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 particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...) of this claim, I offer 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 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. (shrink)

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.

This thesis articulates the resonances between J. M. Coetzee's lifelong engagement with mathematics and his practice as a novelist, critic, and poet. Though the critical discourse surrounding Coetzee's literary work continues to flourish, and though the basic details of his background in mathematics are now widely acknowledged, his inheritance from that background has not yet been the subject of a comprehensive and mathematically- literate account. In providing such an account, I propose that these two strands of his intellectual trajectory not (...) only developed in parallel, but together engendered several of the characteristic qualities of his finest work. The structure of the thesis is essentially thematic, but is also broadly chronological. Chapter 1 focuses on Coetzee's poetry, charting the increasing involvement of mathematical concepts and methods in his practice and poetics between 1958 and 1979. Chapter 2 situates his master's thesis alongside archival materials from the early stages of his academic career, and thus traces the development of his philosophical interest in the migration of quantificatory metaphors into other conceptual domains. Concentrating on his doctoral thesis and a series of contemporaneous reviews, essays, and lecture notes, Chapter 3 details the calculated ambivalence with which he therein articulates, adopts, and challenges various statistical methods designed to disclose objective truth. Chapter 4 explores the thematisation of several mathematical concepts in Dusklands and In the Heart of the Country. Chapter Five considers Waiting for the Barbarians and Foe in the context provided by Coetzee's interest in the attempts of Isaac Newton to bridge the gap between natural language and the supposedly transparent language of mathematics. Finally, Chapter 6 locates in Elizabeth Costello and Diary of a Bad Year a cognitive approach to the use of mathematical concepts in ethics, politics, and aesthetics, and, by analogy, a central aspect of the challenge Coetzee's late fiction poses to the contemporary literary landscape. (shrink)

This dissertation explores the concepts of naturalness, intrinsicality, and duplication. An intrinsic property is had by an object purely in virtue of the way that object is considered in itself. Duplicate objects are exactly similar, considered as they are in themselves. The perfectly natural properties are the most fundamental properties of the world, upon which the nature of the world depends. In this dissertation I develop a theory of intrinsicality, naturalness, and duplication and explore their philosophical applications. Chapter 1 introduces (...) the notions, gives a preliminary survey of some proposed conceptual connections between the notions, and sketches some of their proposed applications. Chapter 2 gives my background assumptions and introduces notational conventions. In chapter 3 I present a theory of naturalness. Although I take ‘natural’ as a primitive, I clarify this notion by distinguishing and explicating various conceptions of naturalness. In chapter 4 I give a theory of various notions related to naturalness, especially intrinsicality and duplication. I show that ‘intrinsic’ and ‘duplicate’ are interde nable, and then give analyses of these and other notions in terms of naturalness. If, as I think likely, naturalness cannot be analyzed, then what is the proper response? David Lewis suggests: accept naturalness as a primitive. I am sympathetic to this proposal, but not to the form Lewis gives it: chapter 5 contains an argument against Lewis’s theory of naturalness. In chapter 6 I reject the idea that naturalness is analyzable in terms of immanent universals. I focus on the work of D. M. Armstrong. I also criticize Armstrong’s arguments against transcendent universals. (shrink)

Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of (...) themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light. (shrink)

This paper distinguishes revolutionary fictionalism from other forms of fictionalism and also from other philosophical views. The paper takes fictionalism about mathematical objects and fictionalism about scientific unobservables as illustrations. The paper evaluates arguments that purport to show that this form of fictionalism is incoherent on the grounds that there is no tenable distinction between believing a sentence and taking the fictionalist's distinctive attitude to that sentence. The argument that fictionalism about mathematics is ‘comically immodest’ is also evaluated. In place (...) of those arguments, an argument against fictionalism about abstract objects of any kind is presented in the last section. This argument takes the form of a trilemma against the fictionalist. (shrink)

In his "Discussion" (1984), Elliott Sober offers some criticisms of the view about species--pluralistic realism--advocated in my 1984. Sober's comments divide into three parts. He attempts to show that species are not sets; he responds to my critique of David Hull's thesis that species are individuals; and he offers some arguments for the claim that species are "chunks of the genealogical nexus." I consider each of these objections in turn, arguing that each of them fails. I attempt to use Sober's (...) insightful critique to explain and defend pluralistic realism more fully. (shrink)

Peacocke’s recent The Primacy of Metaphysics covers a wide range of topics. This critical discussion focuses on the book’s novel account of extensive magnitudes and numbers. First, I further develop and defend Peacocke’s argument against nominalistic approaches to magnitudes and numbers. Then, I argue that his view is more Aristotelian than Platonist because reified magnitudes and numbers are accounted for via corresponding properties and these properties’ application conditions, and because the mentioned objects have a “shallow nature” relative to the corresponding (...) properties. The result is an asymmetric conception of abstraction, which contrasts with the neo-Fregeans’ but has important tenets in common with an approach that I have recently developed. (shrink)

Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly on Frege's corresponding remarks in The Foundations of Arithmetic (1884), supplemented by a few asides on Basic Laws of (...) Arithmetic (1893/1903) and "Thoughts" (1918). My goal is to clarify in which sense the Frege of Foundations and Basic Laws is a platonist concerning the natural numbers.1.. (shrink)

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible.The second section examines the problem as it was posed by Benacerraf in Mathematical Truth and the next section presents a way (...) of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them. (shrink)

The view that mathematical objects are indefinite in nature is presented and defended, hi the first section, Field's argument for fictionalism, given in response to Benacerraf's problem of identification, is closely examined, and it is contended that platonists can solve the problem equally well if they take the view that mathematical objects are indefinite. In the second section, two general arguments against the intelligibility of objectual indefiniteness are shown erroneous, hi the final section, the view is compared to mathematical structuralism, (...) and it is shown that a version of structuralism should be understood as embracing the same view. (shrink)

One of the most significant discoveries of early twentieth century mathematical logic was a workable definition of ‘ordered pair’ totally within set theory. Norbert Wiener, and independently Casimir Kuratowski, are usually credited with this discovery. A definition of ‘ordered pair’ held the key to the precise formulation of the notions of ‘relation’ and ‘function’ — both of which are probably indispensable for an understanding of the foundations of mathematics. The set-theoretic definition of ‘ordered pair’ thus turned out to be a (...) key victory for logicism, providing one admits set theory is logic. The definition also was instrumental in achieving the appearance of ontological economy — since it seemed only sets were needed — although this feature was emphasized only later. (shrink)