•  951
    What is mathematical logic?
    Philosophia 8 (1): 79-94. 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.
  •  762
    The good, the bad and the ugly
    with Philip Ebert
    Synthese 170 (3): 415-441. 2009.
    This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present …Read more
  •  494
    Actual and Potential Infinity
    Noûs 53 (1): 160-191. 2017.
    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.
  •  446
    This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century…Read more
  •  435
    Hume’s Principle, Bad Company, and the Axiom of Choice
    Review of Symbolic Logic 16 (4): 1158-1176. 2023.
    One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true…Read more
  •  402
    Mathematics and reality
    Philosophy of Science 50 (4): 523-548. 1983.
    The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical 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
  •  370
    Resolving Frege’s Other Puzzle
    Philosophica Mathematica 30 (1): 59-87. 2022.
    Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we …Read more
  •  332
    Identity, indiscernibility, and Ante Rem structuralism: The tale of I and –I
    Philosophia Mathematica 16 (3): 285-309. 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 non-trivial 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
  •  326
    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 self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are u…Read more
  •  285
    Epistemology of mathematics: What are the questions? What count as answers?
    Philosophical Quarterly 61 (242): 130-150. 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
  •  264
    The Objectivity of Mathematics
    Synthese 156 (2): 337-381. 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.
  •  250
    Computability, Notation, and de re Knowledge of Numbers
    with Eric Snyder and Richard Samuels
    Philosophies 1 (7). 2022.
    Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship betwe…Read more
  •  249
    New V, ZF and Abstraction
    with Alan Weir
    Philosophia Mathematica 7 (3): 293-321. 1999.
    We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist 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
  •  246
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are ope…Read more
  •  238
    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 analytic-synthetic distinction. This logic-friendly 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
  •  227
    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 meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or …Read more
  •  219
    Conservativeness and incompleteness
    Journal of Philosophy 80 (9): 521-531. 1983.
  •  218
    Higher-Order Logic or Set Theory: A False Dilemma
    Philosophia Mathematica 20 (3): 305-323. 2012.
    The purpose of this article is show that second-order 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 second-order 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?
  •  214
    The classical continuum without points
    Review of Symbolic Logic 6 (3): 488-512. 2013.
    We develop a point-free 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
  •  209
    Do not claim too much: Second-order logic and first-order logic
    Philosophia Mathematica 7 (1): 42-64. 1999.
    The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.
  •  209
    Philosophy of mathematics: structure and ontology
    Oxford 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
  •  206
    Aristotelian Continua
    Philosophia Mathematica 24 (2): 214-246. 2016.
    In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, and we show th…Read more
  •  205
    Mathematical structuralism
    Philosophia Mathematica 4 (2): 81-82. 1996.
    STEWART SHAPIRO; Mathematical Structuralism, Philosophia Mathematica, Volume 4, Issue 2, 1 May 1996, Pages 81–82, https://doi.org/10.1093/philmat/4.2.81.
  •  194
  •  192
    Modality and ontology
    Mind 102 (407): 455-481. 1993.
  •  190
    The central contention of this book is that second-order 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 higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages ar…Read more
  •  190
    Hofweber’s Nominalist Naturalism
    with Eric Snyder and Richard Samuels
    In Gianluigi Oliveri, Claudio Ternullo & Stefano Boscolo (eds.), Objects, Structures, and Logics. pp. 31-62. 2022.
    In this paper, we outline and critically evaluate Thomas Hofweber’s solution to a semantic puzzle he calls Frege’s Other Puzzle. After sketching the Puzzle and two traditional responses to it—the Substantival Strategy and the Adjectival Strategy—we outline Hofweber’s proposed version of Adjectivalism. We argue that two key components—the syntactic and semantic components—of Hofweber’s analysis both suffer from serious empirical difficulties. Ultimately, this suggests that an altogether different…Read more