God and numbers provide two challenges to metaphysical naturalism–the former if God exists and is a supernatural being, the latter if numbers exist and mathematical Platonism is true. Evolutionary theory is often described as having a commitment to naturalism, but this is doubly wrong. The theory is neutral on the question of whether God exists and mathematical evolutionary theory entails that numbers exist. The chapter develops the point about theistic neutrality by considering what evolutionary biologists mean when they say that (...) mutations are “unguided.” Evolutionary theory is logically compatible with deism and also with various interventionist theologies. In connection with mathematical evolutionary theory, the chapter discusses and criticizes the indispensability argument for mathematical Platonism developed by Quine and Putnam. The chapter criticizes their epistemological holism by considering ideas in Bayesian confirmation theory. (shrink)

The ontological dependence of one domain on another is compatible with the explanatory autonomy of the less basic domain. That autonomy results from the fact that the relationship between two domains can be very complex. In this paper I distinguish two different types of complexity, two ways the relationship between domains can fail to be transparent, both of which are relevant to evolutionary biology. Sometimes high level explanations preserve a certain type of causal or counterfactual information which would be lost (...) at the lower level; I argue that this is central to the proper understanding of the adaptationist program. Sometimes high level kinds are multiply realised by lower level kinds: I argue that this is central to the understanding of macroevolution. (shrink)

Conflicting accounts of the role of mathematics in our physical theories can be traced to two principles. Mathematics appears to be both (1) theoretically indispensable, as we have no acceptable non-mathematical versions of our theories, and (2) metaphysically dispensable, as mathematical entities, if they existed, would lack a relevant causal role in the physical world. I offer a new account of a role for mathematics in the physical sciences that emphasizes the epistemic benefits of having mathematics around when we do (...) science. This account successfully reconciles theoretical indispensability and metaphysical dispensability and has important consequences for both advocates and critics of indispensability arguments for platonism about mathematics. (shrink)

Miller's reply to Nelson misses the point because it does not attend to the difference between identifying the truth conditions for a proposition and explaining why those conditions are the ones in which the proposition is true.

Alexander Miller has recently considered an ingenious extension of Frank Jackson and Philip Pettit's account of 'program explanation' as a way of defending non-reductive naturalist versions of moral realism against Harman's explanatory criticism. Despite the ingenuity of this extension, Miller concludes that program explanation cannot help such moral realists in their attempt to defend moral properties. Specifically, he argues that such moral program explanations are dispensable from an epistemically unlimited point of view. I show that Miller's argument for this negative (...) claim is inadequate, and that he has, in spite of himself, identified a promising defence of moral realism. (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)

Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be.

Some properties are causally relevant for a certain effect, others are not. In this paper we describe a problem for our understanding of this notion and then offer a solution in terms of the notion of a program explanation.

Contains an overall account of morality in its philosophical format particularly with regard to problems of observation, evidence, and truth.

We defend Joseph Melia's thesis that the role of mathematics in scientific theory is to 'index' quantities, and that even if mathematics is indispensable to scientific explanations of concrete phenomena, it does not explain any of those phenomena. This thesis is defended against objections by Mark Colyvan and Alan Baker.

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)

The Eleatic Principle or causal criterion is a causal test that entities must pass in order to gain admission to some philosophers’ ontology.1 This principle justifies belief in only those entities to which causal power can be attributed, that is, to those entities which can bring about changes in the world. The idea of such a test is rather important in modern ontology, since it is neither without intuitive appeal nor without influential supporters. Its supporters have included David Armstrong (1978, (...) Vol 2, 5), Brian Ellis (1990, 22) and Hartry Field2 (1989, 68) to name but a few. (shrink)

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.

Philosophers of mathematics have become increasingly interested in the explanatory role of mathematics in empirical science, in the context of new versions of the Quinean ‘Indispensability Argument’ which employ inference to the best explanation for the existence of abstract mathematical objects. However, little attention has been paid to analysing the nature of the explanatory relation involved in these mathematical explanations in science (MES). In this paper, I attack the only articulated account of MES in the literature (an account sketched by (...) Mark Steiner), according to which a genuine MES incorporates an explanatory proof of the mathematical result being used. The central case study involves an explanation for why bees build the cells of their honeycombs in the shape of hexagons. I make a distinction between two kinds of MES, mathematics-driven explanation in science and science-driven mathematical explanation, and argue that it is the second category which is both scientifically and philosophical more central. I conclude that the explanatory relation involved in MES is genuinely scientific and hence that the phenomenon of MES poses a challenge to general accounts of scientific explanation. (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)

