Intrinsic Explanation and Field’s Dispensabilist Strategy

Abstract

Hartry Field defended the importance of his nominalist reformulation of Newtonian Gravitational Theory, as a response to the indispensability argument, on the basis of a general principle of intrinsic explanation. In this paper, I argue that this principle is not sufficiently defensible, and can not do the work for which Field uses it. I argue first that the model for Field’s reformulation, Hilbert’s axiomatization of Euclidean geometry, can be understood without appealing to the principle. Second, I argue that our desires to unify our theories and explanations undermines Field’s principle. Third, the claim that extrinsic theories seem like magic is, in this case, really just a demand for an account of the applications of mathematics in science. Finally, even if we were to accept the principle, it would not favor the fictionalism that motivates Field’s argument, since the indispensabilist’s mathematical objects are actually intrinsic to scientific theory.     

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.     

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).     

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.     

The holistic presumptions of the indispensability argument

The indispensability argument is sometimes seen as weakened by its reliance on a controversial premise of confirmation holism. Recently, some philosophers working on the indispensability argument have developed versions of the argument which, they claim, do not rely on holism. Some of these writers even claim to have strengthened the argument by eliminating the controversial premise. I argue that the apparent removal of holism leaves a lacuna in the argument. Without the holistic premise, or some other premise which facilitates the transfer of evidence to mathematical portions of scientific theories, the argument is implausible.     

Lucky achievement: virtue epistemology on the value of knowledge

Virtue epistemology argues that knowledge is more valuable than Gettierized belief because knowledge is an achievement, but Gettierized belief is not. The key premise in the achievement argument is that achievement is apt (successful because competent) and Gettierized belief is inapt (successful because lucky). I first argue that the intuition behind the achievement argument is based wrongly on the fact that ‘being successful because lucky’ implicates ‘being not competent enough’. I then offer an argument from moral luck to argue that virtue epistemologists should maintain that knowledge is no more valuable than Gettierized belief.     

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.     

The skeptical paradox and the indispensability of knowledge-beliefs

Some philosophers understand epistemological skepticism as merely presenting a paradox to be solved, a paradox given rise to by some apparently forceful arguments. I argue that such a view needs to be justified, and that the best way to do so is to show that we cannot help seeing skepticism as obviously false. The obviousness (to us) of the falsity of skepticism is, I suggest, explained by the fact that we cannot live without knowledge-beliefs (a knowledge-belief about the world is a belief that a person or a group of people know that p, where p is an empirical proposition about the world). I then go on to argue for the indispensability of knowledge-beliefs. The first line of argument appeals to the practical aspects of our employment of the concept of knowledge, and the second line of argument draws on some Davidsonian ideas concerning understanding and massive agreement.     

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.     

Conceptual analysis as armchair psychology: in defense of methodological naturalism

Three proponents of the Canberra Plan, namely Jackson, Pettit, and Smith, have developed a collective functionalist program—Canberra Functionalism—spanning from philosophical psychology to ethics. They argue that conceptual analysis is an indispensible tool for research on cognitive processes since it reveals that there are some folk concepts, like belief and desire, whose functional roles must be preserved rather than eliminated by future scientific explanations. Some naturalists have recently challenged this indispensability argument, though the point of that challenge has been blunted by a mutual conflation of metaphysical and methodological strands of naturalism. I argue that the naturalist’s challenge to the indispensability argument, like naturalism itself, ought to be reformulated as a strictly methodological thesis. So understood, the challenge succeeds by showing (1) that we cannot know a priori on the basis of conceptual analysis of folk platitudes that something must occupy the functional roles specified for beliefs and desires, and (2) that proponents of Canberra Functionalism sometimes tacitly concede this point by treating substantive psychological theories as the deliverances of a priori platitudes analysis.     

The Kantian Moral Hazard Argument for religious fictionalism

In this paper I do three things. Firstly, I defend the view that in his most familiar arguments about morality and the theological postulates, the arguments which appeal to the epistemological doctrines of the first Critique, Kant is as much of a fictionalist as anybody not working explicitly with that conceptual apparatus could be: his notion of faith as subjectively and not objectively grounded is precisely what fictionalists are concerned with in their talk of nondoxastic attitudes. Secondly, I reconstruct a logically distinct argument to a fictionalist conclusion which I argue Kant also gives us, this time an argument to the conclusion that it is a good thing if our commitment to the existence of God is nondoxastic. And finally, I argue that this argument is of continuing interest, to Kantians and non-Kantians alike, not only because it raises interesting questions about the relation of morality to belief in God (which go in the opposite direction to most discussions, which focus on whether and to what extent belief in God can be an aid to morality), but also because this ‘Moral Hazard Argument’ seems to be available to religious realists and non-realists alike, thus suggesting that religious fictionalism is not by any means just an interesting version of religious non-realism.     

Why inference to the best explanation doesn’t secure empirical grounds for mathematical platonism

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I object that this inference depends on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist.

