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Defending the Axioms: On the Philosophical Foundations of Set Theory

Oxford, England: Oxford University Press (2011)

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  1. Fictionalism and Mathematical Objectivity.Iulian D. Toader - 2012 - In Mircea Dumitru, Mircea Flonta & Valentin Muresan (eds.), Metaphysics and Science. Dedicated to professor Ilie Pârvu. Universty of Bucharest Press. pp. 137-158.
    This paper, written in Romanian, compares fictionalism, nominalism, and neo-Meinongianism as responses to the problem of objectivity in mathematics, and then motivates a fictionalist view of objectivity as invariance.
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  • (1 other version)Multiverse Conceptions in Set Theory.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo - 2015 - Synthese 192 (8):2463-2488.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
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  • Knowledge of Mathematics without Proof.Alexander Paseau - 2015 - British Journal for the Philosophy of Science 66 (4):775-799.
    Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...)
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  • Mathematical Progress — On Maddy and Beyond.Simon Weisgerber - 2023 - Philosophia Mathematica 31 (1):1-28.
    A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly (...)
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  • The Price of Mathematical Scepticism.Paul Blain Levy - 2022 - Philosophia Mathematica 30 (3):283-305.
    This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. -/- Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.
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  • What is Absolute Undecidability?†.Justin Clarke-Doane - 2012 - Noûs 47 (3):467-481.
    It is often supposed that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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  • Three conceptions of explaining how possibly—and one reductive account.Johannes Persson - 2011 - In Henk W. De Regt, Stephan Hartmann & Samir Okasha (eds.), EPSA Philosophy of Science: Amsterdam 2009. Springer. pp. 275--286.
    Philosophers of science have often favoured reductive approaches to how-possibly explanation. This article identifies three alternative conceptions making how-possibly explanation an interesting phenomenon in its own right. The first variety approaches “how possibly X?” by showing that X is not epistemically impossible. This can sometimes be achieved by removing misunderstandings concerning the implications of one’s current belief system but involves characteristically a modification of this belief system so that acceptance of X does not result in contradiction. The second variety offers (...)
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  • (1 other version)Abstract objects.Gideon Rosen - 2008 - Stanford Encyclopedia of Philosophy.
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  • (1 other version)Recalcitrant Disagreement in Mathematics: An “Endless and Depressing Controversy” in the History of Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2023 - Global Philosophy 33 (38):1-29.
    If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...)
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  • What analytic metaphysics can do for scientific metaphysics.Chanwoo Lee - 2023 - Ratio 36 (3):192-203.
    The apparent chasm between two camps in metaphysics, analytic metaphysics and scientific metaphysics, is well recognized. I argue that the relationship between them is not necessarily a rivalry; a division of labour that resembles the relationship between pure mathematics and science is possible. As a case study, I look into the metaphysical underdetermination argument for ontic structural realism, a well‐known position in scientific metaphysics, together with an argument for the position in analytic metaphysics known as ontological nihilism. I argue that (...)
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  • Virtue theory of mathematical practices: an introduction.Andrew Aberdein, Colin Jakob Rittberg & Fenner Stanley Tanswell - 2021 - Synthese 199 (3-4):10167-10180.
    Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...)
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  • (1 other version)Plato’s Intellectual Development and Visual Thinking.Susanna Saracco - 2021 - Plato Journal 21:21-42.
    Plato has devised texts which call the readers to collaborate cognitively with them. An important epistemic stimulation is the schematization, the line segment, which summarizes Plato’s idea of intellectual development. In this research, visual thinking will help us to make the most of the Platonic invitation to investigate further cognitive growth. It will be analyzed how visual discoveries are rendered possible by mental number lines, realizing the epistemological importance of visualization. Thanks to visualization, structuralism will be grasped. It will reveal (...)
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  • Mathematizing as a virtuous practice: different narratives and their consequences for mathematics education and society.Deborah Kant & Deniz Sarikaya - 2020 - Synthese 199 (1-2):3405-3429.
    There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy is (...)
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  • Methodology in the ontology of artworks: exploring hermeneutic fictionalism.Elisa Caldarola - 2020 - In Concha Martinez Vidal & José Luis Falguera Lopez (ed.), Abstract Objects: For and Against.
    There is growing debate about what is the correct methodology for research in the ontology of artworks. In the first part of this essay, I introduce my view: I argue that semantic descriptivism is a semantic approach that has an impact on meta-ontological views and can be linked with a hermeneutic fictionalist proposal on the meta-ontology of artworks such as works of music. In the second part, I offer a synthetic presentation of the four main positive meta-ontological views that have (...)
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  • Set-theoretic justification and the theoretical virtues.John Heron - 2020 - Synthese 199 (1-2):1245-1267.
    Recent discussions of how axioms are extrinsically justified have appealed to abductive considerations: on such accounts, axioms are adopted on the basis that they constitute the best explanation of some mathematical data, or phenomena. In the first part of this paper, I set out a potential problem caused by the appeal made to the notion of mathematical explanation and suggest that it can be remedied once it is noted that all the justificatory work is done by appeal to the theoretical (...)
