About this topic
Summary Philosophical discussions about mathematics have a long history, which basically coincides with the history of philosophy. The main historiographic divisions are thus the same as for philosophy in general, i.e. there is philosophy of mathematics in Ancient Philosophy, in Medieval Philosophy, in Early Modern Philosophy (16th-18th centuries), and in Late Modern Philosophy (19th-20th centuries). For a general introduction to the topic, including source material, see R. Marcus and M. McEvoy, eds., A Historical Introduction to the Philosophy of Mathematics: A Reader (Bloomsbury, 2016). For excerpts and translations from crucial authors since Kant, compare W. Ewald, ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vols. I-II (Oxford University Press, 1996).  And for the late 19th and the first half of the 20th centuries, see P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics: Selected Readings (2nd ed., Cambridge University Press, 1984).
Key works Logicism, formalism, intuitionism, structuralism, foundations, logic, proof, truth, axioms, infinity.
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  1. Frege's Basic Law V and Cantor's Theorem.Manuel Bremer - manuscript
    The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological (...)
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  2. Lectures on Complex Numbers and their Functions, Part I: Theory of Complex Number Systems.Hermann Hankel & Richard Lawrence - manuscript - Translated by Richard Lawrence.
    A transcription and translation of Hermann Hankel's 1867 Vorlesungen über die complexen Zahlen und ihre Functionen, I. Theil: Theorie der Complexen Zahlensysteme, a textbook on complex analysis that played an important role in the transition to modern mathematics in nineteenth century Germany.
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  3. On some historical aspects of the theory of Riemann zeta function.Giuseppe Iurato - manuscript
    This comprehensive historical account concerns that non-void intersection region between Riemann zeta function and entire function theory, with a view towards possible physical applications.
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  4. Wittgenstein's philosophy of mathematics.Victor Rodych - unknown - Stanford Encyclopedia of Philosophy.
  5. Conceptions of infinity and set in Lorenzen’s operationist system.Carolin Antos - forthcoming - In Logic, Epistemology and the Unity of Science. Springer.
    In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...)
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  6. Ms.Natasha Bailie - forthcoming - British Journal for the History of Mathematics.
    The reception of Newton's Principia in 1687 led to the attempt of many European scholars to ‘mathematicise' their field of expertise. An important example of this ‘mathematicisation' lies in the work of Irish-Scottish philosopher Francis Hutcheson, a key figure in the Scottish Enlightenment. This essay aims to discuss the mathematical aspects of Hutcheson's work and its impact on British thought in the following centuries, providing a case in point for the importance of the interactions between mathematics and philosophy throughout time.
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  7. Brouwer's Intuition of Twoity and Constructions in Separable Mathematics.Bruno Bentzen - forthcoming - History and Philosophy of Logic:1-21.
    My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of (...)
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  8. Russell Marcus and Mark McEvoy, eds. An Historical Introduction to the Philosophy of Mathematics: A Reader.James Robert Brown - forthcoming - Philosophia Mathematica:nkw033.
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  9. On the Depth of Gödel’s Incompleteness Theorems.Yong Cheng - forthcoming - Philosophia Mathematica.
    ABSTRACT We use Gödel’s incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel’s incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel’s incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel’s incompleteness theorems.
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  10. Hilbert on number, geometry and continuity.M. Hallett - forthcoming - Bulletin of Symbolic Logic.
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  11. Wittgenstein and Other Philosophers: His Influence on Historical and Contemporary Analytic Philosophers (Volume II).Ali Hossein Khani & Gary Kemp (eds.) - forthcoming - Routledge.
    This edited volume includes 49 Chapters, each of which discusses the influence of a philosopher's reading of Wittgenstein in his/her philosophical works and the way such Wittgensteinian ideas have manifested themselves in those works.
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  12. Frege and Peano on definitions.Edoardo Rivello - forthcoming - In Proceedings of the "Frege: Freunde und Feinde" conference, held in Wismar, May 12-15, 2013.