     

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.     

Schellenberg on the epistemic force of experience

According to Schellenberg, our perceptual experiences have the epistemic force they do because they are exercises of certain sorts of capacity, namely capacities to discriminate particulars—objects, property-instances and events—in a sensory mode. She calls her account the “capacity view.” In this paper, I will raise three concerns about Schellenberg’s capacity view. The first is whether we might do better to leave capacities out of our epistemology and take content properties as the fundamental epistemically relevant features of experiences. I argue we would. The second is whether Schellenberg’s appeal to factive and phenomenal evidence accommodates the intuitive verdicts about the bad case that she claims it does. I argue it does not. The third is whether Schellenberg’s account of factive evidence is adequate to capture nuances concerning the justification for singular but nondemonstrative perceptual beliefs, such as the belief that’s NN, where NN is a proper name. I argue it is not. If I am right, these points suggest a mental-state-first account of perceptual justification, rather than a capacity-first account, and one which treats the good and bad cases alike in respect of justification and complicates the relation between perceptual content and what one is justified in believing.     

Against Doxastic Compatibilism

William Alston has argued that the so‐called deontological conception of epistemic justification, on which epistemic justification is to be spelled out in terms of blame, responsibility, and obligations, is untenable. The basic idea of the argument is that this conception is untenable because we lack voluntary control over our beliefs and, therefore, cannot have any obligations to hold certain beliefs. If this is convincing, however, the argument threatens the very idea of doxastic responsibility. For, how can we ever be responsible for our beliefs if we lack control over them? Several philosophers have argued that the idea that we bear responsibility for our beliefs can be saved, because absence of voluntary control over our beliefs is perfectly compatible with having obligations to hold particular beliefs. With others, I call this view ‘doxastic compatibilism’. It comes in two varieties. On the first variety, doxastic obligations do not require any kind of doxastic control whatsoever. I argue that this variety of doxastic compatibilism fails because it confuses doxastic responsibility with other closely related phenomena. On the second variety, doxastic obligations do not require voluntary doxastic control, but only compatibilist doxastic control (roughly, reason‐responsiveness) and we do in fact have such control. I grant that we have such control, but also argue that having such control is insufficient for bearing doxastic responsibility. The plausibility of the examples put forward by doxastic compatibilists in support of the claim that it is sufficient for doxastic responsibility derives from the fact that in these examples, the subjects have control over factors that influence what they believe rather than control over those beliefs themselves.     

Optimus prime: paraphrasing prime number talk

Baker (Mind 114:223–238, 2005; Brit J Philos Sci 60:611–633, 2009) has recently defended what he calls the “enhanced” version of the indispensability argument for mathematical Platonism. In this paper I demonstrate that the nominalist can respond to Baker’s argument. First, I outline Baker’s argument in more detail before providing a nominalistically acceptable paraphrase of prime-number talk. Second, I argue that, for the nominalist, mathematical language is used to express physical facts about the world. In endorsing this line I follow moves made by Saatsi (Brit J Philos Sci 62(1):143–154, 2011). But, unlike Saatsi, I go on to argue that the nominalist requires a paraphrase of prime-number talk, for otherwise we lack an account of what that ‘physical fact’ is in the case of mathematics that seemingly makes reference to prime numbers.     

A defence of Owens’ exclusivity objection to beliefs having aims

In this paper we argue that Steglich-Petersen’s response to Owens’ Exclusivity Objection does not work. Our first point is that the examples Steglich-Petersen uses to demonstrate his argument do not work because they employ an undefended conception of the truth aim not shared by his target (and officially eschewed by Steglich-Petersen himself). Secondly we will make the point that deliberating over whether to form a belief about p is not part of the belief forming process. When an agent enters into this process of deliberation, he has not, contra Steglich-Petersen, already adopted the truth aim with regard to p. In closing, we further suggest that proponents of the truth aim hypothesis need to focus on aim-guidance, not mere aim attribution, for their approach to have explanatory utility so underlining the significance of Owens’ argument.     

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.     

Inferentialism and knowledge: Brandom’s arguments against reliabilism

I take issue with Robert Brandom’s claim that on an analysis of knowledge based on objective probabilities it is not possible to provide a stable answer to the question whether a belief has the status of knowledge. I argue that the version of the problem of generality developed by Brandom doesn’t undermine a truth-tracking account of noninferential knowledge that construes truth-tacking in terms of conditional probabilities. I then consider Sherrilyn Roush’s claim that an account of knowledge based on probabilistic tracking faces a version of the problem of generality. I argue that the problems she raises are specific to her account, and do not affect the version of the view that I have advanced. I then consider Brandom’s argument that the cases that motivate reliabilist epistemologies are in principle exceptional. I argue that he has failed to make a cogent case for this claim. I close with the suggestion that the representationalist approach to knowledge that I endorse and Brandom rejects is in principle compatible with the kind of pragmatist approach to belief and truth that both Brandom and I endorse.