LOT 2: The Language of Thought Revisited

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can do to make an explanation mathematical; realists think something can be done, but they are not clear about what that something is.

I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism (in this journal). I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist.     

Why inference to the best explanation doesn’t secure empirical grounds for mathematical platonism

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I object that this inference depends on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist.

     

Importance and Explanatory Relevance: The Case of Mathematical Explanations

A way to argue that something (e.g. mathematics, idealizations, moral properties, etc.) plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part (one that if removed affects the explanation) is an explanatorily relevant part. This can be very useful in many ontological debates. My aim in this paper is twofold. First of all, I will try to assess how this view on explanatory relevance can affect the recent ontological debate in the philosophy of mathematics—as I will argue, contrary to how it may appear at first glance, it does not help very much the mathematical realists. Second of all, I will show that there are big problems with it.     

The shape of things to come: Psychoneural reduction and the future of psychology

I contrast Bickle's new wave reductionismwith other relevant views about explanation across intertheoretic contexts. I then assess Bickle's empirical argument for psychoneural reduction. Bickle shows that psychology is not autonomous from neuroscience, and concludes that at least some versions of nonreductive physicalism are false. I argue this is not sufficient to establish his further claim that psychology reduces to neuroscience. Examination of Bickle's explanations reveals that they do not meet his own reductive standard. Furthermore, there are good empirical reasons to doubt that the cognitive approach to mind should be abandoned. I suggest that the near future will not see a reduction of psychology to neuroscience, so much as a replacement of both sciences by an improved form of neuropsychology.     

Evidence,explanation and enhanced indispensability

In this paper I shall adopt a possible reading of the notions of ‘explanatory indispensability’ and ‘genuine mathematical explanation in science’ on which the Enhanced Indispensability Argument (EIA) proposed by Alan Baker is based. Furthermore, I shall propose two examples of mathematical explanation in science and I shall show that, whether the EIA-partisans accept the reading I suggest, they are easily caught in a dilemma. To escape this dilemma they need to adopt some account of explanation and offer a plausible answer to the following ‘question of evidence’: What is a genuine mathematical explanation in empirical science and on what basis do we consider it as such? Finally, I shall suggest how a possible answer to the question of evidence might be given through a specific account of mathematical explanation in science. Nevertheless, the price of adopting this standpoint is that the genuineness of mathematical explanations of scientific facts turns out to be dependent on pragmatic constraints and therefore cannot be plugged in EIA and used to establish existential claims about mathematical objects.     

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.     

Hypothetical identities: Explanatory problems for the explanatory argument

Recently, several philosophers have defended an explanatory argument that supposedly provides novel empirical grounds for accepting the type identity theory of phenomenal consciousness. They claim that we are justified in believing that the type identity thesis is true because it provides the best explanation for the correlations between physical properties and phenomenal properties. In this paper, I examine the actual role identities play in science and point out crucial shortcomings in the explanatory argument. I show that the supporters of the argument have failed to show that the identity thesis provides a satisfactory explanation for the correlations between physical and phenomenal properties. Hence, the explanatory argument, as it stands, does not provide new grounds for accepting the type identity theory.     

Proving quadratic reciprocity: explanation,disagreement, transparency and depth

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late eighteenth century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.

     

What inductive explanations could not be

Marc Lange argues that proofs by mathematical induction are generally not explanatory because inductive explanation is irreparably circular. He supports this circularity claim by presenting two putative inductive explanantia that are one another’s explananda. On pain of circularity, at most one of this pair may be a true explanation. But because there are no relevant differences between the two explanantia on offer, neither has the explanatory high ground. Thus, neither is an explanation. I argue that there is no important asymmetry between the two cases because they are two presentations of the same explanation. The circularity argument requires a problematic notion of identity of proofs. I argue for a criterion of proof individuation that identifies the two proofs Lange offers. This criterion can be expressed in two equivalent ways: one uses the language of homotopy type theory, and the second assigns algebraic representatives to proofs. Though I will concentrate on one example, a criterion of proof identity has much broader consequences: any investigation into mathematical practice must make use of some proof-individuation principle.     

