The desire to minimize the number of individual new entities postulated is often referred to as quantitative parsimony. Its influence on the default hypotheses formulated by scientists seems undeniable. I argue that there is a wide class of cases for which the preference for quantitatively parsimonious hypotheses is demonstrably rational. The justification, in a nutshell, is that such hypotheses have greater explanatory power than less parsimonious alternatives. My analysis is restricted to a class of cases I shall refer to as (...) additive. Such cases involve the postulation of a collection of qualitatively identical individual objects which collectively explain some particular observed phenomenon. Especially clear examples of this sort occur in particle physics. 1 Introduction 2 Particle physics: a case study 3 Three kinds of simplicity 4 Explanatory power 5 Explanation and non-observation 6 Parsimony and scientific methodology 7 Conclusions. (shrink)

Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The Element's (...) first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively. (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)

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)

According to one popular nominalist picture, even when mathematics features indispensably in scientific explanations, this mathematics plays only a purely representational role: physical facts are represented, and these exclusively carry the explanatory load. I think that this view is mistaken, and that there are cases where mathematics itself plays an explanatory role. I distinguish two kinds of explanatory generality: scope generality and topic generality. Using the well-known periodical-cicada example, and also a new case study involving bicycle gears, I argue that (...) what is picked out by the mathematics are structural facts that go beyond any specific physical facts. (shrink)

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)

The desire to minimize the number of individual new entities postulated is often referred to as quantitative parsimony. Its influence on the default hypotheses formulated by scientists seems undeniable. I argue that there is a wide class of cases for which the preference for quantitatively parsimonious hypotheses is demonstrably rational. The justification, in a nutshell, is that such hypotheses have greater explanatory power than less parsimonious alternatives. My analysis is restricted to a class of cases I shall refer to as (...) additive. Such cases involve the postulation of a collection of qualitatively identical individual objects which collectively explain some particular observed phenomenon. Especially clear examples of this sort occur in particle physics.1Introduction2Particle physics: a case study3Three kinds of simplicity4Explanatory power5Explanation and non‐observation6Parsimony and scientific methodology7Conclusions. (shrink)

Indispensability-based arguments for mathematical platonism are typically motivated by drawing an analogy between abstract mathematical objects and concrete scientific posits. In this paper, I argue that mathematics can sometimes help to reduce our concrete ontological, ideological, and structural commitments. My focus is on optimization explanations, and in particular the case study involving periodical cicadas. I argue that in this case, stronger mathematical apparatus yields explanations that have fewer concrete commitments. The nominalist cannot accept these more parsimonious explanations without embracing the (...) stronger mathematics, and this poses a challenge for the nominalist position. (shrink)

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)

The rise of the field of “ experimental mathematics” poses an apparent challenge to traditional philosophical accounts of mathematics as an a priori, non-empirical endeavor. This paper surveys different attempts to characterize experimental mathematics. One suggestion is that experimental mathematics makes essential use of electronic computers. A second suggestion is that experimental mathematics involves support being gathered for an hypothesis which is inductive rather than deductive. Each of these options turns out to be inadequate, and instead a third suggestion is (...) considered according to which experimental mathematics involves calculating instances of some general hypothesis. The paper concludes with the examination of some philosophical implications of this characterization. (shrink)

One recent trend in the philosophy of mathematics has been to approach the central epistemological and metaphysical issues concerning mathematics from the perspective of the applications of mathematics to describing the world, especially within the context of empirical science. A second area of activity is where philosophy of mathematics intersects with foundational issues in mathematics, including debates over the choice of set-theoretic axioms, and over whether category theory, for example, may provide an alternative foundation for mathematics. My central claim is (...) that these latter issues are of direct relevance to philosophical arguments connected to the applicability of mathematics. In particular, the possibility of there being distinct alternative foundations for mathematics blocks the standard argument from the indispensable role of mathematics in science to the existence of specific mathematical objects. (shrink)

The aim of the paper is to introduce some of the history and key concepts of network science to a philosophical audience, and to highlight a crucial—and often problematic—presumption that underlies the network approach to complex systems. Network scientists often talk of “the structure” of a given complex system or phenomenon, which encourages the view that there is a unique and privileged structure inherent to the system, and that the aim of a network model is to delineate this structure. I (...) argue that this sort of naïve realism about structure is not a coherent or plausible position, especially given the multiplicity of types of entities and relations that can feature as nodes and links in complex networks. (shrink)

Defenders of the enhanced indispensability argument argue that the most effective route to platonism is via the explanatory role of mathematical posits in science. Various compelling cases of mathematical explanation in science have been proposed, but a satisfactory general philosophical account of such explanations is lacking. In this paper, I lay out the framework for such an account based on the notion of “the mathematical stance.” This is developed by analogy with Dennett’s well-known concept of “the intentional stance.” Roughly, adopting (...) the mathematical stance towards a particular physical phenomenon involves treating it as an abstract mathematical structure for the purposes of prediction and explanation. Interestingly, Dennett himself frequently draws analogies between his intentional stance towards beliefs and desires and scientists’ stance towards centers of gravity. I explore the theoretical role played by centers of gravity within science and discuss how an indispensabilist platonist ought to categorize the ontological status of this type of posit. I conclude with some thoughts on how an approach based on the mathematical stance might be developed into a more general philosophical account of the application of mathematics in science. (shrink)

