Quine's Weak and Strong Indispensability Argument

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. This revised version was published online in August 2006 with corrections to the Cover Date.     

B-theory old and new: on ontological commitment

The most important argument against the B-theory of time is the paraphrase argument. The major defense against that argument is the “new” tenseless theory of time, which is built on what I will call the “indexical reply” to the paraphrase argument. The move from the “old” tenseless theory of time to the new is most centrally a change of viewpoint about the nature and determiners of ontological commitment. Ironically, though, the new tenseless theorists have generally not paid enough sustained, direct attention to that notion. I will defend a general criterion of ontological commitment and apply it to generate a version of the new tenseless theory of time. I will argue that many of the extant versions of the new tenseless theory of time (specifically, all those which seek to identify tenseless truth-conditions of tensed sentences as a way out of apparent ontological commitment to tensed features of reality) are unsatisfactory because their general criterion of ontological commitment is inadequate. Those versions of the new tenseless theory which are adequate (specifically, those which identify tenseless truthmakers for tensed sentences) actually entail the criterion of ontological commitment that I defend, despite appearances to the contrary.     

Ontological realism and sentential form

The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: (i) some mathematical claims are true, and (ii) the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny (i). Those who deny (ii) typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for (ii). Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have the same grammatical form. He makes this claim on the grounds that the two sentences are superficially similar. I argue that these grounds are not sufficient. Other sentences with the same superficial form appear to have different grammatical forms. I offer two plausible interpretations of Benacerraf’s number sentence that make use of plural quantification. These interpretations appear not to incur ontological commitments to distinctively mathematical objects, even assuming Quine’s criterion. Such interpretations open a new, plural strategy for the mathematical nominalist.     

Resolving Scheffler and Chomsky’s Problems on Quine’s Criterion of Ontological Commitments

Journal of Indian Council of Philosophical Research - This paper resolves the problems raised by Israel Scheffler and Noam Chomsky against Quine’s criterion of ontological commitment. I call...     

When Best Theories Go Bad1

It is common for contemporary metaphysical realists to adopt Quine’s criterion of ontological commitment while at the same time repudiating his ontological pragmatism. 2 Drawing heavily from the work of others—especially Joseph Melia and Stephen Yablo—I will argue that the resulting approach to meta‐ontology is unstable. In particular, if we are metaphysical realists, we need not accept ontological commitment to whatever is quantified over by our best first‐order theories.     

Contrastive Empiricism and Indispensability

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. This revised version was published online in July 2006 with corrections to the Cover Date.     

Against Mathematical Convenientism

Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine–Putnam indispensability argument and Baker’s enhanced indispensability argument.     

Quine, Putnam, and the ‘Quine–Putnam’ Indispensability Argument

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine’s argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam’s writings reveals some neglected arguments for platonism which may be superior to the indispensability argument.

---

RELATIVISM,COMMENSURABILITY AND TRANSLATABILITY

This paper discusses conceptual relativism. The main focus is on the contrasting ideas of Wittgenstein and Davidson, with Quine, Kuhn, Feyerabend and Hacker in supporting roles. I distinguish conceptual from alethic and ontological relativism, defend a distinction between conceptual scheme and empirical content, and reject the Davidsonian argument against the possibility of alternative conceptual schemes: there can be conceptual diversity without failure of translation, and failure of translation is not necessarily incompatible with recognizing a practice as linguistic. Conceptual relativism may be untenable, but not for the hermeneutic reasons espoused by Davidson.     

Giving the Ontological Argument Its Due

In this paper, I shall present and defend an ontological argument for the existence of God. The argument has two premises: (1) possibly, God exists, and (2) necessary existence is a perfection. I then defend, at length, arguments for both of these premises. Finally, I shall address common objections to ontological arguments, such as the Kantian slogan (‘existence is not a real predicate’), and Gaunilo-style parodies, and argue that they do not succeed. I conclude that there is at least one extant ontological argument that is plausibly sound.     

A Defense of Emergence

I defend a physicalistic version of ontological emergence; qualia emerge from the brain, but are physical properties nevertheless. First, I address the following questions: what are the central tenets of physicalistic ontological emergentism; what are the relationships between these tenets; what is the relationship between physicalistic ontological emergentism and non-reductive physicalism; and can there even be a physicalistic version of ontological emergentism? This discussion is merely an attempt to clarify exactly what a physicalistic version of ontological emergentism must claim, and to show that the view is at least coherent. I then defend the view from objections, for example, Kim’s (Philos Stud 95:3–36, 1999) attempt to apply a version of his exclusion argument to ontological emergentism. I conclude by offering a positive argument for the view: given certain empirical evidence concerning the organization of the brain, physicalism might have to endorse ontological emergentism to avoid epiphenomenalism.     

Defending the Indispensability Argument: Atoms, Infinity and the Continuum

This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that these new versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science.     

Ontology: minimalism and truth-conditions

In this paper, I develop a criticism to a method for metaontology, namely, the idea that a discourse’s or theory’s ontological commitments can be read off its sentences’ truth-conditions. Firstly, I will put forward this idea’s basis and, secondly, I will present the way Quine subscribed to it (not actually for hermeneutical or historic interest, but as a way of exposing the idea). However, I distinguish between two readings of Quine’s famous ontological criterion, and I center the focus on (assuming without further discussion the other one to be mistaken) the one currently dubbed “ontological minimalism”, a kind of modern Ockhamism applied to the mentioned metaontological view. I show that this view has a certain application via Quinean thesis of reference inscrutability but that it is not possible to press that application any further and, in particular, not for the ambitious metaontological task some authors try to employ. The conclusion may sound promising: having shown the impossibility of a semantic ontological criterion, intentionalist or subjectivist ones should be explored.     

