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)
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)
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)
In this paper I examine a strategy which aims to bypass the technicalities of the indispensability debate and to offer a direct route to nominalism. The starting-point for this alternative nominalist strategy is the claim that--according to the platonist picture--the existence of mathematical objects makes no difference to the concrete, physical world. My principal goal is to show that the 'Makes No Difference' (MND) Argument does not succeed in undermining platonism. The basic reason why not is that the makes-no-difference claim (...) which the argument is based on is problematic. Arguments both for and against this claim can be found in the literature; I examine three such arguments, uncovering flaws in each one. In the second half of the paper, I take a more direct approach and present an analysis of the counterfactual which underpins the makes-no-difference claim. What this analysis reveals is that indispensability considerations are in fact crucial to the proper evaluation of the MND Argument, contrary to the claims of its supporters. (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)
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)
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 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)
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)
Science and mathematics: the scope and limits of mathematical fictionalism Content Type Journal Article Category Book Symposium Pages 1-26 DOI 10.1007/s11016-011-9640-3 Authors Christopher Pincock, University of Missouri, 438 Strickland Hall, Columbia, MO 65211-4160, USA Alan Baker, Department of Philosophy, Swarthmore College, Swarthmore, PA 19081, USA Alexander Paseau, Wadham College, Oxford, OX1 3PN UK Mary Leng, Department of Philosophy, University of York, Heslington, York, YO10 5DD UK Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
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)
This book outlines the realist and pluralist philosophy of John Anderson, Australia's most original thinker. His teaching at Sydney University and his arti6es have deeply influenced Australian intellectual life. Several main themes run through his work, but Anderson never gave an overall account of his views. This is remedied here: exhibiting the range of Anderson's thought from logic, epistemology and theory of mind, to language and social theory, this volume sketches realism as a systematic philosophical position, while showing something of (...) the history of ideas in Australia. (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)
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)
We agree with Jones & Love (J&L) that much of Bayesian modeling has taken a fundamentalist approach to cognition; but we do not believe in the potential of Bayesianism to provide insights into psychological processes. We discuss the advantages of associative explanations over Bayesian approaches to causal induction, and argue that Bayesian models have added little to our understanding of human causal reasoning.
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: firstname.lastname@example.org. (shrink)
This paper discusses the “blended identity” of online rock fans to show that the standard dichotomy between anonymous and real life personas is an inadequate description of self-presentation in online communities. Using data from an ethnographic, exploratory study of an online community and comparison groups including interviews, an online questionnaire, fan discussion boards, and participant/observation, the research analyzes fan identity online and then offline. Rolling Stones fans often adopt names that illustrate their allegiance to the band, along with avatars. Issues (...) of gender and the technological change of software platform also affect types of online self-presentations and their construction. Fans engage in “role embracement”, merging their individual selves with the role of Stones fans, demonstrated by reactions of friends and family. Connections between offline and online settings occur, with band affiliation of fans expressed through choice of apparel offline, and usernames from online filtering into the offline interactions among fans. (shrink)
Counter-terrorism officials in the USA and the UK responded to the events of 11 September 2001 and 7 July 2005 with an increasing resort to the use of ?intelligence-led policing? methods such as racial and religious profiling. Reliance on intelligence, to the effect that most people who commit a certain crime have a certain ethnicity, can lead to less favourable treatment of an individual with that ethnicity because of his membership in that group, not because of any act he is (...) suspected or known to have committed. This paper explains the context in which intelligence-led policing flourishes, and how this discussion contributes to the profiling debate in both the USA and the UK, and then sets out two key contentions. First, we argue that Article 14 ECHR as applied under the UK Human Rights Act has a more protective, and less ?prosecutorial?, conception of discrimination than has the US Equal Protection Clause, meaning that judges need not find a discriminatory motive to find that discrimination has occurred. Second, we contend that Article 14 provides the judiciary with the key tool of proportionality, which, when properly applied, makes it harder for discrimination to stand up to scrutiny. (shrink)
This case study extends the findings of Pickering et al. 2009 to the domain of conversational humor. We find that, as was the case in humorous narratives, conversational humor is not marked by higher pitch or volume, increased speech rate, or significant pauses. Unlike narrative humor, conversational humor is not produced at a lower pitch and slower rate than non-humorous parts of the text. We find that smiling and laughter tend to occur with humor.
Mitchell et al.'s claim, that their propositional theory is a single-process theory, is illusory because they relegate some learning to a secondary memory process. This renders the single-process theory untestable. The propositional account is not a process theory of learning, but rather, a heuristic that has led to interesting research.