Is Mathematics a Domain for Philosophers of Explanation?

In this paper we discuss three interrelated questions. First: is explanation in mathematics a topic that philosophers of mathematics can legitimately investigate? Second: are the specific aims that philosophers of mathematical explanation set themselves legitimate? Finally: are the models of explanation developed by philosophers of science useful tools for philosophers of mathematical explanation? We argue that the answer to all these questions is positive. Our views are completely opposite to the views that Mark Zelcer has put forward recently. Throughout this paper, we show why Zelcer’s arguments fail.     

Morality is Not Like Mathematics: The Weakness of the Math‐Moral Analogy

In both the early modern period and in contemporary debates, philosophers have argued that there are analogies between mathematics and morality that imply that the ontology and epistemology of morality are crucially similar to the ontology and epistemology of mathematics. I describe arguments for the math‐moral analogy in four early modern philosophers (Locke, Cudworth, Clarke, and Balguy) and in three contemporary philosophers (Clarke‐Doane, Peacocke, and Roberts). I argue that these arguments fail to establish important ontological and epistemological similarities between morality and mathematics. There are analogies between the two areas, but the disanalogies are more significant, undermining the attempt to confer on morality the same ontological and epistemological status that mathematics possesses.     

Philosophy of mathematics: Prospects for the 1990s

For some time now, academic philosophers of mathematics have concentrated on intramural debates, the most conspicuous of which has centered on Benacerraf's epistemological challenge. By the late 1980s, something of a consensus had developed on how best to respond to this challenge. But answering Benacerraf leaves untouched the more advanced epistemological question of how the axioms are justified, a question that bears on actual practice in the foundations of set theory. I suggest that the time is ripe for philosophers of mathematics to turn outward, to take on a problem of real importance for mathematics itself.     

The Necessity of Mathematics

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.     

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.     

On three arguments against categorical structuralism

Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are dispensable. The goal of this paper is to show that these criticisms are misguided by arguing that category theory is entirely autonomous from set theory.     

Scientific Explanation and Moral Explanation*

Moral philosophers are, among other things, in the business of constructing moral theories. And moral theories are, among other things, supposed to explain moral phenomena. Consequently, one's views about the nature of moral explanation will influence the kinds of moral theories one is willing to countenance. Many moral philosophers are (explicitly or implicitly) committed to a deductive model of explanation. As I see it, this commitment lies at the heart of the current debate between moral particularists and moral generalists. In this paper I argue that we have good reasons to give up this commitment. In fact, I show that an examination of the literature on scientific explanation reveals that we are used to, and comfortable with, non‐deductive explanations in almost all areas of inquiry. As a result, I argue that we have reason to believe that moral explanations need not be grounded in exceptionless moral principles.     

The Implicit Definition of the Set-Concept

Once Hilbert asserted that the axioms of a theory `define` theprimitive concepts of its language `implicitly'. Thus whensomeone inquires about the meaning of the set-concept, thestandard response reads that axiomatic set-theory defines itimplicitly and that is the end of it. But can we explainthis assertion in a manner that meets minimum standards ofphilosophical scrutiny? Is Jané (2001) wrong when hesays that implicit definability is ``an obscure notion''? Doesan explanation of it presuppose any particular view on meaning?Is it not a scandal of the philosophy of mathematics that no answersto these questions are around? We submit affirmative answers to allquestions. We argue that a Wittgensteinian conception of meaninglooms large beneath Hilbert's conception of implicit definability.Within the specific framework of Horwich's recent Wittgensteiniantheory of meaning called semantic deflationism, we explain anexplicit conception of implicit definability, and then go on toargue that, indeed, set-theory, defines the set-conceptimplicitly according to this conception. We also defend Horwich'sconception against a recent objection from the Neo-Fregeans Hale and Wright (2001). Further, we employ the philosophicalresources gathered to dissolve all traditional worries about thecoherence of the set-concept, raisedby Frege, Russell and Max Black, and whichrecently have been defended vigorously by Hallett (1984) in hismagisterial monograph Cantorian set-theory and limitationof size. Until this day, scandalously, these worries havebeen ignored too by philosophers of mathematics.     

What's Wrong With Indispensability?

For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is indispensable in the wrong way), and, taking my cue from Mark Colyvan's (1998) attempt to provide a Quinean account of unapplied mathematics as `recreational', suggest that, if one approaches the problem from a Quinean naturalist starting point, one must conclude that all mathematics is recreational in this way.     

Arithmetic,set theory,reduction and explanation

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important and well-known case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer some evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate and successful foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.     

The pernicious influence of mathematics upon philosophy

We shall argue that the attempt carried out by certain philosophers in this century to parrot the language, the method, and the results of mathematics has harmed philosophy. Such an attempt results from a misunderstanding of both mathematics and philosophy, and has harmed both subjects.Portions of the present text have previously appeared inThe Review of Metaphysics 44 (1990), 259–271, are reprinted with permission.     

Ernst Cassirer's Neo-Kantian Philosophy of Geometry

One of the most important philosophical topics in the early twentieth century – and a topic that was seminal in the emergence of analytic philosophy – was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant.     

