A Truthmaker Indispensability Argument

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.     

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.

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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.     

Dispensability in 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 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.     

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.     

Quine's truth

W. V. Quine has made statements about truth which are not obviously compatible, and his statements have been interpreted in more than one way. For example, Donald Davidson claims that Quine has an epistemic theory of truth, but Quine himself often says that truth is just disquotational. This paper argues that Quine should recognize two different notions of truth. One of these is disquotational, the other is empiricist. There is nothing wrong with recognizing two different notions of truth. Both may be perfectly legitimate, even though, to some extent, they may be applicable in different contexts. Roughly speaking, a sentence is true in the empiricist sense if it belongs to a theory which entails all observation sentences which would be assented to by the speakers of the language in question (and no observation sentences which would be dissented from by these speakers). Various objections to this idea are discussed and rejected.     

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.     

A priori truth

Conclusion There are several epistemic distinctions among truths that I have argued for in this paper. First, there are those truths which holdof every rationally accessible conceptual scheme (class A truths). Second, there are those truths which holdin every rationally accessible conceptual scheme (class B truths). And finally, there are those truths whose truthvalue status isindependent of the empirical sciences (class C truths). The last category broadly includes statementsabout systems and the statements they contain, as well as statements true by virtue of the rules of language itself.At the risk of anachronism, I'll describe the positions of Carnap (1956); Quine (1980); Grice et al. (1956); the various Putnam's and myself in terms of the above distinctions: both Carnap and Quine (pretty much) think there are no class A or class B truths. Both Putnam (1975) and Putnam (1983c) think there are class A and class B truths, and that these classes overlap. I deny there are class B truths but affirm the existence of class A truths (although I haven't given explicit examples of the latter here). Finally, everyone here but Quine (1980) thinks there are class C truths (of one sort or another). Putnam (1975) attempts to show that certain class C truths are simultaneously class A and class B truths. Grice et al. (1956) take pains to distinguish the claim that there are class C truths from the claim that there are class A truths, and claim (against Quine, 1980) that no argument showing there are no class A truths shows there are no class C truths.On my interpretation of Quine (1980) he thinks that the nonexistence of class A truths shows there are no class C truths-given the extra bit of argument that a notion of true by convention or true by virtue of meaning without epistemic content, is a distinction without significance. But that issue, which is the one Grice et al. (1956) are concerned with, has not been the focus in this paper-and so in a sense I have shifted the terms of the original debate.Here I have been primarily concerned to distinguish epistemic notions and sort out how and in what ways they relate to each other. A primary tool in this exercise has been the explicit recognition that formal models of truth make universality an unlikely property of our conceptual schemes. If I have not convinced anyone that the epistemic notions sort out the way I think they do, I hope at least that some burden-shifting has occurred: that philosophers do not either take it for granted that certain notionsmust be expressible in any conceptual scheme or treat the fact that conceptual schemes must be (in some sense) limited as of little (philosophical) moment.On the other hand, if I am right about the epistemology, it follows that previous attempts to mark out the necessary structures in rationally accessible conceptual schemes via a priori truths is hopeless. What I think must replace their role, what I call globally incorrigible sets of sentences, is a topic for another time.My thanks to Arnold Koslow and Mark Richard for their helpful suggestions. I also want to thank the City University of New York Graduate Center for inviting me to be a visiting scholar academic year 1989–90, during which time this paper was written. While there I was partially supported by a Mellon fellowship from Tufts University, for which I am grateful.     

On A Neglected Path to Intuitionism

According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard (Heyting) semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the dispute between classicists and intuitionists. I argue for this by presenting an account of consequence, and a cognate semantic theory for the language of the propositional calculus, which (a) respects the meanings of the connectives as embodied in the familiar classical truth-tables, (b) does not presuppose Bivalence, and with respect to which (c) the rules of the intuitionist propositional calculus are sound and complete. Thus the disagreement between classicists and intuitionists, at least, need not stem from their attaching different senses to the connectives; one may deny the doctrine without changing the subject. The basic notion of my semantic theory is truth at a possibility, where a possibility is a way that (some) things might be, but which differs from a possible world in that the way in question need not be fully specific or determinate. I compare my approach with a previous theory of truth at a possibility due to Lloyd Humberstone, and with a previous attempt to refute Quine’s thesis due to John McDowell.     

Kant's transcendental idealism and contemporary anti‐realism

This paper compares Kant's transcendental idealism with three main groups of contemporary anti‐realism, associated with Wittgenstein, Putnam, and Dummett, respectively. The kind of anti‐realism associated with Wittgenstein has it that there is no deep sense in which our concepts are answerable to reality. Associated with Putnam is the rejection of four main ideas: theoryindependent reality, the idea of a uniquely true theory, a correspondence theory of truth, and bivalence. While there are superficial similarities between both views and Kant's, I find more significant differences. Dummettian anti‐realism, too, clearly differs from Kant's position: Kant believes in verification‐transcendent reality, and transcendental idealism is not a theory of meaning or truth. However, I argue that part of the Dummettian position is extremely useful for understanding part of Kant's position – his idealism about the appearances of things. I argue that Kant's idealism about appearances can be expressed as the rejection of experiencetranscendent reality with respect to appearances.     

