Mathematical Truth

Edited by Mark Balaguer (California State University, Los Angeles)
Assistant editor: Sam Roberts (University of Sheffield)
About this topic
Summary

The topic of mathematical truth is importantly tied to the ontology of mathematics.  In particular, a central question is what kinds of objects we commit ourselves to when we endorse the truth of ordinary mathematical sentences, like ‘4 is even’ and ‘There are infinitely many prime numbers.’   But there are other important philosophical questions about mathematical truth as well.  For instance: Is there any plausible way to maintain that mathematical truths are analytic, i.e., true solely in virtue of meaning?  And given that most ordinary mathematical sentences (e.g., the two sentences listed above) follow from the axioms of our various mathematical theories (e.g., from sentences like ‘0 is a number’), how can we account for the truth of the axioms?  And how can we account for the objectivity of mathematics (i.e., for the fact that some mathematical sentences are objectively correct and others are objectively incorrect)?  Can we do this without endorsing the existence of mathematical objects?  Do mathematical objects even help?  And so on.

Key works

Some key works on these topics include the following: Carnap 1950; Benacerraf 1973; Putnam 1980; Field 1993; Field 1998; Wright & Hale 1992; Gödel 1964; Maddy 1988; and Maddy 1988.

Introductions

Introductory works include Shapiro 2000 and Colyvan 2012.

Related

Contents
301 found
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  1. Internalism and the Determinacy of Mathematics.Lavinia Picollo & Daniel Waxman - 2023 - Mind 132 (528):1028-1052.
    A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably (...)
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  2. Metamathematik der Elementarmathematik.Erwin Engeler - 1983 - New York: Springer Verlag.
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  3. Numbers as properties.Melisa Vivanco - 2023 - Synthese 202 (4):1-23.
    Although number sentences are ostensibly simple, familiar, and applicable, the justification for our arithmetical beliefs has been considered mysterious by the philosophical tradition. In this paper, I argue that such a mystery is due to a preconception of two realities, one mathematical and one nonmathematical, which are alien to each other. My proposal shows that the theory of numbers as properties entails a homogeneous domain in which arithmetical and nonmathematical truth occur. As a result, the possibility of arithmetical knowledge is (...)
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  4. Towards a Computational Ontology for the Philosophy of Wittgenstein: Representing Aspects of the Tractarian Philosophy of Mathematics.Jakub Gomułka - 2023 - Analiza I Egzystencja 63:27-54.
    The present paper concerns the Wittgenstein ontology project: an attempt to create a Semantic Web representation of Ludwig Wittgenstein’s philosophy. The project has been in development since 2006, and its current state enables users to search for information about Wittgenstein-related documents and the documents themselves. However, the developers have much more ambitious goals: they attempt to provide a philosophical subject matter knowledge base that would comprise the claims and concepts formulated by the philosopher. The current knowledge representation technology is not (...)
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  5. Not So Simple.Colin R. Caret - 2023 - Asian Journal of Philosophy 2 (2):1-16.
    In a recent series of articles, Beall has developed the view that FDE is the formal system most deserving of the honorific “Logic”. The Simple Argument for this view is a cost-benefit analysis: the view that FDE is Logic has no drawbacks and it has some benefits when compared with any of its rivals. In this paper, I argue that both premises of the Simple Argument are mistaken. I use this as an opportunity to further reflect on how such arguments (...)
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  6. Theory of fuzzy computation.Apostolos Syropoulos - 2014 - New York: Springer.
    The book provides the first full length exploration of fuzzy computability. It describes the notion of fuzziness and present the foundation of computability theory. It then presents the various approaches to fuzzy computability. This text provides a glimpse into the different approaches in this area, which is important for researchers in order to have a clear view of the field. It contains a detailed literature review and the author includes all proofs to make the presentation accessible. Ideas for future research (...)
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  7. A beginner's guide to mathematical logic.Raymond M. Smullyan - 2014 - Mineola, New York: Dover Publications.
    Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts. He highlights each subject with inventive explanations and unique problems. Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs. Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and graduate (...)
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  8. Deductivism in the Philosophy of Mathematics.Alexander Paseau & Fabian Pregel - 2023 - Stanford Encyclopedia of Philosophy 2023.
