theory') that they aren't spatio-temporally located, aren't engendered by facts of physics and don't enter into causal/explanatory relations with other facts.
7. But this apology for naturalism is inaccurate in two related respects. In the first place, what is given up for its sake is not justly described as 'local theoretical simplicity'. For what must be denied are data – epistemologically basic convictions. It is blindingly obvious to us … that Julius Caesar wan't a number. […] And no less obviously false are certain implications of every one of the sceptical 'error theories' (i.e. denials of existence) and strained reductive analyses aimed at safeguarding naturalism from the threats posed by numbers…
8. And, in the second place, the norm of simplicity, as it is deployed in science, is not in fact a licence to reject recalcitrant data … A scientist is obliged to respect all relevant data, and when they don't conform to a simple pattern, that reality must be accepted. [EPR: 125-6]
The thought that facts about an abstract domain of numbers are simply data, I would hazard a guess, is an important motivation for contemporary platonists, and explains why arguments for nominalism are often simply written off on the grounds that they entail an unacceptable conclusion. But are we right to see these as data? Horwich seems to hold that everyone is (or was, at some point) a pre-theoretical platonist, and adopted nominalism for the sake of metaphysical simplicity. I suggested before that the second part of this claim is false, but the first part is also, at least to some extent, inaccurate. I for one was a pre-theoretical nominalist: I didn't realise that anyone believed in the existence of numbers until I took a class in metaphysics, and when I made mathematical claims I didn't take the purpose of this practice to reside in describing a domain of abstract objects. It would be interesting to see some stats on the topic, but at the very least, a number of people don't take facts about an abstract domain of numbers to be data (I've met a few). So, not everyone has epistemologically basic convictions about the existence of mathematical objects, and to treat these as data is question-begging within the context of this debate.
Besides being question-begging in the current context, there is a kind of hard-line stance against the appeal to "epistemologically basic convictions" of this sort—on the grounds that it is unduly conservative: insulating views from criticism—as articulated, with characteristic understatement, by Kant:
To appeal to ordinary common sense when insight and science run short, and not before, is one of the subtle discoveries of recent times, whereby the dullest windbag can confidently take on the most profound thinker and hold his own with him. So long as a small residue of insight remains, however, one would do well to avoid resorting to this emergency help. And seen in the light of day, this appeal is nothing other than a call to the judgement of the multitude; applause at which the philosopher blushes, but which the popular wag becomes triumphant and defiant. [Prolegmonena: 4:259]
Now, Horwich isn't exactly a dull windbag or a popular wag (nor, for that matter, was Reid); and, in any given debate, something will be treated as data, if only temporally, since possessing some common commitments is a precondition of debate in the first place. So what sorts of things ought one allow as data in the debate between platonists and nominalists? A plausible supposition is that the division between platonists and nominalists coincides with a division over the kind of things that one allows as data. Platonists will take the data to be true sentences about mathematical objects, whilst nominalists will take the data to be mathematical practices. I take it that the latter view is the correct one (and usefully non-question-begging in the context of this debate): what must be accounted for are facts about how (e.g.) solving differential equations and carrying out measurement procedures allows us to track features of and make predictions about concrete systems. This would explain why nominalists (like me) are drawn to nominalism: it is very hard to see how the existence of abstract objects would be required to explain any practice. The take-home claim (that I've not really defended here with any rigour): there are deep connections between nominalism and pragmatist methodology.