Confirmation Theory and Indispensability

In this paper I examine Quine's indispensability argument, with particular emphasis on what is meant by 'indispensable'. I show that confirmation theory plays a crucial role in answering this question and that once indispensability is understood in this light, Quine's argument is seen to be a serious stumbling block for any scientific realist wishing to maintain an anti-realist position with regard to mathematical entities.     

Hilary Putnam's Dialectical Thinking: An Application to Fallacy Theory

In recent and not so recent years, fallacy theory has sustained numerous challenges, challenges which have seen the theory charged with lack of systematicity as well as failure to deliver significant insights into its subject matter. In the following discussion, I argue that these criticisms are subordinate to a more fundamental criticism of fallacy theory, a criticism pertaining to the lack of intelligibility of this theory. The charge of unintelligibility against fallacy theory derives from a similar charge against philosophical theories of truth and rationality developed by Hilary Putnam. I examine how Putnam develops this charge in the case of the conception of rationality pursued by logical positivism. Following that examination, I demonstrate the significance of this charge for how we proceed routinely to analyse one informal fallacy, the fallacy of petitio principii. Specifically, I argue that the significance of this charge lies in its issuance of a rejection of the urge to theorise in fallacy inquiry in general and petitio inquiry in particular. My conclusion takes the form of guidelines for the post-theoretical pursuit of fallacy inquiry.     

Black,White and Gray: Quine on Convention

This paper examines Quine’s web of belief metaphor and its role in his various responses to conventionalism. Distinguishing between two versions of conventionalism, one based on the under-determination of theory, the other associated with a linguistic account of necessary truth, I show how Quine plays the two versions of conventionalism against each other. Some of Quine’s reservations about conventionalism are traced back to his 1934 lectures on Carnap. Although these lectures appear to endorse Carnap’s conventionalism, in exposing Carnap’s failure to provide an explanatory account of analytic truth, they in fact anticipate Quine’s later critique of conventionalism. I further argue that Quine eventually deconstructs both his own metaphor and the thesis of under-determination it serves to illustrate. This enables him to hold onto under-determination, but at the cost of depleting it of any real epistemic significance. Lastly, I explore the implications of this deconstruction for Quine’s indeterminacy of translation thesis.     

A Coherence Theory of Truth in Ethics

Quine argues, in “On the Nature of Moral Values” that a coherence theory of truth is the “lot of ethics”. In this paper, I do a bit of work from within Quinean theory. Specifically, I explore precisely what a coherence theory of truth in ethics might look like and what it might imply for the study of normative value theory generally. The first section of the paper is dedicated to the exposition of a formally correct coherence truth predicate, the possibility of which has been the subject of some skepticism. In the final two sections of the paper, I claim that a coherence theory in ethics does not reduce the practice of moral inquiry to absurdity, in practice as well as in principle.     

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.     

Default reasonableness and the mathoids

In this paper I will argue that (principled) attempts to ground a priori knowledge in default reasonable beliefs cannot capture certain common intuitions about what is required for a priori knowledge. I will describe hypothetical creatures who derive complex mathematical truths like Fermat’s last theorem via short and intuitively unconvincing arguments. Many philosophers with foundationalist inclinations will feel that these creatures must lack knowledge because they are unable to justify their mathematical assumptions in terms of the kind of basic facts which can be known without further argument. Yet, I will argue that nothing in the current literature lets us draw a principled distinction between what these creatures are doing and paradigmatic cases of good a priori reasoning (assuming that the latter are to be grounded in default reasonable beliefs). I will consider, in turn, appeals to reliability, coherence, conceptual truth and indispensability and argue that none of these can do the job.     

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).     

Duhem–Quine virtue epistemology

The Duhem?CQuine Thesis is the claim that it is impossible to test a scientific hypothesis in isolation because any empirical test requires assuming the truth of one or more auxiliary hypotheses. This is taken by many philosophers, and is assumed here, to support the further thesis that theory choice is underdetermined by empirical evidence. This inquiry is focused strictly on the axiological commitments engendered in solutions to underdetermination, specifically those of Pierre Duhem and W. V. Quine. Duhem resolves underdetermination by appealing to a cluster of virtues called ??good sense??, and it has recently been argued by Stump (Stud Hist Philos Biol Biomed Sci, 18(1):149?C159, 2007) that good sense is a form of virtue epistemology. This paper considers whether Quine, who??s philosophy is heavily influenced by the very thesis that led Duhem to the virtues, is also led to a virtue epistemology in the face of underdetermination. Various sources of Quinian epistemic normativity are considered, and it is argued that, in conjunction with other normative commitments, Quine??s sectarian solution to underdetermination amounts to a skills based virtue epistemology. The paper also sketches formal features of the novel form of virtue epistemology common to Duhem and Quine that challenges the adequacy of epistemic value truth-monism and blocks any imperialist naturalization of virtue epistemology, as the epistemic virtues are essential to the success of the sciences themselves.