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  • Maddy On The Multiverse.Claudio Ternullo - 2019 - In Stefania Centrone, Deborah Kant & Deniz Sarikaya (eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Springer Verlag. pp. 43-78.
    Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...)
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  • Conceptual evaluation: epistemic.Alejandro Pérez Carballo - 2019 - In Alexis Burgess, Herman Cappelen & David Plunkett (eds.), Conceptual Engineering and Conceptual Ethics. New York, USA: Oxford University Press. pp. 304-332.
    On a view implicitly endorsed by many, a concept is epistemically better than another if and because it does a better job at ‘carving at the joints', or if the property corresponding to it is ‘more natural' than the one corresponding to another. This chapter offers an argument against this seemingly plausible thought, starting from three key observations about the way we use and evaluate concepts from en epistemic perspective: that we look for concepts that play a role in explanations (...)
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  • Epistemic injustice in mathematics.Colin Jakob Rittberg, Fenner Stanley Tanswell & Jean Paul Van Bendegem - 2020 - Synthese 197 (9):3875-3904.
    We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept (...)
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  • A naturalistic justification of the generic multiverse with a core.Matteo de Ceglie - 2018 - Contributions of the Austrian Ludwig Wittgenstein Society 26:34-36.
    In this paper, I argue that a naturalist approach in philosophy of mathematics justifies a pluralist conception of set theory. For the pluralist, there is not a Single Universe, but there is rather a Multiverse, composed by a plurality of universes generated by various set theories. In order to justify a pluralistic approach to sets, I apply the two naturalistic principles developed by Penelope Maddy (cfr. Maddy (1997)), UNIFY and MAXIMIZE, and analyze through them the potential of the set theoretic (...)
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  • Regarding the ‘Hole Argument’.James Owen Weatherall - 2018 - British Journal for the Philosophy of Science 69 (2):329-350.
    I argue that the hole argument is based on a misleading use of the mathematical formalism of general relativity. If one is attentive to mathematical practice, I will argue, the hole argument is blocked. _1._ Introduction _2._ A Warmup Exercise _3._ The Hole Argument _4._ An Argument from Classical Spacetime Theory _5._ The Hole Argument Revisited.
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  • Ipotesi del Continuo.Claudio Ternullo - 2017 - Aphex 16.
    L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...)
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  • Formal analyticity.Zeynep Soysal - 2018 - Philosophical Studies 175 (11):2791-2811.
    In this paper, I introduce and defend a notion of analyticity for formal languages. I first uncover a crucial flaw in Timothy Williamson’s famous argument template against analyticity, when it is applied to sentences of formal mathematical languages. Williamson’s argument targets the popular idea that a necessary condition for analyticity is that whoever understands an analytic sentence assents to it. Williamson argues that for any given candidate analytic sentence, there can be people who understand that sentence and yet who fail (...)
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  • II—Persistent Philosophical Disagreement.Chris Daly - 2017 - Proceedings of the Aristotelian Society 117 (1):23-40.
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  • The varieties of indispensability arguments.Marco Panza & Andrea Sereni - 2016 - Synthese 193 (2):469-516.
    The indispensability argument comes in many different versions that all reduce to a general valid schema. Providing a sound IA amounts to providing a full interpretation of the schema according to which all its premises are true. Hence, arguing whether IA is sound results in wondering whether the schema admits such an interpretation. We discuss in full details all the parameters on which the specification of the general schema may depend. In doing this, we consider how different versions of IA (...)
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  • Mathematical Depth.Alasdair Urquhart - 2015 - Philosophia Mathematica 23 (2):233-241.
    The first part of the paper is devoted to surveying the remarks that philosophers and mathematicians such as Maddy, Hardy, Gowers, and Zeilberger have made about mathematical depth. The second part is devoted to the question of whether we can make the notion precise by a more formal proof-theoretical approach. The idea of measuring depth by the depth and bushiness of the proof is considered, and compared to the related notion of the depth of a chess combination.
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  • Naturalizing Badiou: mathematical ontology and structural realism.Fabio Gironi - 2014 - New York: Palgrave-Macmillan.
    This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...)
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  • Axioms in Mathematical Practice.Dirk Schlimm - 2013 - Philosophia Mathematica 21 (1):37-92.
    On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...)
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  • Robert Goldblatt. Quantifiers, propositions and identity: Admissible semantics for quantified modal and substructural logics. Lecture notes in logic; 38. cambridge: Cambridge university press, 2011. Isbn 978-1-107-01052-9. Pp. XIII + 282. [REVIEW]R. Jones - 2013 - Philosophia Mathematica 21 (1):123-127.
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  • Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics.Julian C. Cole - 2013 - Philosophia Mathematica 21 (1):9-36.
    I contend that mathematical domains are freestanding institutional entities that, at least typically, are introduced to serve representational functions. In this paper, I outline an account of institutional reality and a supporting metaontological perspective that clarify the content of this thesis. I also argue that a philosophy of mathematics that has this thesis as its central tenet can account for the objectivity, necessity, and atemporality of mathematics.