    Deductivism says that a mathematical sentence s should be understood as expressing the claim that s deductively follows from appropriate axioms. For instance, deductivists might construe “2+2=4” as “the sentence ‘2+2=4’ deductively follows from the axioms of arithmetic”. Deductivism promises a number of benefits. It captures the fairly common idea that mathematics is about “what can be deduced from the axioms”; it avoids an ontology of abstract mathematical objects; and it maintains that our access to mathematical truths requires nothing beyond (...)
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  9. Nothing Infinite: A Summary of Forever Finite.Kip Sewell - 2023 - Rond Media Library.
    In 'Forever Finite: The Case Against Infinity' (Rond Books, 2023), the author argues that, despite its cultural popularity, infinity is not a logical concept and consequently cannot be a property of anything that exists in the real world. This article summarizes the main points in 'Forever Finite', including its overview of what debunking infinity entails for conceptual thought in philosophy, mathematics, science, cosmology, and theology.
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  10. On Radical Enactivist Accounts of Arithmetical Cognition.Markus Pantsar - 2022 - Ergo: An Open Access Journal of Philosophy 9.
    Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical enactivist view. (...)
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  11. Arithmetical truth and hidden higher-order concepts.Daniel Isaacson - 1987 - In Logic Colloquium '85: Proceedings of the Colloquium held in Orsay, France July 1985 (Studies in Logic and the Foundations of Mathematics, Vol. 122.). Amsterdam, New York, Oxford, Tokyo: North-Holland. pp. 147-169.
    The incompleteness of formal systems for arithmetic has been a recognized fact of mathematics. The term “incompleteness” suggests that the formal system in question fails to offer a deduction which it ought to. This chapter focuses on the status of a formal system, Peano Arithmetic, and explores a viewpoint on which Peano Arithmetic occupies an intrinsic, conceptually well-defined region of arithmetical truth. The idea is that it consists of those truths which can be perceived directly from the purely arithmetical content (...)
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  12. Métamathématique.Paul Lorenzen - 1967 - Paris,: Mouton.
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  13. Wittgenstein on Mathematical Advances and Semantical Mutation.André Porto - 2023 - Philósophos.
    The objective of this article is to try to elucidate Wittgenstein’s ex-travagant thesis that each and every mathematical advancement involves some “semantical mutation”, i.e., some alteration of the very meanings of the terms involved. To do that we will argue in favor of the idea of a “modal incompati-bility” between the concepts involved, as they were prior to the advancement, and what they become after the new result was obtained. We will also argue that the adoption of this thesis profoundly (...)
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  14. Fondements des mathématiques.Michel Combès - 1971 - Paris,: Presses universitaires de France.
  15. Human Thought, Mathematics, and Physical Discovery.Gila Sher - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Berlin: Springer. pp. 301-325.
    In this paper I discuss Mark Steiner’s view of the contribution of mathematics to physics and take up some of the questions it raises. In particular, I take up the question of discovery and explore two aspects of this question – a metaphysical aspect and a related epistemic aspect. The metaphysical aspect concerns the formal structure of the physical world. Does the physical world have mathematical or formal features or constituents, and what is the nature of these constituents? The related (...)
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  16. Sur les principes des mathématiques chez Aristote et Euclide.Ian Mueller - 1991 - In Jules Vuillemin & Rushdī Rāshid (eds.), Mathématiques et philosophie de l'antiquité à l'age classique: hommage à Jules Vuillemin. Diffusion, Presses du CNRS.
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  17. Einführung in die formale Logik unter der besonderen Fragestellung: was ist Wahrheit allein aufgrund der Form?Manfred Buth - 1996 - New York: Lang.
    Der Autor hat sich das Ziel gesetzt, verschiedene Ansatze zur formalen Logik unter einer einheitlichen Fragestellung darzustellen und zu erortern. Das Buch beginnt deshalb mit der Aristotelischen Syllogistik als einer historisch bedeutsamen Auspragung der formalen Logik. Dann wird durch die schrittweise Erweiterung der einfachen Aussagenlogik zur klassischen Quantorenlogik in einen wesentlichen Teil der mathematischen Logik eingefuhrt. Es folgen einige Anwendungen der formalen Logik, Ansatze zu einer Theorie der Argumentation, ein kurzer Abriss der Geschichte der formalen Logik und ein Uberblick uber (...)
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  18. The Paradigm Shift in the 19th-century Polish Philosophy of Mathematics.Paweł Polak - 2022 - Studia Historiae Scientiarum 21:217-235.
