
581What is mathematical logic?Philosophia 8 (1): 7994. 1978.This review concludes that if the authors know what mathematical logic is they have not shared their knowledge with the readers. This highly praised book is replete with errors and incoherency.

337Mathematics and realityPhilosophy of Science 50 (4): 523548. 1983.The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and nonmathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) sur…Read more

263We hold these truths to be selfevident: But what do we mean by that?: We hold these truths to be selfevidentReview of Symbolic Logic 2 (1): 175207. 2009.At the beginning of Die Grundlagen der Arithmetik [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of selfevidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are u…Read more

249Epistemology of mathematics: What are the questions? What count as answers?Philosophical Quarterly 61 (242): 130150. 2011.A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietis…Read more

217Identity, indiscernibility, and Ante Rem structuralism: The tale of I and –IPhilosophia Mathematica 16 (3): 285309. 2008.Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a nontrivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one o…Read more

206The Objectivity of MathematicsSynthese 156 (2): 337381. 2007.The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.

178Categories, Structures, and the FregeHilbert Controversy: The Status of Metamathematics &daggerPhilosophia Mathematica 13 (1): 6177. 2005.There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of metamathematics in an algebraic or structuralist approach to mathematics. Can metamathematics itself be understood in algebraic or structural terms? Or …Read more

173New V, ZF and abstractiontPhilosophia Mathematica 7 (3): 293321. 1999.We examine George Boolos's proposed abstraction principle for extensions based on the limitationofsize conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neologicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New …Read more

170Where in the (world wide) web of belief is the law of noncontradiction?Noûs 41 (2). 2007.It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analyticsynthetic distinction. This logicfriendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so…Read more

168The classical continuum without pointsReview of Symbolic Logic 6 (3): 488512. 2013.We develop a pointfree construction of the classical one dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually ex…Read more

161So truth is safe from paradox: now what?Philosophical Studies 147 (3): 445455. 2010.The article is part of a symposium on Hartry Field’s “Saving truth from paradox”. The book is one of the most significant intellectual achievements of the past decades, but it is not clear what, exactly, it accomplishes. I explore some alternatives, relating the developed view to the intuitive, pretheoretic notion of truth.

159Truth, function and paradoxAnalysis 71 (1): 3844. 2011.Michael Lynch’s Truth as One and Many is a contribution to the large body of philosophical literature on the nature of truth. Within that genre, advocates of truthascorrespondence, advocates of truthascoherence, and the like, all hold that truth has a single underlying metaphysical nature, but they sharply disagree as to what this nature is. Lynch argues that many of these views make good sense of truth attributions for a limited stretch of discourse, but he adds that each of the contenders …Read more

157HigherOrder Logic or Set Theory: A False DilemmaPhilosophia Mathematica 20 (3): 305323. 2012.The purpose of this article is show that secondorder logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify secondorder logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?

144Do not claim too much: Secondorder logic and firstorder logicPhilosophia Mathematica 7 (1): 4264. 1999.The purpose of this article is to delimit what can and cannot be claimed on behalf of secondorder logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.

142Frege meets zermelo: A perspective on ineffability and reflection: A perspective on ineffability and reflectionReview of Symbolic Logic 1 (2): 241266. 2008._1. Philosophical background: iteration, ineffability, reflection._ There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, firstorder Zermelo–Fraenkel set theory with the axiom of choice : the iterative conception and limitation of size. Each strand provides a rather hospitable environment for the hypothesis that the settheoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.

134

128Actual and Potential InfinityNoûs 53 (1): 160191. 2019.The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

119Frege Meets Aristotle: Points as AbstractsPhilosophia Mathematica. 2015.There are a number of regionsbased accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neologicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at sta…Read more

118Philosophy of Mathematics: Structure and OntologyOxford University Press. 1997.Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests re…Read more

116Introduction to special issue: Abstraction and NeoLogicismPhilosophia Mathematica 8 (2): 9799. 2000.

116Secondorder languages and mathematical practiceJournal of Symbolic Logic 50 (3): 714742. 1985.

112Logical consequence: Models and modalityIn Matthias Schirn (ed.), The Philosophy of Mathematics Today, Clarendon Press. pp. 131156. 1998.

110The guru, the logician, and the deflationist: Truth and logical consequenceNoûs 37 (1). 2003.The purpose of this paper is to present a thought experiment and argument that spells trouble for “radical” deflationism concerning meaning and truth such as that advocated by the staunch nominalist Hartry Field. The thought experiment does not sit well with any view that limits a truth predicate to sentences understood by a given speaker or to sentences in (or translatable into) a given language, unless that language is universal. The scenario in question concerns sentences that are not under…Read more

104Structures and Logics: A Case for (a) RelativismErkenntnis 79 (S2): 309329. 2014.In this paper, I use the cases of intuitionistic arithmetic with Church’s thesis, intuitionistic analysis, and smooth infinitesimal analysis to argue for a sort of pluralism or relativism about logic. The thesis is that logic is relative to a structure. There are classical structures, intuitionistic structures, and (possibly) paraconsistent structures. Each such structure is a legitimate branch of mathematics, and there does not seem to be an interesting logic that is common to all of them. One …Read more

103Foundations Without Foundationalism: A Case for SecondOrder LogicOxford University Press. 1991.The central contention of this book is that secondorder logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higherorder logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of secondorder concepts in mathematics and the extent to which mathematical ideas can be formulated in higherorder logic. He also shows how firstorder languages ar…Read more
Columbus, Ohio, United States of America
Areas of Specialization
Philosophy of Language 
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Areas of Interest
Philosophy of Language 
Logic and Philosophy of Logic 
Philosophy of Mathematics 