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  • On the Contemporary Practice of Philosophy of Mathematics.Colin Jakob Rittberg - 2019 - Acta Baltica Historiae Et Philosophiae Scientiarum 7 (1):5-26.
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  • Unleashing the Constructive Potential of Emotions: Some Critical Comments on Risk, Technology and Moral Emotions by Sabine Roeser.Steffen Steinert - 2020 - Science and Engineering Ethics 26 (4):1913-1920.
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  • Philosophy’s Loss of Logic to Mathematics: An Inadequately Understood Take-Over.Woosuk Park - 2018 - Cham, Switzerland: Springer Verlag.
    This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to understand (...)
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  • Mathematical Structuralism, Modal Nominalism, and the Coherence Principle.James S. J. Schwartz - 2015 - Philosophia Mathematica 23 (3):367-385.
    According to Stewart Shapiro's coherence principle, structures exist whenever they can be coherently described. I argue that Shapiro's attempts to justify this principle are circular, as he relies on criticisms of modal nominalism which presuppose the coherence principle. I argue further that when the coherence principle is not presupposed, his reasoning more strongly supports modal nominalism than ante rem structuralism.
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  • (1 other version)On Tins and Tin-Openers.Michael Liston - 2011 - In Henk W. De Regt, Stephan Hartmann & Samir Okasha (eds.), EPSA Philosophy of Science: Amsterdam 2009. Springer. pp. 151--160.
    Most science requires applied mathematics. This truism underlies the Quine-Putnam indispensability argument: we cannot be mathematical nominalists without rejecting whole swaths of good science that are seamlessly linked with mathematics. One style of response accepts the challenge head-on and attempts to show how to do science without mathematics. There is some consensus that the response fails because the nominalistic apparatus deployed either is not extendible to all of mathematical physics or is merely a deft reconstrual equivalent to standard mathematics. A (...)
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  • Social constructivism in mathematics? The promise and shortcomings of Julian Cole’s institutional account.Jenni Rytilä - 2021 - Synthese 199 (3-4):11517-11540.
    The core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting for the (...)
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  • Intellectual humility in mathematics.Colin Jakob Rittberg - unknown - Synthese 199 (3-4):5571-5601.
    In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of (...)
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  • Husserl’s Transcendentalization of Mathematical Naturalism.Mirja Hartimo - 2020 - Journal of Transcendental Philosophy 1 (3):289-306.
    The paper aims to capture a form of naturalism that can be found “built-in” in phenomenology, namely the idea to take science or mathematics on its own, without postulating extraneous normative “molds” on it. The paper offers a detailed comparison of Penelope Maddy’s naturalism about mathematics and Husserl’s approach to mathematics in Formal and Transcendental Logic. It argues that Maddy’s naturalized methodology is similar to the approach in the first part of the book. However, in the second part Husserl enters (...)
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  • A theory of implicit commitment.Mateusz Łełyk & Carlo Nicolai - 2022 - Synthese 200 (4):1-26.
    The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still lacking. We (...)
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  • No Magic: From Phenomenology of Practice to Social Ontology of Mathematics.Mirja Hartimo & Jenni Rytilä - 2023 - Topoi 42 (1):283-295.
    The paper shows how to use the Husserlian phenomenological method in contemporary philosophical approaches to mathematical practice and mathematical ontology. First, the paper develops the phenomenological approach based on Husserl's writings to obtain a method for understanding mathematical practice. Then, to put forward a full-fledged ontology of mathematics, the phenomenological approach is complemented with social ontological considerations. The proposed ontological account sees mathematical objects as social constructions in the sense that they are products of culturally shared and historically developed practices. (...)
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  • Ontological realism and sentential form.Eileen S. Nutting - 2018 - Synthese 195 (11):5021-5036.
    The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: some mathematical claims are true, and the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny. Those who deny typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for. Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have (...)
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  • Non-deductive methods in mathematics.Alan Baker - 2010 - Stanford Encyclopedia of Philosophy.
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  • (1 other version)Conceptual (and Hence Mathematical) Explanation, Conceptual Grounding and Proof.Francesca Poggiolesi & Francesco Genco - 2021 - Erkenntnis:1-27.
    This paper studies the notions of conceptual grounding and conceptual explanation (which includes the notion of mathematical explanation), with an aim of clarifying the links between them. On the one hand, it analyses complex examples of these two notions that bring to the fore features that are easily overlooked otherwise. On the other hand, it provides a formal framework for modeling both conceptual grounding and conceptual explanation, based on the concept of proof. Inspiration and analogies are drawn with the recent (...)
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  • Regarding the ‘Hole Argument’.James Owen Weatherall - 2016 - British Journal for the Philosophy of Science:axw012.
    I argue that the Hole Argument is based on a misleading use of the mathematical formalism of general relativity. If one is attentive to mathematical practice, I will argue, the Hole Argument is blocked.
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  • Kurt gödel.Juliette Kennedy - 2008 - Stanford Encyclopedia of Philosophy.
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