    Frege and Peano started in 1896 a debate where they contrasted the respective conceptions on the theory and practice of mathematical definitions. Which was (if any) the influence of the Frege-Peano debate on the conceptions by the two authors on the theme of defining in mathematics and which was the role played by this debate in the broader context of their scientific interaction?
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  13. Ian Hacking, Why Is There Philosophy of Mathematics at All? [REVIEW]Max Harris Siegel - forthcoming - Mind 124.
  14. Du Châtelet’s Philosophy of Mathematics.Aaron Wells - forthcoming - In Fatema Amijee (ed.), The Bloomsbury Handbook of Du Châtelet. Bloomsbury.
    I begin by outlining Du Châtelet’s ontology of mathematical objects: she is an idealist, and mathematical objects are fictions dependent on acts of abstraction. Next, I consider how this idealism can be reconciled with her endorsement of necessary truths in mathematics, which are grounded in essences that we do not create. Finally, I discuss how mathematics and physics relate within Du Châtelet’s idealism. Because the primary objects of physics are partly grounded in the same kinds of acts as yield mathematical (...)
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  15. Competing Roles of Aristotle's Account of the Infinite.Robby Finley - 2024 - Apeiron 57 (1):25-54.
    There are two distinct but interrelated questions concerning Aristotle’s account of infinity that have been the subject of recurring debate. The first of these, what I call here the interpretative question, asks for a charitable and internally coherent interpretation of the limited pieces of text where Aristotle outlines his view of the ‘potential’ (and not ‘actual’) infinite. The second, what I call here the philosophical question, asks whether there is a way to make Aristotle’s notion of the potential infinite coherent (...)
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  16. Cultures of Mathematical Practice in Alexandria in Egypt: Claudius Ptolemy and His Commentators (Second–Fourth Century CE).Alberto Bardi - 2023 - In Handbook of the History and Philosophy of Mathematical Practice.
    Claudius Ptolemy’s mathematical astronomy originated in Alexandria in Egypt under Roman rule in the second century CE and held for more than a millennium, even beyond the Copernican theories (sixteenth century). To trace the flourishing of such mathematical creativity requires an understanding of Ptolemy’s philosophy of mathematical practice, the ancient commentators of Ptolemaic works, and the historical context of Alexandria in Egypt, a multicultural city which became a cradle of cultures of mathematical practices and blossomed into the Ptolemaic system and (...)
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  17. The Turning Point in Wittgenstein’s Philosophy of Mathematics: Another Turn.Yemima Ben-Menahem - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Springer. pp. 377-393.
    According to Mark Steiner, Wittgenstein’s intense work in the philosophy of mathematics during the early 1930s brought about a distinct turning point in his philosophy. The crux of this transition, Steiner contends, is that Wittgenstein came to see mathematical truths as originating in empirical regularities that in the course of time have been hardened into rules. This interpretation, which construes Wittgenstein’s later philosophy of mathematics as more realist than his earlier philosophy, challenges another influential interpretation which reads Wittgenstein as moving (...)
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  18. Listy Gottloba Fregego. Uwagi o polskim wydaniu [rec. Gottlob Frege: Korespondencja naukowa]. [REVIEW]Krystian Bogucki - 2023 - Folia Philosophica 48:1-24. Translated by Andrzej Painta, Marta Ples-Bęben, Mateusz Jurczyński & Lidia Obojska.
    The present article reviews the Polish-language edition of Gottlob Frege’s scientific correspondence. In the article, I discuss the material hitherto unpublished in Polish in relation to the remainder of Frege’s works. First of all, I inquire into the role and nature of definitions. Then, I consider Frege’s recognition criteria for sameness of thoughts. In the article’s third part, I study letters devoted to the principle of semantic compositionality, while in the fourth part I discuss Frege’s remarks concerning the context principle.
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  19. Diagrams, Visual Imagination, and Continuity in Peirce's Philosophy of Mathematics.Vitaly Kiryushchenko - 2023 - New York, NY, USA: Springer.
    This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of (...)