Absence perception and the philosophy of zero

Synthese - Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of...     

Causal Relations and Explanatory Strategies in Physics

Many philosophers now regard causal approaches to explanation as highly promising, even in physics. This is due in large part to James Woodward's influential argument that a wide variety of scientific explanations are causal, based on his interventionist approach to causation. This article argues that some derivations describing causal relations and satisfying Woodward's criteria for causal explanation fail to be explanatory. Further, causal relations are unnecessary for a range of explanations, widespread in physics, involving highly idealized models. These constitute significant limitations on the scope of causal explanation. We have good reason to doubt that causal explanation is as widespread or important in physics as Woodward and other proponents maintain.     

Hume’s Solution of the Goodman Paradox and the Reliability Riddle (Mill’s problem)

Many solutions of the Goodman paradox have been proposed but so far no agreement has been reached about which is the correct solution. However, I will not contribute here to the discussion with a new solution. Rather, I will argue that a solution has been in front of us for more than two hundred years because a careful reading of Hume’s account of inductive inferences shows that, contrary to Goodman’s opinion, it embodies a correct solution of the paradox. Moreover, the account even includes a correct answer to Mill’s question of why in some cases a single instance is sufficient for a complete induction, since Hume gives a well-supported explanation of this reliability phenomenon. The discussion also suggests that Bayesian theory by itself cannot explain this phenomenon. Finally, we will see that Hume’s explanation of the reliability phenomenon is surprisingly similar to the explanation given lately by a number of naturalistic philosophers in their discussion of the Goodman paradox.     

Philosophy Makes No Progress,So What Is the Point of It?

Philosophy makes no progress. It fails to do so in the way science and mathematics make progress. By “no progress” is meant that there is no successive advance of a well‐established body of knowledge—no views are definitively established or definitively refuted. Yet philosophers often talk and act as if the subject makes progress, and that its point and value lies in its doing so, while in fact they also approach the subject in ways that clearly contradict any claim to progress. This article presents evidence for, and a theoretical explanation of, the view that philosophy makes no progress, concluding with an account of what philosophy is and what the point and value of it is. Philosophy should not be shy about being what it is, nor should it pretend to be what it is not. What it is should be reflected in philosophizing and the way it is taught.     

Theories of apparent motion

Apparent motion is an illusion in which two sequentially presented and spatially separated stimuli give rise to the experience of one moving stimulus. This phenomenon has been deployed in various philosophical arguments for and against various theories of consciousness, time consciousness and the ontology of time. Nevertheless, philosophers have continued working within a framework that does not reflect the current understanding of apparent motion. The main objectives of this paper are to expose the shortcomings of the explanations provided for apparent motion and to offer an alternative explanation for the phenomenon.     

Essentialist explanation

Recent years have seen an explosion of interest in metaphysical explanation, and philosophers have fixed on the notion of ground as the conceptual tool with which such explanation should be investigated. I will argue that this focus on ground is myopic and that some metaphysical explanations that involve the essences of things cannot be understood in terms of ground. Such ‘essentialist’ explanation is of interest, not only for its ubiquity in philosophy, but for its being in a sense an ultimate form of explanation. I give an account of the sense in which such explanation is ultimate and support it by defending what I call the inessentiality of essence. I close by suggesting that this principle is the key to understanding why essentialist explanations can seem so satisfying.     

Self-Deception Needs No Explaining

How can one deceive oneself if at the same time one knows the truth? The idea of such a thing has puzzled philosophers, and many philosophical efforts have been devoted to explaining the puzzle. Yet all such attempts have been misplaced. For in fact there is nothing distinctive about the way the mind works in self-deception, nothing that needs special explaining. The perception of a puzzle arises from certain mistaken assumptions about how the mind works generally. Once this is explained, we see that the way the mind works in self-deception embodies no deviation from the norm. The aura of paradox then disappears, and we see that self-deception requires no special explanation of its own.     

Kant's Theory of Judgment,and Judgments of Taste: On Henry Allison's Kant's Theory of Taste

An examination of the currently fashionable thesis that scientists, and especially biologists in the wake of the Darwinian Revolution, can now solve the problems that traditional philosophers have only talked about. Past philosophers, for example during the Enlightenment, have themselves made use of contemporary, scientific techniques and theories. The present claim may only be another such move, to be welcomed by philosophers who would distinguish themselves from rhetoricians. Others may prefer to stake out the merely human or subjective world as their field, identifying 'truth' with 'what it's better to believe'. Both moderns and postmoderns must abandon the rational realism that actually sustained Enlightenment endeavours, and Darwinian explanation, on its own, must erode traditional ethical values and the meta-ethical assumptions that sustain them. Universal humanism is only one possible project among many - and Darwinian reasonings suggest that it is hypocritical. In this crisis there may after all be a rôle for traditional, Platonizing philosophers, believing that there is a truth, and that we can find it out. Such a theory is actually better able to explain our scientific successes, and our evolutionary past.