    The Polish philosophy of mathematics in the 19th century had its origins in the Romantic period under the influence of the then-predominant idealist philosophies. The decline of Romantic philosophy precipitated changes in general philosophy, but what is less well known is how it triggered changes in the philosophy of mathematics. In this paper, we discuss how the Polish philosophy of mathematics evolved from the metaphysical approach that had been formed during the Romantic era to the more modern positivistic paradigm. These (...)
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  19. Sūgakuteki shinri no meikyū: kaigi shugi to no kakutō = The labyrinth of mathematical truth: grapplings with scepticism.Chikara Sasaki - 2020 - Sapporo-shi: Hokkaidō Daigaku Shuppankai.
    『不思議の国のアリス』の数学観から、古代ギリシャから現代への懐疑主義思想との格闘をたどって、数学的知識の成立根拠を探る。.
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  20. Propositional logics of logical truth.A. C. Paseau & Owen Griffiths - 2021 - In Gil Sagi & Jack Woods (eds.), The Semantic Conception of Logic : Essays on Consequence, Invariance, and Meaning. Cambridge University Press.
  21. Axiomatics: mathematical thought and high modernism.Alma Steingart - 2023 - Chicago: University of Chicago Press.
    The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century. Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about (...)
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  22. Mathematical explanation doesn't require mathematical truth.Mary Leng - 2020 - In Shamik Dasgupta, Brad Weslake & Ravit Dotan (eds.), Current Controversies in Philosophy of Science. Routledge.
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  23. Part II. Does mathematical explanation require mathematical truth?: Mathematical explanation requires mathematical truth.Christopher Pincock - 2020 - In Shamik Dasgupta, Brad Weslake & Ravit Dotan (eds.), Current Controversies in Philosophy of Science. Routledge.
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  24. The Benacerraf Problem of Mathematical Truth and Knowledge.Eileen S. Nutting - 2022 - Internet Encyclopedia of Philosophy.
    The Benacerraf Problem of Mathematical Truth and Knowledge Before philosophical theorizing, people tend to believe that most of the claims generally accepted in mathematics—claims like “2+3=5” and “there are infinitely many prime numbers”—are true, and that people know many of them. Even after philosophical theorizing, most people remain committed to mathematical truth and mathematical knowledge. … Continue reading The Benacerraf Problem of Mathematical Truth and Knowledge →.
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  25. In defense of Countabilism.David Builes & Jessica M. Wilson - 2022 - Philosophical Studies 179 (7):2199-2236.
    Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...)
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  26. Calculus of Qualia: Introduction to Qualations 7 2 2022.Paul Merriam - manuscript
    The basic idea is to put qualia into equations (broadly understood) to get what might as well be called qualations. Qualations arguably have different truth behaviors than the analogous equations. Thus ‘black’ has a different behavior than ‘ █ ’. This is a step in the direction of a ‘calculus of qualia’. It might help clarify some issues.
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  27. What is the Point of Persistent Disputes? The meta-analytic answer.Alexandre Billon & Philippe Vellozzo - forthcoming - Dialectica.
    Many philosophers regard the persistence of philosophical disputes as symptomatic of overly ambitious, ill-founded intellectual projects. There are indeed strong reasons to believe that persistent disputes in philosophy (and more generally in the discourse at large) are pointless. We call this the pessimistic view of the nature of philosophical disputes. In order to respond to the pessimistic view, we articulate the supporting reasons and provide a precise formulation in terms of the idea that the best explanation of persistent disputes entails (...)
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  28. Beyond the “Formidable Circle”: Race and the Limits of Democratic Inclusion in Tocqueville's Democracy in America.Christine Dunn Henderson - 2021 - Journal of Political Philosophy 30 (1):94-115.
  29. The static model of inventory management without a deficit with Neutrosophic logic.Maissam Jdid, Rafif Alhabib & A. A. Salama - 2021 - International Journal of Neutrosophic Science 16 (1):42-48.
    In this paper, we present an expansion of one of the well-known classical inventory management models, which is the static model of inventory management without a deficit and for a single substance, based on the neutrosophic logic, where we provide through this study a basis for dealing with all data, whether specific or undefined in the field of inventory management, as it provides safe environment to manage inventory without running into deficit , and give us an approximate ideal volume of (...)