Confirming Mathematical Theories: an Ontologically Agnostic Stance

The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the prospects for justifying this move are qualitatively worse in mathematics than they are in science. This revised version was published online in June 2006 with corrections to the Cover Date.     

Indispensability and Holism

It is claimed that the indispensability argument for the existence of mathematical entities (IA) works in a way that allows a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist. This is supposed to be possible by virtue of the appeal to confirmational holism that enters into the formulation of IA. Holism about confirmation is supposed to be motivated in analogy with holism about falsification. I present an account of how holism about falsification is supposed to be motivated. I argue that the argument for holism about falsification is in tension with how we think about confirmation and with two principles suggested by Quine for construing a plausible variety of holism. Finally, I show that one of Quine’s principles does not allow a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist.     

Found Guilty by Association: In Defence of the Quinean Criterion

Much recent work in metaontology challenges the so‐called ‘Quinean tradition’ in metaphysics. Especially prominently, Amie Thomasson argues for a highly permissive ontology over ontologies which eliminate many entities. I am concerned with disputing not her ontological claim, but the methodology behind her rejection of eliminativism – I focus on ordinary objects. Thomasson thinks that by endorsing the Quinean criterion of ontological commitment eliminativism goes wrong; a theory eschewing quantification over a kind may nonetheless be committed to its existence. I argue that, contrary to Thomasson's claims, we should retain the Quinean criterion. Her arguments show that many eliminativist positions are flawed, but their flaws lie elsewhere: the Quinean criterion is innocent. Showing why reveals the importance of pragmatism in ontology. In §1 I compare Thomasson's account and the eliminativist views to which it stands in opposition. In §2 I re‐construct Thomasson's reasons for rejecting the Quinean criterion. In §3 I defend the Quinean criterion, showing that the eliminativists’ flaws are not consequences of applying the Quinean criterion, before explaining the criterion's importance when properly understood. I conclude that Thomasson, though right to criticise the methodology of ordinary‐object eliminativists, is wrong to identify the Quinean criterion as the source of their mistake.     

Evidential Holism and Indispensability Arguments

The indispensability argument is a method for showing that abstract mathematical objects exist (call this mathematical Platonism). Various versions of this argument have been proposed (§1). Lately, commentators seem to have agreed that a holistic indispensability argument (§2) will not work, and that an explanatory indispensability argument is the best candidate. In this paper I argue that the dominant reasons for rejecting the holistic indispensability argument are mistaken. This is largely due to an overestimation of the consequences that follow from evidential holism. Nevertheless, the holistic indispensability argument should be rejected, but for a different reason (§3)—in order that an indispensability argument relying on holism can work, it must invoke an unmotivated version of evidential holism. Such an argument will be unsound. Correcting the argument with a proper construal of evidential holism means that it can no longer deliver mathematical Platonism as a conclusion: such an argument for Platonism will be invalid. I then show how the reasons for rejecting the holistic indispensability argument importantly constrain what kind of account of explanation will be permissible in explanatory versions (§4).     

The Disappearance of Change: Towards a Process Account of Persistence

This paper aims to motivate a new beginning in metaphysical thinking about persistence by drawing attention to the disappearance of change in current accounts of persistence. I defend the claim that the debate is stuck in a dilemma which results from neglecting the constructive role of change for persistence. Neither of the two main competing views, perdurantism and endurantism, captures the idea of persistence as an identity through time. I identify the fundamental ontological reasons for this, namely the shared commitment to what I call ‘thing ontology’: an ontology that gives the ontological priority to static things. I conclude by briefly indicating how switching to a process ontological framework that takes process and change to be ontologically primary may allow for overcoming the dilemma of persistence.     

Grounding and the indispensability argument

There has been much discussion of the indispensability argument for the existence of mathematical objects. In this paper I reconsider the debate by using the notion of grounding, or non-causal dependence. First of all, I investigate what proponents of the indispensability argument should say about the grounding of relations between physical objects and mathematical ones. This reveals some resources which nominalists are entitled to use. Making use of these resources, I present a neglected but promising response to the indispensability argument—a liberalized version of Field’s response—and I discuss its significance. I argue that if it succeeds, it provides a new refutation of the indispensability argument; and that, even if it fails, its failure may bolster some of the fictionalist responses to the indispensability argument already under discussion. In addition, I use grounding to reply to a recent challenge to these responses.     

On the Question 'Do Numbers Exist?'

Since we know that there are four prime numbers less than 8 we know that there are numbers. This 'short argument' is correct but it is not an ontological claim or part of philosophy of mathematics. Both realists (Quine) and nominalists (Field) reject the short argument and adopt the idea that the existence of numbers might be posited to explain known mathematical truths. Philosophers operate with a negative conception of what numbers are: they are not in space and time, not related causally to us, not perceivable, etc. This preliminary outlook does not actually characterize a kind of existing thing at all. It creates the atmosphere of weirdness characteristic of both fictionalism and Platonism. Positing things for the sake of explanation makes sense in empirical contexts, but the intelligibility of positing cannot not survive the move to philosophy of mathematics. Modal realism is a model for the unsatisfactory thinking that generates ontological commitment in mathematics.