Online research in philosophy

Indispensability arguments for realism about mathematical entities have come under serious attack in recent years. To my mind the most profound attack has come from Penelope Maddy, who argues that scientific/mathematical practice doesn't support the key premise of the indispensability argument, that is, that we ought to have ontological commitment to those entities that are indispensable to our best scientific theories. In this paper I defend the Quine/Putnam indispensability argument against Maddy's objections.

The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects which are the subject matter of pure mathematics (e.g. numbers, sets, groups) actually exist. Furthermore, they claim that these objects differ radically from the concrete objects (trees, cats, stars, molecules) which inhabit the material world. We take the standard platonistic position to include the claim that platonic objects lack spatio-temporal location and causal powers. Many (perhaps most) mathematical platonists subscribe to this view.1 But some who call themselves (or might be called) mathematical platonists..

The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light of the indispensability argument, the marriage is seen to be very unstable. Unless one is careful about how the Quine-Putnam argument is disarmed, one can be forced to either mathematical realism or, alternatively, scientific instrumentalism. I will explore the various options: (i) finding a way to reconcile the two partners in the marriage by disarming the indispensability argument (Jody Azzouni [2], Hartry Field [13, 14], Alan Musgrave [18, 19], David Papineau [21]); (ii) embracing mathematical realism (W.V.O. Quine [23], Michael Resnik [25], J.J.C. Smart [27]); and (iii) embracing some form of scientific instrumentalism (Ot´ avio Bueno [7, 8], Bas van Fraassen [30]). Elsewhere [11], I have argued for option (ii) and I won’t repeat those arguments here. Instead, I will consider the difficulties for each of the three options just mentioned, with special attention to option (i). In relation to the latter, I will discuss an argument due to Alan Musgrave [19] for why option (i) is a plausible and promising approach. From the discussion of Musgrave’s argument, it will emerge that the issue of holist versus separatist theories of confirmation plays a curious role in the realism–antirealism debate in the philosophy of mathematics. I will argue that if you take confirmation to be an holistic matter—it’s whole theories (or significant parts thereof) that are confirmed in any experiment—then there’s an inclination to opt for (ii) in order to resolve the marital tension outlined above..

The Quine-Putnam indispensability argument urges us to place mathematical entities on the same ontological footing as (other) theoretical entities of empirical science. Recently this argument has attracted much criticism, and in this paper I address one criticism due to Elliott Sober. Sober argues that mathematical theories cannot share the empirical support accrued by our best scientific theories, since mathematical propositions are not being tested in the same way as the clearly empirical propositions of science. In this paper I defend the Quine-Putnam argument against Sober's objections.

Quine has famously put forward the indispensability argument to force belief in the existence of mathematical objects (such as classes) due to their indis- pensability to our best theories of the world (Quine 1960). Quine has also advocated the indeterminacy of reference argument, according to which ref- erence is dramatically indeterminate: given a language, there’s no unique reference relation for that language (see Quine 1969a). In this paper, I ar- gue that these two arguments are in conflict with each other. Whereas the indispensability argument supports realism about mathematics, the indeter- minacy of reference argument, when applied to mathematics, provides a powerful strategy in support of mathematical anti-realism. I conclude the paper by indicating why the indeterminacy of reference phenomenon should be preferred over the considerations regarding indispensability. In the end, even the Quinean shouldn’t be a realist (platonist) about mathematics.

Recently, nominalists have made a case against the Quine–Putnam indispensability argument for mathematical Platonism by taking issue with Quine’s criterion of ontological commitment. In this paper I propose and defend an indispensability argument founded on an alternative criterion of ontological commitment: that advocated by David Armstrong. By defending such an argument I place the burden back onto the nominalist to defend her favourite criterion of ontological commitment and, furthermore, show that criterion cannot be used to formulate a plausible form of the indispensability argument.

One of the most influential arguments for realism about mathematical objects is the indispensability argument. Simply put, this is the argument that insofar as we are committed to the existence of the physical objects existentially quantified over in our best scientific theories, we are also committed to the mathematical objects existentially quantified over in these theories. Following the Quine–Putnam formulation of the indispensability argument, some proponents of the indispensability argument have made the mistake of taking confirmational holism to be an essential premise of the argument. In this paper, I consider the reasons philosophers have taken confirmational holism to be essential to the argument and argue that, contrary to the traditional view, confirmational holism is dispensable.

Quine's views on indispensability arguments in mathematics are scrutinised. A weak indispensability argument is distinguished from a strong indispensability thesis. The weak argument is the combination of the criterion of ontological commitment, holism and a mild naturalism. It is used to refute nominalism. Quine's strong indispensability thesis claims that one should consider all and only the mathematical entities that are really indispensable. Quine has little support for this thesis. This is even clearer if one takes into account Maddy's critique of Quine's strong indispensability thesis. Maddy's critique does not refute Quine's weak indispensability argument. We are left with a weak and almost unassailable indispensability argument.

In this paper I do two things: (1) I support the claim that there is still some confusion about just what the Quine-Putnam indispensability argument is and the way it employs Quinean meta-ontology and (2) I try to dispel some of this confusion by presenting the argument in a way which reveals its important meta-ontological features, and include these features explicitly as premises. As a means to these ends, I compare Peter van Inwagen’s argument for the existence of properties with Putnam’s presentation of the indispensability argument. Van Inwagen’s argument is a classic exercise in Quinean meta-ontology and yet he claims – despite his argument’s conspicuous similarities to the Quine-Putnam argument – that his own has a substantially different form. I argue, however, that there is no such difference between these two arguments even at a very high level of specificity; I show that there is a detailed generic indispensability argument that captures the single form of both. The arguments are identical in every way except for the kind of objects they argue for – an irrelevant difference for my purposes. Furthermore, Putnam’s and van Inwagen’s presentations make an assumption that is often mistakenly taken to be an important feature of the Quine-Putnam argument. Yet this assumption is only the implicit backdrop against which the argument is typically presented. This last point is brought into sharper relief by the fact that van Inwagen’s list of the four nominalistic responses to his argument is too short. His list is missing an important – and historically popular – fifth option.

In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that abstract mathematical objects exist. I wish to defend a particular objection to such arguments that has become increasingly popular recently. It is called instrumental nominalism. I consider the recent versions of this view and conclude that it has yet to be given an adequate formulation. I provide such a formulation and show that it can be used to answer the indispensability arguments.

There are two main indispensability arguments in the literature, though one has received nearly all of the attention. They correspond to two ways in which we use mathematics in science and in everyday life. We use mathematical language to help us describe non-mathematical reality; and we use mathematical reasoning to help us perform inferences concerning non-mathematical reality using only a feasible amount of cognitive power. The former use is the starting point of the Quine-Putnam indispensability argument ([Quine, 1980a], [Quine, 1980b], [Quine, 1981a], [Quine, 1981b], [Putnam, 1979a], [Putnam, 1979b]); the latter provides the basis for Ketland’s more recent argument ([Ketland, 2005]). I begin by considering the Quine-Putnam argument and introduce instrumental nominalism to defuse it. Then I show that Ketland’s argument can be defused in a similar way.