Robin Donkin was an exceptional scholar in the field of historical geography, particularly concerning Latin America and the domestication of plants and animals globally. His early research was on the effect of the Cistercians on medieval landscape, and he held posts at the University of Edinburgh and the University of Brimingham. Donkin then lectured in Latin American geography at the University of Cambridge. He was a Fellow of Jesus College, Cambridge and was elected as a Fellow of the British Academy (...) in 1985. Obituary by Alan R. H. Baker FBA. (shrink)

In this paper I present a case study of mathematical explanation in science that is new to the philosophical literature, and that arises in the context of estimating the energetic costs of running in bipedal animals. I refer to this as the Bipedal Gait Costs explanation. I argue that it is important for examples of applied mathematics to be driven not just by philosophical and mathematical concerns but also by scientific concerns. After a detailed presentation of the BGC case study, (...) I discuss ways in which it is different from standard examples of MES. One distinctive aspect of BGC is that substantive mathematics occurs at two different levels. I argue for a distinction to be drawn between the mathematical modeling that occurs at one level, which is non-explanatory, and the mathematical theorem which is applied to results gleaned from these models, which is explanatory. I conclude by drawing some connections with broader indispensability-based arguments for platonism in the philosophy of mathematics. (shrink)

The core thesis of Malebranche’s doctrine of occasionalism is that God is the sole true cause, where a true cause is one that has the power to initiate change and for which the mind perceives a necessary connection between it and its effects. Malebranche gives two separate arguments for his core thesis, T, based on necessary connection and on divine power respectively. The standard view is that these two arguments are necessary to establish T. I argue for a reinterpretation of (...) Malebranche’s strategy, according to which the Necessary Connection Argument alone is sufficient to establish T. The Divine Power Argument, which is anyway weaker, is needed not to support T but to bridge the gap between T and full-fledged occasionalism. Specifically, it is needed to rule out the existence of causal powers in nature, a scenario which is consistent with T but inconsistent with occasionalism. (shrink)

The core thesis of Malebranche’s doctrine of occasionalism is that God is the sole true cause, where a true cause is one that has the power to initiate change and for which the mind perceives a necessary connection between it and its effects. Malebranche gives two separate arguments for his core thesis, T, based on necessary connection and on divine power respectively. The standard view is that these two arguments are necessary to establish T. I argue for a reinterpretation of (...) Malebranche’s strategy, according to which the Necessary Connection Argument alone is sufficient to establish T. The Divine Power Argument, which is anyway weaker, is needed not to support T but to bridge the gap between T and full-fledged occasionalism. Specifically, it is needed to rule out the existence of causal powers in nature, a scenario which is consistent with T but inconsistent with occasionalism. (shrink)

According to Malebranche’s occasionalism, all cases of causation in the world are due to the action of God’s will. These actions are divided into "particular volitions" and "general volitions". There has been sharp disagreement in the secondary literature concerning the nature of general volitions, for Malebranche. One side claims that general volitions are volitions of wide scope which cover a multiplicity of potential situations. The other side claims that general volitions are specific but fall under the scope of broader laws (...) of nature. I draw on the quantifier apparatus of modern predicate logic to highlight the differences between the two views, which I refer to as the Universality view and the Regularity view respectively. I also outline a third ‘Hybrid’ view that borrows elements of each. I conclude that there is a fundamental ambiguity in Malebranche’s account of general volitions which prevents any one account from fully fitting with Malebranche’s remarks. (shrink)

In a 2005 paper, John Burgess and Gideon Rosen offer a new argument against nominalism in the philosophy of mathematics. The argument proceeds from the thesis that mathematics is part of science, and that core existence theorems in mathematics are both accepted by mathematicians and acceptable by mathematical standards. David Liggins (2007) criticizes the argument on the grounds that no adequate interpretation of “acceptable by mathematical standards” can be given which preserves the soundness of the overall argument. In this discussion (...) I offer a defense of the Burgess-Rosen argument against Liggins’s objection. I show how plausible versions of the argument can be constructed based on either of two interpretations of mathematical acceptability, and I locate the argument in the space of contemporary anti-nominalist views. (shrink)

In their paper, “Vexing Expectations,” Nover and Hájek (2004) present an allegedly paradoxical betting scenario which they call the Pasadena Game (PG). They argue that the silence of standard decision theory concerning the value of playing PG poses a serious problem. This paper provides a threefold response. First, I argue that the real problem is not that decision theory is “silent” concerning PG, but that it delivers multiple conflicting verdicts. Second, I offer a diagnosis of the problem based on the (...) insight that standard decision theory is, rightly, sensitive to order. Third, I describe a new betting scenario—the Alternating St. Petersburg Game—which is genuinely paradoxical. Standard decision theory is silent on the value of playing this game even if restrictions are placed on the order in which the various alternative payoffs are summed. †To contact the author, please write to: Department of Philosophy, Swarthmore College, Swarthmore, PA 19081; e-mail: [email protected]. (shrink)

Book Information The Foundations of Mathematics in the Theory of Sets. The Foundations of Mathematics in the Theory of Sets J. P. Mayberry Cambridge Cambridge University Press 2000 xx + 424 Hardback US$80.00 By J. P. Mayberry. Cambridge University Press. Cambridge. Pp. xx + 424. Hardback:US$80.00.

