This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

Hartry Field has shown us a way to be nominalists: we must purge our scientific theories of quantification over abstracta and we must prove the appropriate conservativeness results. This is not a path for the faint hearted. Indeed, the substantial technical difficulties facing Field's project have led some to explore other, easier options. Recently, Jody Azzouni, Joseph Melia, and Stephen Yablo have argued that it is a mistake to read our ontological commitments simply from what the quantifiers of our best (...) scientific theories range over. In this paper, I argue that all three arguments fail and they fail for much the same reason; would-be nominalists are thus left facing Field's hard road. (shrink)

In this paper I reply to Jody Azzouni, Otávio Bueno, Mary Leng, David Liggins, and Stephen Yablo, who offer defences of so-called ‘ easy road ’ nominalist strategies in the philosophy of mathematics.

There is a long tradition comparing moral knowledge to mathematical knowledge. In this paper, I discuss apparent similarities and differences between knowledge in the two areas, realistically conceived. I argue that many of these are only apparent, while others are less philosophically significant than might be thought. The picture that emerges is surprising. There are definitely differences between epistemological arguments in the two areas. However, these differences, if anything, increase the plausibility of moral realism as compared to mathematical realism. It (...) is hard to see how one might argue, on epistemological grounds, for moral antirealism while maintaining commitment to mathematical realism. But it may be possible to do the opposite. (shrink)

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, D—the (...) challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)

The present paper advances an analogy between cases of extra-mathematical explanation and cases of what might be termed ‘extra-logical explanation’: the explanation of a physical fact by a logical fact. A particular case of extra-logical explanation is identified that arises in the philosophical literature on time travel. This instance of extra-logical explanation is subsequently shown to be of a piece with cases of extra-mathematical explanation. Using this analogy, we argue extra-mathematical explanation is part of a broader class of non-causal explanation. (...) This has important implications for extra-mathematical explanation, for time travel and for theories of explanation more generally. (shrink)

The indispensability argument seeks to establish the existence of mathematical objects. The success of the indispensability argument turns on finding cases of genuine extra- mathematical explanation. In this paper, I identify a new case of extra- mathematical explanation, involving the search patterns of fully-aquatic marine predators. I go on to use this case to predict the prevalence of extra- mathematical explanation in science.

The aim of this paper is to open a new front in the debate between platonism and nominalism by arguing that the degree of explanatory entanglement of mathematics in science is much more extensive than has been hitherto acknowledged. Even standard examples, such as the prime life cycles of periodical cicadas, involve a penumbra of mathematical features whose presence can only be explained using relatively sophisticated mathematics. I introduce the term ‘mathematical spandrel’ to describe these penumbral properties, and focus on (...) the property that cicada period lengths are expressible as the sum of two perfect squares. I argue that mathematical spandrels pose a particular problem for nominalism because of the way in which they are entangled with scientific explanations. (shrink)

Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...) potential future directions of research for each side in the debate over the existence of abstract mathematical objects. (shrink)

Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...) issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)

The official model of explanation proposed by the logical empiricists, the covering law model, is subject to familiar objections. The goal of the present paper is to explore an unofficial view of explanation which logical empiricists have sometimes suggested, the view of explanation as unification. I try to show that this view can be developed so as to provide insight into major episodes in the history of science, and that it can overcome some of the most serious difficulties besetting the (...) covering law model. (shrink)

This book, based on Jaegwon Kim's 1996 Townsend Lectures, presents the philosopher's current views on a variety of issues in the metaphysics of the mind...

This is a major work in moral philosophy, the long-awaited follow-up to Parfit's 1984 classic Reasons and Persons, a landmark of twentieth-century philosophy. Parfit now presents a powerful new treatment of reasons and a critical examination of the most prominent systematic moral theories, leading to his own ground-breaking conclusion.

In this book, T. M. Scanlon offers new answers to these questions, as they apply to the central part of morality that concerns what we owe to each other.

Mark Colyvan (2010) raises two problems for ‘easy road’ nominalism about mathematical objects. The first is that a theory’s mathematical commitments may run too deep to permit the extraction of nominalistic content. Taking the math out is, or could be, like taking the hobbits out of Lord of the Rings. I agree with the ‘could be’, but not (or not yet) the ‘is’. A notion of logical subtraction is developed that supports the possibility, questioned by Colyvan, of bracketing a theory’s (...) mathematical aspects to obtain, as remainder, what it says ‘mathematics aside’. The other problem concerns explanation. Several grades of mathematical involvement in physical explanation are distinguished, by analogy with Quine’s three grades of modal involvement. The first two grades plausibly obtain, but they do not require mathematical objects. The third grade is likelier to require mathematical objects. But it is not clear from Colyvan’s example that the third grade really obtains. (shrink)

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are vulnerable (...) to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

Moral Realism is a systematic defence of the idea that there are objective moral standards. Russ Shafer-Landau argues that there are moral principles that are true independently of what anyone, anywhere, happens to think of them. His central thesis, as well as the many novel supporting arguments used to defend it, will spark much controversy among those concerned with the foundations of ethics.

