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.     

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.

     

Butler’s Stone and Ultimate Psychological Hedonism

This paper discusses psychological hedonism with a special reference to the writings of Bishop Butler, and Elliot Sober and David Sloan Wilson. Contrary to philosophical orthodoxy, Sober and Wilson have claimed that Butler failed to refute psychological hedonism. In this paper it is argued: (1) that there is a difference between reductive and ultimate psychological hedonism; (2) that Butler failed to refute ultimate psychological hedonism, but that he succeeded in refuting reductive psychological hedonism; and, finally and more importantly, (3) that Butler’s criticism of reductive hedonism can be used as a stepping-stone in another argument showing the implausibility of ultimate psychological hedonism as well.     

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.     

Antiskeptical Conditionals

Empirical knowledge exists in the form of antiskeptical conditionals , which are propositions like [if I am not undetectably deceived, then I am holding a pen]. Such conditionals, despite their trivial appearance, have the same essential content as the categorical propositions that we usually discuss, and can serve the same functions in science and practical reasoning. This paper sketches out two versions of a general response to skepticism that employs these conditionals. The first says that our ordinary knowledge attributions can safely be replaced by statements using antiskeptical conditionals, which provides a way around the standard sort of skeptical argument while accepting its soundness with respect to the usual targets. The second analyzes the objects of our ordinary knowledge attributions as antiskeptical conditionals, which allows us to refute, not just evade, the skeptic's argument. Both versions compare favorably to the best-known current approaches to skepticism, including semantic contextualism.     

Functionalism and propositions

Conclusion Some have argued, following Stalnaker, that a plausible functionalist account of belief requires coarse-grained propositions. I have explored a class of functionalist accounts, and my argument has been that, in this class, there is no account which meetsall of the following conditions: it is plausible, noncircular, and allows for the validity of the argument to coarse-grained propositions. In producing this argument, I believe that I have shown that it might be open to a functionalist to adopt fine-grained propositions; thus, one might be a functionalist without holding that all mathematical beliefs are about strings of symbols (and that the belief that all bachelors are unmarried men is a belief about words).My project in this paper has been minimal in the following sense. I havenot argued thatno functionalist account of belief which meets the three conditions can be produced; rather, I have simply explored the inadequacies of certain sorts of accounts. I think that this is useful insofar as it makes clear the challenges to be met by an account of belief which can play the required role in the argument to coarse-grained propositions. It is compatible with my position that such an account is forthcoming, insofar as I have not produced a functionalist theory of belief which is clearly non-circular, plausible, and which yields fine-grained propositions. Of course, it is also compatible with my position that no plausible, non-circular functionalist account of belief of any sort can be produced. My argument has been that,if one construes such mental states as belief as functional states, no convincing argument has yet been produced that they require coarse-grained objects.     

Defending the piggyback principle against Shapiro and Sober’s empirical approach

Jaegwon Kim’s supervenience/exclusion argument attempts to show that non-reductive physicalism is incompatible with mental causation. This influential argument can be seen as relying on the following principle, which I call “the piggyback principle”: If, with respect to an effect, E, an instance of a supervenient property, A, has no causal powers over and above, or in addition to, those had by its supervenience base, B, then the instance of A does not cause E (unless A is identical with B). In their “Epiphenomenalism: The Dos and the Don’ts,” Larry Shapiro and Elliott Sober employ a novel empirical approach to challenge the piggyback principle. Their empirical approach pulls from the experiments of August Weismann regarding the inheritance of acquired characteristics. Through an examination of Weismann’s experiments, Shapiro and Sober extract lessons in reasoning about the epiphenomenalism of a property. And according to these empirically drawn lessons, the piggyback principle is a don’t. My primary aim in this paper is to defend the piggyback principle against Shapiro and Sober’s empirical approach.     

Van Inwagen and the Quine-Putnam 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.

Can the new indispensability argument be saved from Euclidean rescues?

The traditional formulation of the indispensability argument for the existence of mathematical entities (IA) has been criticised due to its reliance on confirmational holism. Recently a formulation of IA that works without appeal to confirmational holism has been defended. This recent formulation is meant to be superior to the traditional formulation in virtue of it not being subject to the kind of criticism that pertains to confirmational holism. I shall argue that a proponent of the version of IA that works without appeal to confirmational holism will struggle to answer a challenge readily answered by proponents of a version of IA that does appeal to confirmational holism. This challenge is to explain why mathematics applied in falsified scientific theories is not considered to be falsified along with the rest of the theory. In cases where mathematics seemingly ought to be falsified it is saved from falsification, by a so called ??Euclidean rescue??. I consider a range of possible answers to this challenge and conclude that each answer fails.     

Apriority and applied mathematics

I argue that we need not accept Quine's holistic conception of mathematics and empirical science. Specifically, I argue that we should reject Quine's holism for two reasons. One, his argument for this position fails to appreciate that the revision of the mathematics employed in scientific theories is often related to an expansion of the possibilities of describing the empirical world, and that this reveals that mathematics serves as a kind of rational framework for empirical theorizing. Two, this holistic conception does not clearly demarcate pure mathematics from applied mathematics. In arguing against Quine, I present a formal account of applied mathematics in which the mathematics employed in an empirical theory plays a role that is analogous to the epistemological role Kant assigned synthetic a priori propositions. According to this account, it is possible to insulate pure mathematics from empirical falsification, and there is a sense in which applied mathematics can also be labeled as a priori.I am especially indebted to Michael Friedman for his valuable suggestions and criticisms of this paper. I would also like to thank Mark Wilson, Anil Gupta, and Arthur Fine for their comments and encouragement.     

