Alan Baker

In my 2007 paper, “Is There a Problem of Induction for Mathematics?” I rejected the idea that enumerative induction has force for mathematical claims. My core argument was based on the fact that we are restricted to examining relatively small numbers, so our samples are always biased, and hence they carry no inductive weight. In recent years, I have come to believe that this argument is flawed. In particular, while arithmetical samples are indeed biased, my new view is that this bias actually strengthens the inductive support that accrues from them. The reason is that small numbers typically provide a more severe test of general arithmetical claims due to the greater frequency of significant properties and boundary cases among such numbers. In this paper, I describe and defend this new view, which a call Positive Bias Pro-Inductivism.

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.

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].

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.