- There are two influential schools of thought within the philosophy of maths.
- One of these, championed by Bob Hale and Crispin Wright, follows in the tradition of Fregean logicism. Within this tradition we find a common argument for the conclusion that we should be necessitarian Platonists: we should think that abstract mathematical objects exist, and necessarily so.
- A different school, championed by, among others, Hartry Field and Mark Colyvan, follows in the tradition of Quinean empiricism. Within this tradition we find a common argument for the conclusion that we should be contingent Platonists (or contingent fictionalists); we should think that mathematical objects exist in some, but not other worlds.
- Hale, Wright, Field, and Colyvan agree about some things. They agree that mathematical discourse is truth apt, and that it is true only if the entities that it quantifies over, namely mathematical objects, exist. And they agree that if mathematical objects exist, then, necessarily, they are abstracta: they agree, for instance, that non-error-theoretic brands of nominalism, such as those that identify mathematical objects with sets of concreta, are false. Hale, Wright, and Field agree about something else. Each is, broadly speaking, a minimalist about truth, though a minimalist of a somewhat different stripe. Field is a minimalist of the Quinean disquotationalist variety. Hale and Wright endorse a more Fregean style minimalism. But they agree that truth is not a substantial property and that sentences are not true in virtue of corresponding in the appropriate way to the way the world is. While, like Field, Colyvan is working within a broadly Quinean paradigm, he is best thought of as a maximalist about truth. This difference will turn out to be important.
- This chapter is interested in how disagreement about the nature of abstract mathematical objects, and about the modal status of those objects, can be traced to adoption of one or the other of the neo-Fregean or Quinean frameworks. The chapter
The three arguments in question are
- explores the relationship between three arguments for the existence of mathematical objects that issue from these paradigms,
- it explores the nature of the mathematical objects that those arguments yield, and
- it explores the modal status of those objects.
- the argument from singular referring terms (from the neo-Fregean paradigm)
- the original indispensability argument (from the Quinean paradigm) and
- the new indispensability argument (in part from the Quinean paradigm).
- This chapter seeks to show that the argument from singular referring terms and the original indispensability argument yield what we might think of as minimal mathematical objects: they yield a commitment to mathematical objects whose nature is ontologically thin because it is exhausted either by the totality of true sentences that jointly make up scientific theory, or by the identity conditions used to stipulatively introduce those objects via an abstraction principle. It follows then, that for a wide range of properties mathematical objects neither have, nor lack, those properties.
- If successful, the argument from singular referring terms and the original indispensability argument yield apparently Platonist conclusions (at least with respect to some worlds). But in each case we might reasonably ask whether the entities to which we end up committed as so minimal that anything like true Platonism is really vindicated.
- Section 2 introduces the original indispensability argument and shows why it yields the conclusion that minimal mathematical objects exist contingently. Though it is not the primary focus of the chapter, along the way I point out some of the more problematic aspects of the Quinean framework that underpins this argument. In Section 2.1, I explicate the new indispensability argument and argue that, despite its proponents' claims, the modal force of its conclusion is unclear. The new indispensability argument rejects some of the minimalist assumptions of the original Quinean framework, and ultimately, I argue, its proponents are faced with a dilemma. Either they must accept the strong naturalism that is part of the Quinean paradigm, or they must reject that claim. If they reject strong naturalism they almost entirely jettison the Quinean framework, and in doing so concede that we must in part look to metaphysics to discern our ontological commitments. As I see it, the view would then collapse into the common view that our ontology should be guided by our metaphysical commitments, and would no longer count as an indispensability argument. Given this, I take it that proponents of the new indispensability argument must endorse strong naturalism. But if they do so, I argue, they must distribute their credences between the view that mathematical objects are minimal entities, and the view that they are non-minimal entities whose nature we can in principle never come to know, and can never have reason to suppose is one way rather than another. Either our ontology, or our epistemology, then, is thin.
- Section 3 outlines the neo-Fregean framework and shows why it yields a commitment to necessarily existing minimal mathematical objects. Finally, in Section 4, I explore the relationship between the neo-Fregean framework and the mathematical objects that issue from it, and the Quinean framework and the mathematical objects that issue from it. I suggest that there is a good deal in common between the two frameworks, and that this is what accounts for the fact that what we get, at the end, are minimal objects and that those who are predisposed towards Platonism have good reason to be suspicious that such objects are too minimal for the views that posit them to count as Platonist at all.
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