According to many, that the normative supervenes on the non-normative is a truism of normative discourse. This chapter argues that those committed to more specific moral, aesthetic, and epistemic supervenience theses should also hold : As a matter of conceptual necessity, whenever something has a normative property, it has a base property or collection of base properties that metaphysically necessitates the normative one. The main aim in this chapter is to show that none of the available arguments establish, or indeed (...) the relevant epistemic, aesthetic, and moral supervenience theses. is not a conceptual truth. This has considerable dialectical importance. One interesting upshot is that it affords non-reductivists and non-naturalists a novel way of resisting certain prominent supervenience-based objections to their views, including objections that formulate supervenience as a purely metaphysical thesis. (shrink)

Statements about a person's good slip into and out of our ordinary discourse about the world with nary a ripple. Such statements are objects of belief and assertion, they obey the rules of logic, and they are often defended by evidence and argument. They even participate in common-sense explanations, as when we say of some person that he has been less subject to wild swings of enthusiasm and disappointment now that, with experience, he has gained a clearer idea of what (...) is good for him. Statements about a person's good present themselves as being about something with respect to which our beliefs can be true or false, warranted or unwarranted. Let us speak of these features as the descriptive side of discourse about a person's good. (shrink)

In this brief book one of the most distinguished living American philosophers takes up the question of whether ethical judgments can properly be considered ...

A defence of non-causal explanations of events is presented in cases where explanation is understood as modal explanation. In such cases the source of modal information is crucial. All explanations implicitly use contrast classes, and relative to a particular contrast we can privilege some difference makers over others. Thinking about changes in these privileged “actual” difference makers is then the source of modal information for any given explanandum. If an actual difference maker is non-causal, then we have a principled definition (...) of a non-causal explanation of an event regardless of how much causal information is also contained in the explanans. A demarcation of explanation into causal and non-causal in this way recovers ordinary language about explanation as well as reflecting genuine differences in practice, such as the in-principle evidential base for any modal claim. (shrink)

According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set of (...) sentences in the nominalist language such theories entail. However, I argue that what such theories show is that mathematics can enable us to express possibilities about the concrete world that may not be expressible in nominalistically acceptable language. While I grant that this may make quantification over abstracta indispensable, I deny that such indispensability is a reason for accepting them into our ontology. I urge that the nominalist should be allowed to quantify over abstracta whilst denying their existence and I explain how this apparently contradictory practice (a practice I call 'weaseling') is in fact coherent, unproblematic and rational. Finally, I examine the view that platonistic theories are simpler or more attractive than their nominalistic reformulations, and thus that abstract ought to be accepted into our ontology for the same sorts of reasons as other theoretical objects. I argue that, at least in the case of numbers, functions and sets, such arguments misunderstand the kind of simplicity and attractiveness we seek. (shrink)

David Malament argued that Hartry Field's nominalisation program is unlikely to be able to deal with non-space-time theories such as phase-space theories. We give a specific example of such a phase-space theory and argue that this presentation of the theory delivers explanations that are not available in the classical presentation of the theory. This suggests that even if phase-space theories can be nominalised, the resulting theory will not have the explanatory power of the original. Phase-space theories thus raise problems for (...) nominalists that go beyond Malament's initial concerns. Thanks to Mark Steiner, Jens Christian Bjerring, Ben Fraser, John Mathewson, and two anonymous referees for helpful comments on an earlier draft of this paper. CiteULike Connotea Del.icio.us What's this? (shrink)

Counterfactuals is David Lewis' forceful presentation of and sustained argument for a particular view about propositions which express contrary to fact conditionals, including his famous defense of realism about possible worlds and his theory of laws of nature.

This paper argues that the exclusion problem for mental causation can be solved by a variant of non-reductive physicalism that takes the mental not merely to supervene on, but to be grounded in, the physical. A grounding relation between events can be used to establish a principle that links the causal relations of grounded events to those of grounding events. Given this principle, mental events and their physical grounds either do not count as overdetermining physical effects, or they do so (...) in a way that is not objectionable. (shrink)

Somewhat loose arguments that non-reductive physicalist realism is untenable. Anomalous monism makes the mental irrelevant, functionalism is compatible with species-specific reduction, and supervenience is weak or reductive.

"This is a fine volume that clarifies, defends, and moves beyond the views that Kim presented in Mind in a Physical World.