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  30. Reading Davis and Fanon: A Creolizing Approach to Race, Gender and Sexuality.Gamal Abdel-Shehid - 2021 - Philosophy and Global Affairs 1 (2):343-350.
    The paper uses insights from Jane Anna Gordon’s Creolizing Political Theory to come up with a different way to read the work of Frantz Fanon in general and his discussion of gender and sexuality in particular. The paper argues against a hermetic reading of Fanon, one which reads him outside of context and influences. Instead of this close, or primary reading of Fanon, I offer a “conversation” between Fanon and the early work of Angela Y. Davis. The paper shows that (...)
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  31. Creolizing Theory in Conversation with Theorizing Race in the Americas.Juliet Hooker - 2021 - Philosophy and Global Affairs 1 (2):277-281.
    This review essay situates Jane Anna Gordon’s in light of methodological debates about the nature and role of “comparison.” Gordon repurposes the concept of “creolization” as a means for political theory to grapple with heterogeneity and mixture, not as discrete sets of thinkers and traditions, but as co-constituting. Gordon’s use of creolizing is then read alongside Hooker’s concept of juxtaposition as an alternative to comparison.
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  32. Race, Ethnicity and a Post-racial/ethnic Future.Ovett Nwosimiri - 2021 - Filosofia Theoretica 10 (2):159-174.
    Ethnicity and racial identity formation are elements of our social world. In recent years, there has been numerous works on ethnicity and race. Both concepts are controversial in different disciplines. The controversies around these concepts have been heated up by scholars who have devoted their time to the discourse of ethnicity and race, and to understand the ascription of both concepts. Ethnicity and race have been causes of conflict, prejudice and discrimination among various ethnic and racial groups around the world. (...)
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  33. Infinite Reasoning.Jared Warren - 2020 - Philosophy and Phenomenological Research 103 (2):385-407.
    Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I (...)
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  34. The Dream of Recapture.Carlo Nicolai - manuscript
    As a response to the semantic and logical paradoxes, theorists often reject some principles of classical logic. However, classical logic is entangled with mathematics, and giving up mathematics is too high a price to pay, even for nonclassical theorists. The so-called recapture theorems come to the rescue. When reasoning with concepts such as truth/class membership/property instantiation, if ones is interested in consequences of the theory that only contain mathematical vocabulary, nothing is lost by reasoning in the nonclassical framework. It is (...)
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  35. Taking Stock: Hale, Heck, and Wright on Neo-Logicism and Higher-Order Logic.Crispin Wright - 2021 - Philosophia Mathematica 29 (3): 392--416.
    ABSTRACT Four philosophical concerns about higher-order logic in general and the specific demands placed on it by the neo-logicist project are distinguished. The paper critically reviews recent responses to these concerns by, respectively, the late Bob Hale, Richard Kimberly Heck, and myself. It is argued that these score some successes. The main aim of the paper, however, is to argue that the most serious objection to the applications of higher-order logic required by the neo-logicist project has not been properly understood. (...)
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  36. William Boos. Metamathematics and the Philosophical Tradition.Brendan Larvor - 2021 - Philosophia Mathematica 29 (2):292-293.
  37. Mathematical Selves and the Shaping of Mathematical Modernism: Conflicting Epistemic Ideals in the Emergence of Enumerative Geometry.Nicolas Michel - 2021 - Isis 112 (1):68-92.
  38. Paradoxes and Inconsistent Mathematics.Zach Weber - 2021 - New York, NY: Cambridge University Press.
    Logical paradoxes – like the Liar, Russell's, and the Sorites – are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses “dialetheic paraconsistency” – a formal framework where some contradictions can be true without absurdity – as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly (...)
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  39. How (and Why) the Conservation of a Circle is the Core (and only) Dynamic in Nature.Ilexa Yardley - 2021 - Https://Medium.Com/the-Circular-Theory/.
    Solving Navier-Stokes and integrating it with Bose-Einstein. Moving beyond ‘mathematics’ and ‘physics.’ And, philosophy. Integrating 'point' 'line' 'circle.' (Euclid with 'reality.').
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  40. Hermann Cohen’s Principle of the Infinitesimal Method: A Defense.Scott Edgar - 2020 - Hopos: The Journal of the International Society for the History of Philosophy of Science 10 (2):440-470.
