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Volume 25, 2008
Philosophy of Mathematics

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part i: set theory, inconsistency, and measuring theories
1. ProtoSociology: Volume > 25
Douglas Patterson Representationalism and Set-Theoretic Paradox
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I defend the “settist” view that set theory can be done consistently without any form of distinction between sets and “classes” (by whatever name), if we think clearly about belief and the expression of belief—and this, furthermore, entirely within classical logic. Standard arguments against settism in classical logic are seen to fail because they assume, falsely, that expressing commitment to a set theory is something that must be done in a meaningful language, the semantics of which requires, on pain of Russellian paradox, a more powerful set theory. I explore the consequences of this response to the standard argument against “classical logic settism” for our notion of belief, and argue that what is revealed is that representationalist theories of belief cannot be right as long as it is possible to believe that every set is self-identical.
2. ProtoSociology: Volume > 25
Mark Colyvan Who’s Afraid of Inconsistent Mathematics?
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Contemporary mathematical theories are generally thought to be consistent. But it hasn’t always been this way; there have been times in the history of mathematics when the consistency of various mathematical theories has been called into question. And some theories, such as naïve set theory and (arguably) the early calculus, were shown to be inconsistent. In this paper I will consider some of the philosophical issues arising from inconsistent mathematical theories.
3. ProtoSociology: Volume > 25
Andrew Arana Logical and Semantic Purity
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I distinguish two different views on what makes a proof of a theorem ‘pure’, firstly by characterizing them abstractly, and secondly by showing that in practice the views differ on what proofs qualify as pure.
4. ProtoSociology: Volume > 25
Wilhelm K. Essler On Using Measuring Numbers according to Measuring Theories
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It was shown by Frege that four of the five axioms of Peano can be regarded as analytical truths; and it was shown by Russell that the remaining axiom cannot be regarded as being analytically true or even as being analytically false, that this axiom thus is to be regarded as a synthetic statement. In using the concept of apriority in the sense of Reichenbach, it can be shown that this synthetic axiom is to be regarded as an apriorical truth within the usual background theory of measuring theories, which are used not as generalizations of empirical results but as— not moreover provable— preconditions of receiving measuring results and of ordering these results. Furthermore, the systems of numbers, starting with the natural numbers, are developed in a way such that the pre-rational numbers— but not the rational ones— turn out to be those ones which are used in performing measurements according to such theories, while the pre-real numbers— but not the real ones— then turn out to be those ones which are used in using such measuring theories together with their background theories for purely theoretical reasons.
part ii: the challenge of nominalism
5. ProtoSociology: Volume > 25
Jody Azzouni The Compulsion to Believe: Logical Inference and Normativity
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The interaction between intuitions about inference, and the normative constraints that logical principles applied to mechanically-recognizable derivations impose on (informal) inference, is explored. These intuitions are evaluated in a clear testcase: informal mathe­matical proof. It is argued that formal derivations are not the source of our intuitions of validity, and indeed, neither is the semantic recognition of validity, either as construed model-theoretically, or as driven by the subject-matter such inferences are directed towards. Rather, psychologically-engrained inference-packages (often opportunistically used by mathematicians) are the source of our sense of validity. Formal derivations, or the semantic construal of such, are after-the-fact norms imposed on our inference practices.
6. ProtoSociology: Volume > 25
Otávio Bueno Nominalism and Mathematical Intuition
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As part of the development of an epistemology for mathematics, some Platonists have defended the view that we have (i) intuition that certain mathematical principles hold, and (ii) intuition of the properties of some mathematical objects. In this paper, I discuss some difficulties that this view faces to accommodate some salient features of mathematical practice. I then offer an alternative, agnostic nominalist proposal in which, despite the role played by mathematical intuition, these difficulties do not emerge.
7. ProtoSociology: Volume > 25
Yvonne Raley Jobless Objects: Mathematical Posits in Crisis
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This paper focuses on an argument against the existence of mathematical objects called the “Makes No Difference Argument” (MND). Roughly, MND claims that whether or not mathematical objects exist makes no difference, and that therefore, we have no reason to believe in them. The paper analyzes four different versions of MND for their merits. It concludes that the defender of the existence of mathematical objects (the mathematical Platonist) does have some retorts to the first three versions of MND, but that no adequate reply is possible to the fourth, and most crucial, version of MND. That version argues that mathematical objects make no difference to our epistemic processes: they play no role in the process of obtaining mathematical knowledge.