Theism and other minds: On the falsifiability of non-theories

In this paper I consider three necessary conditions for a proposition counting as a theory: that the proposition be posited for its explanatory power; that it derive its feasibility from the extent to which it provides such explanatory power; and that it be empirically falsifiable. I then argue that some propositions might fail as theories because they do not satisfy the first two conditions, yet still satisfy the third condition. Such propositions I label falsifiable non-theories. I offer folk psychology (the proposition that beliefs, desires, and other intentional phenomena exist and play essential motivational and causal roles in many human actions) as a paradigm example of a falsifiable non-theory. I then argue that theism is in an analogous position. Like folk psychology, it fails to satisfy the first two conditions above for most theists. However, the empirical implications that theism has do make it susceptible to falsification. I demonstrate such falsifiability by an extreme scenario from Keith Yandell. Then I argue that recent work by Paul Draper demonstrates how a well articulated empirical argument from evil might threaten just such falsification.     

The problem of universals and its perceptual correlates

This paper deals with the philosophical questions which gave rise to the traditional realist theories of universals. The main thesis is that these same questions may also be interpreted as scientific-empirical questions. The study of these problems has begun only very recently and the relevance of the results for the traditional problem of intensional entities has only been remarked by few workers aware of the philosophical problem. The approach adopted here is that of regarding man as a perceptual system situated in an environment and showing certain capacities like perception of universals. The possible ways of access to these entities is discussed on the basis of some recent empirical findings related to perception. One conclusion is that in order to answer the basic philosophical question of how properties, particulars and propositions are perceived may be answered within the domains of science, accepting only the usual presuppositions used in it. Basically, such an approach sees the root of the problem in the great complexity of information processing involved in the actual perception of properties, particulars and propositions.     

Arguments from Expert Opinion and Persistent Bias

Accounts of arguments from expert opinion take it for granted that expert judgments count as (defeasible) evidence for propositions, and so an argument that proceeds from premises about what an expert judges to a conclusion that the expert is probably right is a strong argument. In Mizrahi (Informal Log 33:57–79, 2013), I consider a potential justification for this assumption, namely, that expert judgments are significantly more likely to be true than novice judgments, and find it wanting because of empirical evidence suggesting that expert judgments under uncertainty are not significantly more likely to be true than novice judgments or even chance. In this paper, I consider another potential justification for this assumption, namely, that expert judgments are not influenced by the cognitive biases novice judgments are influenced by, and find it wanting, too, because of empirical evidence suggesting that experts are vulnerable to pretty much the same cognitive biases that novices are vulnerable to. If this is correct, then the basic assumption at the core of accounts of arguments from expert opinion, namely, that expert judgments count as (defeasible) evidence for propositions, remains unjustified.     

Ontology According to van Fraassen: Some Problems with Constructive Empiricism

This paper argues the case for ontological realism as against various present-day forms of conventionalist, instrumentalist, cultural-relativist, or anti-realist doctrine. In particular it takes issue with Richard Rorty's writings on philosophy of science – where these ideas receive their most extreme and provocative statement – and with Bas van Fraassen's more moderate 'constructive empiricist' approach. This latter entails ontological commitment to whatever shows up through trained observation or empirical research. However, it refuses to countenance realist claims concerning the existence of (as yet) unobserved entities and their role in explanatory theories premised on putative laws of nature. I maintain that van Fraassen's position is: (1) inadequate to account for our knowledge of the growth of scientific knowledge; (2) self-refuting since often undermined by examples which he himself supplies; and (3) incapable of mounting resistance to other, more wholesale (e.g., Rortian) varieties of anti-realist argument. Only by combining causal realism with a principle of inference to the best explanation can philosophy of science avoid these kinds of hyperinduced sceptical doubt.     

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.     

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.     

On the Explanatory Role of Correspondence Truth

An intuitive argument for scientific realism suggests that our successes in predicting and intervening would be inexplicable if the theories that generate them were not approximately true. This argument faces many objections, some of which are briefly addressed in this paper, and one of which is treated in more detail. The focal criticism alleges that appeals to success cannot deliver conclusions that parts of science are true in the sense of truth-as-correspondence that realists prefer. The paper responds to that criticism, in versions proposed by Michael Williams, Michael Levin, and, especially, Paul Horwich, by arguing that critics typically stop at a shallow level of psychological explanation. If we probe more deeply we discover a genuine explanatory role for correspondence truth.     

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.     

Gwiazda on the Bayesian Argument for God

Jeremy Gwiazda made two criticisms of my formulation in terms of Bayes’s theorem of my probabilistic argument for the existence of God. The first criticism depends on his assumption that I claim that the intrinsic probabilities of all propositions depend almost entirely on their simplicity; however, my claim is that that holds only insofar as those propositions are explanatory hypotheses. The second criticism depends on a claim that the intrinsic probabilities of exclusive and exhaustive explanatory hypotheses of a phenomenon must sum to 1; however it is only those probabilities plus the intrinsic probability of the non-occurrence of the phenomenon which must sum to 1.