It is widely held, even among nonnaturalists, that the moral supervenes on the natural. This is to say that for any two metaphysically possible worlds w and w′, and for any entities x in w and y in w′, any isomorphism between x and y that preserves the natural properties preserves the moral properties. In this paper, I put forward a conceivability argument against moral supervenience, assuming non-naturalism. First, I argue that though utilitarianism may be true, and the trolley driver (...) is permitted to kill the one to save the five, there is a conceivable scenario that is just like our world in all natural respects, yet at which deontology is true, and the trolly driver is not permitted to kill the one to save the five. I then argue that in the special case of morality, it is possible to infer from the conceivability of such a scenario to its possibility. It follows that supervenience is false. (shrink)

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics.

A theory of ersatz impossible worlds is developed to deal with the problem of counterpossible conditionals. Using only tools standardly in the toolbox of possible worlds theorists, it is shown that we can construct a model for counterpossibles. This model is a natural extension of Lewis's semantics for counterfactuals, but instead of using classical logic as its base, it uses the logic LP.

This new edition of Alexander Miller’s highly readable introduction to contemporary metaethics provides a critical overview of the main arguments and themes in twentieth- and twenty-first-century contemporary metaethics. Miller traces the development of contemporary debates in metaethics from their beginnings in the work of G. E. Moore up to the most recent arguments between naturalism and non-naturalism, cognitivism and non-cognitivism. From Moore’s attack on ethical naturalism, A. J. Ayer’s emotivism and Simon Blackburn’s quasi-realism to anti-realist and best opinion accounts of (...) moral truth and the non-reductionist naturalism of the ‘Cornell realists’, this book addresses all the key theories and ideas in this field. As well as revisiting the whole terrain with revised and updated guides to further reading, Miller also introduces major new sections on the revolutionary fictionalism of Richard Joyce and the hermeneutic fictionalism of Mark Kalderon. The new edition will continue to be essential reading for students, teachers and professional philosophers with an interest in contemporary metaethics. (shrink)

It is often claimed that irreducibly normative truths would have unacceptable metaphysical implications, and are incompatible with a scientific view of the world. The book argues, on the basis of a general account of the relevance of ontological questions, that this claim is mistaken. It is also a mistake to think that interpreting normative judgments as beliefs would make it impossible to explain their connection with action. An agent’s acceptance of a normative judgment can explain that agent’s subsequent action because (...) it is part of being a rational agent that such an agent’s beliefs about reasons normally, but not invariably, make a difference to the agent’s subsequent behavior. Because facts about reasons are not entities existing apart from us, there is no epistemological problem of how we can “be in touch with” such facts. There are serious worries about normative knowledge, but the problems involved are internal to the normative domain itself. The best solution to these problems would be an overall account of the domain of reasons in normative terms, supported by an argument from reflective equilibrium. But no existing account, constructivist, or based on desires or on an idea of rationality, is plausible, and no alternative is likely to succeed. Conclusions about reasons for action must rest on more piecemeal applications of the method of reflective equilibrium. (shrink)

What in our theoretical pronouncements commits us to objects? The Quinean standard for ontological commitment involves (nearly enough) commitments when we utter “there is” or “there are” statements without hope of eliminating these by paraphrase. Coupled with the indispensability of the truth of applied mathematical doctrine, the result is that the ontologically hard-nosed scientist is a Platonist—haplessly commited to abstracta. In this book Azzouni offers a way around the Quinean straitjacket: ontological commitment turns on how theories are (nearly enough) nailed (...) to the world. The specifics of how theories are applied indicates which among the posits of a theory are mere mathematical garb and which are genuine connections to items out there. In the first part of the book Azzouni undercuts the arguments, both actual and possible, in support of Quine’s criterion. An alternative criterion for what exists—ontological independence—is offered, one in sturdy accord with ordinary folk views on the matter. In the second part of the book, a beginning is made of bringing this alternative to bear upon scientific theories with a rich mathematical component. Along the way, old philosophical issues about absolute space and time versus relative space and time, the status of mathematical posits, such as spatial and temporal points, and so on, are illuminated. (shrink)

The type identity theory, according to which types of mental state are identical to types of physical state, fell out of favour for some years but is now being considered with renewed interest. Many philosophers are critically re-examining the arguments which were marshalled against it, finding in the type identity theory both resources to strengthen a comprehensive, physicalistic metaphysics and a useful tool in understanding the relationship between developments in psychology and new results in neuroscience. This volume brings together leading (...) philosophers of mind, whose essays challenge in new ways the standard objections to type identity theory, such as the multiple realizability objection and the modal argument. Other essays show how cognitive science and neuroscience are lending new support to type identity theory and still others provide, extend and improve traditional arguments concerning the theory's explanatory power. (shrink)