Explicating pluralism: Where the mind to molecule pathway gets off the track—Reply to Bickle

It is argued that John Bickle’s Ruthless Reductionism is flawed as an account of the practice of neuroscience. Examples from genetics and linguistics suggest, first, that not every mind-brain link or gene-phenotype link qualifies as a reduction or as a complete explanation, and, second, that the higher (psychological) level of analysis is not likely to disappear as neuroscience progresses. The most plausible picture of the evolving sciences of the mind-brain seems a patchwork of multiple connections and partial explanations, linking anatomy, mechanisms and functions across different domains, levels, and grain sizes. Bickle’s claim that only the molecular level provides genuine explanations, and higher level concepts are just heuristics that will soon be redundant, is thus rejected. In addition, it is argued that Bickle’s recasting of philosophy of science as metascience explicating empirical practices, ignores an essential role for philosophy in reflecting upon criteria for reduction and explanation. Many interesting and complex issues remain to be investigated for the philosophy of science, and in particular the nature of interlevel links found in empirical research requires sophisticated philosophical analysis.     

On the Question 'Do Numbers Exist?'

Since we know that there are four prime numbers less than 8 we know that there are numbers. This 'short argument' is correct but it is not an ontological claim or part of philosophy of mathematics. Both realists (Quine) and nominalists (Field) reject the short argument and adopt the idea that the existence of numbers might be posited to explain known mathematical truths. Philosophers operate with a negative conception of what numbers are: they are not in space and time, not related causally to us, not perceivable, etc. This preliminary outlook does not actually characterize a kind of existing thing at all. It creates the atmosphere of weirdness characteristic of both fictionalism and Platonism. Positing things for the sake of explanation makes sense in empirical contexts, but the intelligibility of positing cannot not survive the move to philosophy of mathematics. Modal realism is a model for the unsatisfactory thinking that generates ontological commitment in mathematics.     

The Born-Einstein debate: Where application and explanation separate

Application in science has its own structure, distinct from the structure of theoretical science, and therefore needs its own philosophy. The covering power of a formal scientific theory is no guide to its explanatory power. Explanation is too much to ask of a fundamental scientific theory. This is seen by considering two strands of the Born-Einstein debate: first the explanatory power of quantum mechanics and second, the reality of unobserved properties. The function of theoretical physics is to describe rather than to explain. Some techniques are a standard part of theory; while some aread hoc to the problems at hand. Very few of the derivations in mathematical physics are explanatory. This shows distinctly separate structures for theory and for application.This paper was written while I was at the Center for Interdisciplinary Research in Bielefeld, West Germany, and I would like to thank the Center, and Lorenz Kruger, and others who made my visit there possible. I would also like to thank Lorenz Kruger and Norton Wise for their philosophical help.     

Reduction: the Cheshire cat problem and a return to roots

In this paper, I propose two theses, and then examine what the consequences of those theses are for discussions of reduction and emergence. The first thesis is that what have traditionally been seen as robust, reductions of one theory or one branch of science by another more fundamental one are a largely a myth. Although there are such reductions in the physical sciences, they are quite rare, and depend on special requirements. In the biological sciences, these prima facie sweeping reductions fade away, like the body of the famous Cheshire cat, leaving only a smile. ... The second thesis is that the “smiles” are fragmentary patchy explanations, and though patchy and fragmentary, they are very important, potentially Nobel-prize winning advances. To get the best grasp of these “smiles,” I want to argue that, we need to return to the roots of discussions and analyses of scientific explanation more generally, and not focus mainly on reduction models, though three conditions based on earlier reduction models are retained in the present analysis. I briefly review the scientific explanation literature as it relates to reduction, and then offer my account of explanation. The account of scientific explanation I present is one I have discussed before, but in this paper I try to simplify it, and characterize it as involving field elements (FE) and a preferred causal model system (PCMS) abbreviated as FE and PCMS. In an important sense, this FE and PCMS analysis locates an “explanation” in a typical scientific research article. This FE and PCMS account is illustrated using a recent set of neurogenetic papers on two kinds of worm foraging behaviors: solitary and social feeding. One of the preferred model systems from a 2002 Nature article in this set is used to exemplify the FE and PCMS analysis, which is shown to have both reductive and nonreductive aspects. The paper closes with a brief discussion of how this FE and PCMS approach differs from and is congruent with Bickle’s “ruthless reductionism” and the recently revived mechanistic philosophy of science of Machamer, Darden, and Craver.     