    In Bertrand Russell's 1903 Principles of Mathematics, he offers an apparently devastating criticism of the neo-Kantian Hermann Cohen's Principle of the Infinitesimal Method and its History (PIM). Russell's criticism is motivated by his concern that Cohen's account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum, and thus threatens the very idea of mathematical truth. This paper defends Cohen against that objection of Russell's, and argues that properly understood, Cohen's views of limits and infinitesimals (...)
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  41. Shadows of Syntax: Revitalizing Logical and Mathematical Conventionalism.Jared Warren - 2020 - New York, USA: Oxford University Press.
    What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...)
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  42. How can necessary facts call for explanation.Dan Baras - 2020 - Synthese 198 (12):11607-11624.
    While there has been much discussion about what makes some mathematical proofs more explanatory than others, and what are mathematical coincidences, in this article I explore the distinct phenomenon of mathematical facts that call for explanation. The existence of mathematical facts that call for explanation stands in tension with virtually all existing accounts of “calling for explanation”, which imply that necessary facts cannot call for explanation. In this paper I explore what theoretical revisions are needed in order to accommodate this (...)
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  43. On the Coherence of Strict Finitism.Auke Alesander Montesano Montessori - 2019 - Kriterion - Journal of Philosophy 33 (2):1-14.
    Strict finitism is the position that only those natural numbers exist that we can represent in practice. Michael Dummett, in a paper called Wang’s Paradox, famously tried to show that strict finitism is an incoherent position. By using the Sorites paradox, he claimed that certain predicates the strict finitist is committed to are incoherent. More recently, Ofra Magidor objected to Dummett’s claims, arguing that Dummett fails to show the incoherence of strict finitism. In this paper, I shall investigate whether Magidor (...)
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  44. The Residual Access Problem.Sharon Berry - manuscript
    A range of current truth-value realist philosophies of mathematics allow one to reduce the Benacerraf Problem to a problem concerning mathematicians' ability to recognize which conceptions of pure mathematical structures are coherent – in a sense which can be cashed out in terms of logical possibility. In this paper I will clarify what it takes to solve this `residual' access problem and then present a framework for solving it.
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  45. MANY 1 - A Transversal Imaginative Journey across the Realm of Mathematics.Jean-Yves Beziau - 2017 - Journal of the Indian Council of Philosophical Research 34 (2):259-287.
    We discuss the many aspects and qualities of the number one: the different ways it can be represented, the different things it may represent. We discuss the ordinal and cardinal natures of the one, its algebraic behaviour as a neutral element and finally its role as a truth-value in logic.
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  46. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
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  47. Cyberpunk entre literatura e matemática: processos comunicacionais da literatura massiva na crítica científica da realidade.Rafael Duarte Oliveira Venancio - 2013 - Conexão 12 (23).
    O presente artigo busca definir o movimento literário cyberpunk a partir da sua influência teórica vinda do campo da matemática. Utilizando a teorização interna ao movimento, centrada em Rudy Rucker, o objetivo aqui é entender como os campos da análise e dos fundamentos da matemática criam uma importante distinção entre os cyberpunks e as demais distopias literárias. Com isso, há a pressuposição de um movimento de uma crítica sociomatemática feita pelos cyberpunks cujos conceitos matemáticos tornam possível criticar o tempo presente, (...)
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  48. Regel und Witz. Wittgensteinsche Perspektiven auf Mathematik, Sprache und Moral. [REVIEW]Ulf Hlobil - 2010 - Zeitschrift für Philosophische Forschung 64 (3):416-419.
    Review of Timo-Peter Ertz's "Regel und Witz. Wittgensteinsche Perspektiven auf Mathematik, Sprache und Moral," Berlin & New York: de Gruyter, 2008.
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  49. Automated Theorem Proving and Its Prospects. [REVIEW]Desmond Fearnley-Sander - 1995 - PSYCHE: An Interdisciplinary Journal of Research On Consciousness 2.
    REVIEW OF: Automated Development of Fundamental Mathematical Theories by Art Quaife. (1992: Kluwer Academic Publishers) 271pp. Using the theorem prover OTTER Art Quaife has proved four hundred theorems of von Neumann-Bernays-Gödel set theory; twelve hundred theorems and definitions of elementary number theory; dozens of Euclidean geometry theorems; and Gödel's incompleteness theorems. It is an impressive achievement. To gauge its significance and to see what prospects it offers this review looks closely at the book and the proofs it presents.
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  50. What Mathematical Truth Could Not Be--1.Paul Benacerraf - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press.
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