8. ProtoSociology: Volume > 25
Susan Vineberg Is Indispensability Still a Problem for Fictionalism?
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For quite some time the indispensability arguments of Quine and Putnam were considered a formidable obstacle to anyone who would reject the existence of mathematical objects. Various attempts to respond to the indispensability arguments were developed, most notably by Chihara and Field. Field tried to defend mathematical fictionalism, according to which the existential assertions of mathematics are false, by showing that the mathematics used in applications is in fact dispensable. Chihara suggested, on the other hand, that mathema­tics makes true existential assertions, but that these can be interpreted so as to remove the commitment to abstract objects. More recently, there have been various attempts to show that the indispensability arguments contain assumptions that are conceptually misguided in ways having little to do with mathematical content. All of this work is of considerable interest, and the result has been a gathering consensus that the indispensability arguments, as put forth by Quine and Putnam, do not provide convincing reason to accept mathematical realism. The focus here will be on the ways of responding to the indispensability arguments, and in particular on the obstacles to fictionalism that remain after the versions of Quine and Putnam are undercut.
part iii: historical background
9. ProtoSociology: Volume > 25
Madeline Muntersbjorn Mill, Frege and the Unity of Mathematics
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This essay discusses the unity of mathematics by comparing the philosophies of Mill and Frege. While Mill is remembered as a progressive social thinker, his contributions to the development of logic are less widely heralded. In contrast, Frege made important and lasting contributions to the development of logic while his social thought, what little is known of it, was very conservative. Two theses are presented in the paper. The first is that in order to pursue Mill’s progressive sociopolitical project, one must embrace Frege’s distinction between logic and psychology. The second thesis is that in order to pursue Frege’s project of accounting for the unity of mathematics, we must understand mathematics as a human activity and consider the role that history and psychology play in the growth of mathematics.
10. ProtoSociology: Volume > 25
Raffaella De Rosa, Otávio Bueno Descartes on Mathematical Essences
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Descartes seems to hold two inconsistent accounts of the ontological status of mathematical essences. Meditation Five apparently develops a platonist view about such essences, while the Principles seems to advocate some form of “conceptualism”. We argue that Descartes was neither a platonist nor a conceptualist. Crucial to our interpretation is Descartes’ dispositional nativism. We contend that his doctrine of innate ideas allows him to endorse a hybrid view which avoids the drawbacks of Gassendi’s conceptualism without facing the difficulties of platonism. We call this hybrid view “quasi-platonism.” Our interpretation explains Descartes’ account of the nature of mathematical essences, dissolves the tension between the two texts, and highlights the benefits of Descartes’ view.
on contemporary philosophy and sociology
11. ProtoSociology: Volume > 25
Nicholas Rescher Presumption and the Judgement of Elites
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12. ProtoSociology: Volume > 25
Steven I. Miller, Marcel Fredericks, Frank J. Perino Social Science Research and Policymaking: Meta-Analysis and Paradox
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The purpose of this article is to explore some of the non-obvious characteristics of the social science research-social policy (SSRSP) paradigm. We examine some of the underlying assumptions of the readily accepted claim that social science research can lead to the creation of rational social policy. We begin by using the framework of meta-analysis as one of the most powerful means of informing policy by way of empirical research findings. This approach is critiqued and found wanting in several ways. Several conceptual and definitional issues connected to the term “policy” are explored as well. A central argument is that even the best social science research is no guarantee of enlightened policymaking because the very (inductive) basis of empirical research militates against the possibility of going from research findings to policy. This claim is explored within the context of a central paradox. This paradox is explored in some depth. Finally, within the SSRSP claim, we analyze related issues such as the possibility of utilizing Mixed Methods and the politics of policymaking. We conclude that the SSRSP framework is, at best, a subjective one which ironically is needed, but one which is constrained by the very methods that is uses to formulate policy.
13. ProtoSociology: Volume > 25
Adam Sennet Hidden Indexicals and Pronouns
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Nikola Kompa Review: Stephen Schiffer, The Things We Mean
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J. Gregory Keller Agency Implies Weakness of Will
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Notions of agency and of weakness of will clearly seem to be related to one another. This essay takes on a rather modest task in relation to current discussion of these topics; it seeks to establish the following claim: If A is a normal human agent, weakness of will is possible for A. The argument relies on demonstrating that certain necessary conditions for normal human agency are at least roughly equivalent to certain sufficient conditions for weakness of will. The connection between agency and weakness of will is made through the use of an extended example that lays bare the links between the two.
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On ProtoSociology
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Published Volumes
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