How Mathematical Concepts Get Their Bodies

When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised intuitions are further conceptualised while poorly intuited concepts are further intuited. In this paper I study this latter process in mathematics during the twentieth century and, more specifically, show the role of set theory and category theory in this process. I use this material for defending the following claims: (1) mathematical intuitions are subject to historical development just like mathematical concepts; (2) mathematical intuitions continue to play their traditional role in today's mathematics and will plausibly do so in the foreseeable future. This second claim implies that the popular view, according to which modern mathematical concepts, unlike their more traditional predecessors, cannot be directly intuited, is not justified.     

Deterministic Chaos and Computational Complexity: the Case of Methodological Complexity Reductions

Some problems rarely discussed in traditional philosophy of science are mentioned: The empirical sciences using mathematico-quantitative theoretical models are frequently confronted with several types of computational problems posing primarily methodological limitations on explanatory and prognostic matters. Such limitations may arise from the appearances of deterministic chaos and (too) high computational complexity in general. In many cases, however, scientists circumvent such limitations by utilizing reductional approximations or complexity reductions for intractable problem formulations, thus constructing new models which are computationally tractable. Such activities are compared with reduction types (more) established in philosophy of science. This revised version was published online in August 2006 with corrections to the Cover Date.     

Two approaches to natural kinds

Philosophical treatments of natural kinds are embedded in two distinct projects. I call these the philosophy of science approach and the philosophy of language approach. Each is characterized by its own set of philosophical questions, concerns, and assumptions. The kinds studied in the philosophy of science approach are projectible categories that can ground inductive inferences and scientific explanation. The kinds studied in the philosophy of language approach are the referential objects of a special linguistic category—natural kind terms—thought to refer directly. Philosophers may hope for a unified account addresses both sets of concerns. This paper argues that this cannot be done successfully. No single account can satisfy both the semantic objectives of the philosophy of language approach and the explanatory projects of the philosophy of science approach. After analyzing where the tensions arise, I make recommendations about assumptions and projects that are best abandoned, those that should be retained, and those that should go their separate ways. I also recommend adopting the disambiguating terminology of “scientific kinds” and “natural kinds” for the different notions of kinds developed in these different approaches.

     

On the distinction between Peirce’s abduction and Lipton’s Inference to the best explanation

I argue against the tendency in the philosophy of science literature to link abduction to the inference to the best explanation (IBE), and in particular, to claim that Peircean abduction is a conceptual predecessor to IBE. This is not to discount either abduction or IBE. Rather the purpose of this paper is to clarify the relation between Peircean abduction and IBE in accounting for ampliative inference in science. This paper aims at a proper classification—not justification—of types of scientific reasoning. In particular, I claim that Peircean abduction is an in-depth account of the process of generating explanatory hypotheses, while IBE, at least in Peter Lipton’s thorough treatment, is a more encompassing account of the processes both of generating and of evaluating scientific hypotheses. There is then a two-fold problem with the claim that abduction is IBE. On the one hand, it conflates abduction and induction, which are two distinct forms of logical inference, with two distinct aims, as shown by Charles S. Peirce; on the other hand it lacks a clear sense of the full scope of IBE as an account of scientific inference.     

Defending the Indispensability Argument: Atoms, Infinity and the Continuum

This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that these new versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science.     

Against Mathematical Explanation

Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “explanation,” the term is not applicable to mathematics as it is in science.