Late in 1990, the Center for the Study of Ethics in the Professions at Illinois Institute of Technology (lIT) received a grant of more than $200,000 from the National Science Foundation to try a campus-wide approach to integrating professional ethics into its technical curriculum.! Enough has now been accomplished to draw some tentative conclusions. I am the grant's principal investigator. In this paper, I shall describe what we at lIT did, what we learned, and what others, especially philosophers, can learn (...) from us. We set out to develop an approach that others could profitably adopt. I believe that we succeeded. (shrink)

The contemporary operating environment for deployed United States military operations largely focuses on deployments to predominantly Islamic countries. The differences in cultural values between d...

The thesis explores and suggests a solution to a problem that I identify in John McDowell’s and Lynne Rudder Baker’s approaches to mental and intention-dependent (ID) causation in the physical world. I begin (chapter 1) with a brief discussion of McDowell’s non-reductive and anti-scientistic account of mind and world, which I believe offers, through its vision of the unbounded conceptual and the world as within the space of reasons, to liberate and renew philosophy. However, I find an inconsistency in (...) McDowell’s criticism of Davidson’s anomalous monism (chapter 2), stemming from a tension between McDowell’s naïve common sense view of mental causation and an understanding of ordinary physical causation which I think McDowell ought to embrace, which portrays it as both objective, in the sense of being recognition- independent, and as belonging within the space of reasons. The question of the relation between these two concepts of causation is an aspect of the more general question of the relation between the space of reasons and the realm of law. In chapter 3 I begin examining the possibility that Baker’s account of material and property constitution could form the basis of a bridge between the two spaces, and find it generally promising. However, I find that her defence of her version of non-reductive monism against Kim’s causal arguments also runs into problems, which I attribute to the fact that she holds a view of causation as secondary to causal explanation. 2 3 In chapter 4 I develop an account of what I call manifest physical causation – of objective causal relations in the world of Sellars’s manifest image. Based upon McDowell’s transcendental empiricism, which takes the existence of the ordinary perceived world as a condition of the possibility of our possession of conceptual capacities, I contend that it is this picture of reality, rather than that of fundamental physics, that should be the starting point of our theorizing. Causation in the manifest image, I argue, covers the behaviour of the familiar physical world as well as that of its well-understood extensions into the special sciences and engineering. Manifest physical causation, on my account, is productive, acts through mechanisms which are almost entirely mechanical, electromagnetic, and/ or chemical, and is causally closed. In my view, normative, semantic, contentful property-instances are not part of the manifest physical causal nexus. In my final chapter I suggest a modification of Baker’s constitution account, which I call Constituted Causation, whereby higher-level – mental and other ID – causal relations are constituted, in favourable circumstances, by lower-level ones. ID causal relations belong in their own causal nexus but are connected to the manifest physical world through constitution, a relation of unity without identity. Causation and constitution are, respectively, intra- and inter-level relations, and they are non-overlapping. The constituted network of rational and normative relations bears, I believe, striking parallels with McDowell’s view of reality. According to Baker’s view of constitution, the essential properties of constituted entities subsume those of their constituters; extending this to my account enables us to say that the real cause and explanation of someone’s action is that they consciously performed it 4 rather than that certain manifest causal processes occurred at the lower level. (shrink)

Bioethicists function in an environment in which their peers - healthcare executives, lawyers, nurses, physicians - assert the integrity of their fields through codes of professional ethics. Is it time for bioethics to assert its integrity by developing a code of ethics? Answering in the affirmative, this paper lays out a case by reviewing the historical nature and function of professional codes of ethics. Arguing that professional codes are aggregative enterprises growing in response to a field's historical experiences, it asserts (...) that bioethics now needs to assert its integrity and independence and has already developed a body of formal statements that could be aggregated to create a comprehensive code of ethics for bioethics. A Draft Model Aggregated Code of Ethics for Bioethicists is offered in the hope that analysis and criticism of this draft code will promote further discussion of the nature and content of a code of ethics for bioethicists. (shrink)

This paper has two related goals. First, we develop an expressivist account of negation which, in the spirit of Alan Gibbard, treats disagreement as semantically primitive. Our second goal is to make progress toward a unified expressivist treatment of modality. Metaethical expressivists must be expressivists about deontic modal claims. But then metaethical expressivists must either extend their expressivism to include epistemic and alethic modals, or else accept a semantics for modal expressions that is radically disjunctive. We propose that expressivists (...) look to Amie Thomasson’s work for a general strategy for offering a unified expressivist account of modality. Modals in general, we propose, are devices for expressing metalinguistic commitments within the object language, with deontic, epistemic, and metaphysical modals all expressing different kinds of metalinguistic commitments. (shrink)