We discuss a recent attempt by Chris Daly and Simon Langford to do away with mathematical explanations of physical phenomena. Daly and Langford suggest that mathematics merely indexes parts of the physical world, and on this understanding of the role of mathematics in science, there is no need to countenance mathematical explanation of physical facts. We argue that their strategy is at best a sketch and only looks plausible in simple cases. We also draw attention to how frequently Daly and (...) Langford find themselves in conflict with mathematical and scientific practice. (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)

In chapter 8 of Miller 2003, I argued against the idea that Jackson and Pettit's notion of program explanation might help Sturgeon's non-reductive naturalist version of moral realism respond to the explanatory challenge posed by Harman. In a recent paper in the AJP[Nelson 2006, Mark Nelson has attempted to defend the idea that program explanation might prove useful to Sturgeon in replying to Harman. In this note, I suggest that Nelson's argument fails.

This introduction provides a highly readable critical overview of the main arguments and themes in twentieth-century and contemporary metaethics. It 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. A highly readable critical overview of the main arguments and themes in twentieth century and contemporary metaethics. Asks: Are there moral facts? Is there such a thing as moral truth? (...) Is moral knowledge possible? Traces the development of contemporary debates in metaethics from their beginnings in the work of G. E. Moore up to the most recent debates between naturalism and non-naturalism, cognitivism and noncognitivism. Provides for the first time a critical survey of famous figures in twentieth century metaethics such as Moore, Ayer and Mackie together with in-depth discussions of contemporary philosophers such as Blackburn, Gibbard, Wright, Harman, Railton, Sturgeon, McDowell and Wiggins. (shrink)

Moral thinking pervades our practical lives, but where did this way of thinking come from, and what purpose does it serve? Is it to be explained by environmental pressures on our ancestors a million years ago, or is it a cultural invention of more recent origin? In The Evolution of Morality, Richard Joyce takes up these controversial questions, finding that the evidence supports an innate basis to human morality. As a moral philosopher, Joyce is interested in whether any implications follow (...) from this hypothesis. Might the fact that the human brain has been biologically prepared by natural selection to engage in moral judgment serve in some sense to vindicate this way of thinking--staving off the threat of moral skepticism, or even undergirding some version of moral realism? Or if morality has an adaptive explanation in genetic terms--if it is, as Joyce writes, "just something that helped our ancestors make more babies"--might such an explanation actually undermine morality's central role in our lives? He carefully examines both the evolutionary "vindication of morality" and the evolutionary "debunking of morality," considering the skeptical view more seriously than have others who have treated the subject.Interdisciplinary and combining the latest results from the empirical sciences with philosophical discussion, The Evolution of Morality is one of the few books in this area written from the perspective of moral philosophy. Concise and without technical jargon, the arguments are rigorous but accessible to readers from different academic backgrounds. Joyce discusses complex issues in plain language while advocating subtle and sometimes radical views. The Evolution of Morality lays the philosophical foundations for further research into the biological understanding of human morality. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Professor Hilary Putnam has been one of the most influential and sharply original of recent American philosophers in a whole range of fields. His most important published work is collected here, together with several new and substantial studies, in two volumes. The first deals with the philosophy of mathematics and of science and the nature of philosophical and scientific enquiry; the second deals with the philosophy of language and mind. Volume one is now issued in a new edition, including an (...) essay on the philosophy of logic first published in 1971. (shrink)

God and numbers provide two challenges to metaphysical naturalism–the former if God exists and is a supernatural being, the latter if numbers exist and mathematical Platonism is true. Evolutionary theory is often described as having a commitment to naturalism, but this is doubly wrong. The theory is neutral on the question of whether God exists and mathematical evolutionary theory entails that numbers exist. The chapter develops the point about theistic neutrality by considering what evolutionary biologists mean when they say that (...) mutations are “unguided.” Evolutionary theory is logically compatible with deism and also with various interventionist theologies. In connection with mathematical evolutionary theory, the chapter discusses and criticizes the indispensability argument for mathematical Platonism developed by Quine and Putnam. The chapter criticizes their epistemological holism by considering ideas in Bayesian confirmation theory. (shrink)