Alan Richard Baker

Mathematical Explanation in Science

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. 1. Introduction: Mathematical Explanation2. Indispensability and Explanation3. Is the Mathematics Indispensable to the Explanation? 3.1. Object-level arbitrariness3.2. Concept-level arbitrariness3.3. Theory-level arbitrariness4. Is the Explanandum ‘Purely Physical’?5. Is the Mathematics Explanatory in Its Own Right?6. Does Inference to the Best Explanation Apply to Mathematics? 6.1. Lengs first argument6.2. Lengs second argument6.3. Lengs third argument7. Conclusions Introduction: Mathematical Explanation Indispensability and Explanation Is the Mathematics Indispensable to the Explanation? 3.1. Object-level arbitrariness3.2. Concept-level arbitrariness3.3. Theory-level arbitrariness Object-level arbitrariness Concept-level arbitrariness Theory-level arbitrariness Is the Explanandum ‘Purely Physical’? Is the Mathematics Explanatory in Its Own Right? Does Inference to the Best Explanation Apply to Mathematics? 6.1. Lengs first argument6.2. Lengs second argument6.3. Lengs third argument Lengs first argument Lengs second argument Lengs third argument Conclusions

Quantitative Parsimony and Explanatory Power

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. 1Introduction 2Particle physics: a case study 3Three kinds of simplicity 4Explanatory power 5Explanation and non‐observation 6Parsimony and scientific methodology 7Conclusions

The Indispensability Argument and Multiple Foundations for Mathematics

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.

Mathematical Accidents and the End of Explanation

A conspicuous difference between “traditional” philosophy of science and “traditional” philosophy of mathematics concerns the relative importance of the notion of explanation. Explanation has long featured centrally in debates in the philosophy of science, for at least two reasons. Firstly, explanation has been viewed as playing an important role in the methodology of science, principally due to the inductive character of scientific method. This has led to a focus on giving a philosophical model of scientific explanation, whose leading candidates have included Hempel’s deductive-nomological model, the causal model promoted by Lewis, van Fraassen’s pragmatic model, and the unification models of Kitcher and Friedman. Secondly, explanatory considerations have been an important feature of philosophical debates over scientific realism and anti-realism. This has led to a focus on inference to the best explanation and the conditions under which this mode of inference can underpin robust ontological conclusions.

Simulation-Based Definitions Of Emergence

One approach to characterizing the elusive notion of emergence is to define that a property is emergent if and only if its presence can be derived but only by simulation. In this paper I investigate the pros and cons of this approach, focusing in particular on whether an appropriately distinct boundary can be drawn between simulation-based and non-simulation-based methods. I also examine the implications of this definition for the epistemological role of emergent properties in prediction and in explanation.

Mathematical Explanation in Biology

Biology has proved to be a rich source of examples in which mathematics plays a role in explaining some physical phenomena. In this paper, two examples from evolutionary biology, one involving periodical cicadas and one involving bee honeycomb, are examined in detail. I discuss the use of such examples to defend platonism about mathematical objects, and then go on to distinguish several different varieties of mathematical explanation in biology. I also connect these discussions to issues concerning generality in biological explanation, and to the question of how to pick out which mathematical properties are explanatorily relevant.

Parsimony and inference to the best mathematical explanation

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.

Putting Expectations in Order

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.

Explaining the Applicability of Mathematics in Science

Abstract Philosophical discussion of applied mathematics has tended to focus on a cluster of related ‘problems of applicability’ that are often conflated. In this paper, I argue that several of these problems raise no significant philosophical issues. I then identify two particular aspects of the involvement of mathematics in science: that mathematics is indispensable for science, and that mathematics sometimes plays an explanatory role in science. I argue that there are genuine questions surrounding these two features of applied mathematics, and in particular that the issue of the role of mathematics in scientific explanation has ramifications for the debate between realist and anti-realist views in the philosophy of mathematics.

No Reservations Required? Defending Anti-Nominalism

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.