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  20. Wittgenstein, Russell, and Our Concept of the Natural Numbers.Saul A. Kripke - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Springer. pp. 137-155.
    Wittgenstein gave a clearly erroneous refutation of Russell’s logicist project. The errors were ably pointed out by Mark Steiner. Nevertheless, I was motivated by Wittgenstein and Steiner to consider various ideas about the natural numbers. I ask which notations for natural numbers are ‘buck-stoppers’. For us it is the decimal notation and the corresponding verbal system. Based on the idea that a proper notation should be ‘structurally revelatory’, I draw various conclusions about our own concept of the natural numbers.
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  21. Frege, Thomae, and Formalism: Shifting Perspectives.Richard Lawrence - 2023 - Journal for the History of Analytical Philosophy 11 (2):1-23.
    Mathematical formalism is the the view that numbers are "signs" and that arithmetic is like a game played with such signs. Frege's colleague Thomae defended formalism using an analogy with chess, and Frege's critique of this analogy has had a major influence on discussions in analytic philosophy about signs, rules, meaning, and mathematics. Here I offer a new interpretation of formalism as defended by Thomae and his predecessors, paying close attention to the mathematical details and historical context. I argue that (...)
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  22. Elaine Landry.*Plato Was Not a Mathematical Platonist.Colin McLarty - 2023 - Philosophia Mathematica 31 (3):417-424.
    This book goes far beyond its title. Landry indeed surveys current definitions of “mathematical platonism” to show nothing like them applies to Socrates in Plat.
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  23. 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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  24. За игрой в карты с чертиком Визинга.Brian Rabern & Landon Rabern - 2023 - Kvant 2023 (10):2-6.
    We analyze a solitaire game in which a demon rearranges some cards after each move. The graph edge coloring theorems of K˝onig (1931) and Vizing (1964) follow from the winning strategies developed.
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  25. Leibniz on Number Systems.Lloyd Strickland - 2023 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 1-31.
    This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that were (...)
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  26. Why Did Leibniz Invent Binary?Lloyd Strickland - 2023 - In Wenchao Li, Charlotte Wahl, Sven Erdner, Bianca Carina Schwarze & Yue Dan (eds.), »Le present est plein de l’avenir, et chargé du passé«. Hannover: Gottfried-Wilhelm-Leibniz-Gesellschaft e.V.. pp. 354-360.
  27. Why Did Thomas Harriot Invent Binary?Lloyd Strickland - 2023 - Mathematical Intelligencer 46 (1):57-62.
    From the early eighteenth century onward, primacy for the invention of binary numeration and arithmetic was almost universally credited to the German polymath Gottfried Wilhelm Leibniz (1646–1716). Then, in 1922, Frank Vigor Morley (1899–1980) noted that an unpublished manuscript of the English mathematician, astronomer, and alchemist Thomas Harriot (1560–1621) contained the numbers 1 to 8 in binary. Morley’s only comment was that this foray into binary was “certainly prior to the usual dates given for binary numeration”. Almost thirty years later, (...)
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  28. Paul Cohen’s philosophy of mathematics and its reflection in his mathematical practice.Roy Wagner - 2023 - Synthese 202 (2):1-22.
    This paper studies Paul Cohen’s philosophy of mathematics and mathematical practice as expressed in his writing on set-theoretic consistency proofs using his method of forcing. Since Cohen did not consider himself a philosopher and was somewhat reluctant about philosophy, the analysis uses semiotic and literary textual methodologies rather than mainstream philosophical ones. Specifically, I follow some ideas of Lévi-Strauss’s structural semiotics and some literary narratological methodologies. I show how Cohen’s reflections and rhetoric attempt to bridge what he experiences as an (...)
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  29. “In Nature as in Geometry”: Du Châtelet and the Post-Newtonian Debate on the Physical Significance of Mathematical Objects.Aaron Wells - 2023 - In Wolfgang Lefèvre (ed.), Between Leibniz, Newton, and Kant: Philosophy and Science in the Eighteenth Century. Springer Verlag. pp. 69-98.
    Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in this (...)
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  30. A Reassessment of Cantorian Abstraction based on the $$\varepsilon $$ ε -operator.Nicola Bonatti - 2022 - Synthese 200 (5):1-26.
    Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s \-operator. Moreover, it is argued that the BK system and the \-operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first (...)
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  31. Numbers and Narratives — or : When Russell meets Zhu Shijie to discuss Philosophy of Mathematics.Andrea Bréard - 2022 - Laval Théologique et Philosophique 78 (3):365-384.
    Andrea Bréard Même si aucune source métadiscursive sur les mathématiques elles-mêmes n’a été transmise de la Chine ancienne et prémoderne, des réflexions ont été menées sur les objets mathématiques et sur la boîte à outils des praticiens. Cet article montre comment elles sont insérées dans les traités mêmes en prenant en particulier l’exemple d’un domaine de la théorie des nombres qui a évolué en Chine du premier à la fin du xixe siècle. Ces réflexions sont dispersées entre textes, paratextes et (...)
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  32. Descartes et ses mathématiques.Olivia Chevalier (ed.) - 2022 - Paris: Classiques Garnier.
    Dans cet ouvrage, il s'agira non seulement d'aborder différentes facettes de l'activité mathématique de Descartes, assez peu connues, mais également diverses dimensions de sa pensée mathématique.
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  33. Degeneration and Entropy.Eugene Y. S. Chua - 2022 - Kriterion - Journal of Philosophy 36 (2):123-155.
    [Accepted for publication in Lakatos's Undone Work: The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, special issue of Kriterion: Journal of Philosophy. Edited by S. Nagler, H. Pilin, and D. Sarikaya.] Lakatos’s analysis of progress and degeneration in the Methodology of Scientific Research Programmes is well-known. Less known, however, are his thoughts on degeneration in Proofs and Refutations. I propose and motivate two new criteria for degeneration based on the discussion in Proofs and Refutations (...)
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  34. V.A. Yankov on Non-Classical Logics, History and Philosophy of Mathematics.Alex Citkin & Ioannis M. Vandoulakis (eds.) - 2022 - Springer, Outstanding Contributions To Logic (volume 24).
    This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, the theory (...)
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  35. Teológia a matematika v kontexte paradigmatických zmien renesančnej a ranonovovekej kozmológie a fyziky.Gašpar Fronc - 2022 - Bratislava: Univerzita Komenského v Bratislave.
    The publication offers an interdisciplinary and historical approach to the questions of exploration of the world with an emphasis on paradigm changes during the Renaissance and early modern times, leading to new concepts that we can accept as the beginning of the natural sciences in our current understanding. The main goal is to point out the connections between the paradigms of mathematics, theology and natural sciences, the connection of which is for the main protagonists an essential factor in the formation (...)
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  36. Mathematics embodied: Merleau-Ponty on geometry and algebra as fields of motor enaction.Jan Halák - 2022 - Synthese 200 (1):1-28.
    This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for Merleau-Ponty, (...)
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  37. Mirja Hartimo* Husserl and Mathematics.Jairo José da Silva - 2022 - Philosophia Mathematica 30 (3):396-414.
    1. INTRODUCTIONIt has been some time now since the philosophical community has learned to appreciate Husserl’s contribution to the philosophies of logic, mathematics, and science in general, despite still some prejudices and misinterpretations in certain academic circles incapable of reading Husserl beyond the incompetent and malicious review which Frege wrote in 1894 of his Philosophie der Arithmetik (PA) [1891/2003], hereafter Hua XII.Husserl’s philosophy of mathematics, in particular, has been the subject of many articles and books and has attracted the attention (...)
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  38. The collapse of the Hilbert program: A variation on the gödelian theme.Saul A. Kripke - 2022 - Bulletin of Symbolic Logic 28 (3):413-426.
    The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to ∃xA(x), or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent. Here we (...)
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  39. Paul Rusnock and Jan Šebestík. Bernard Bolzano: His Life and His Work.Sandra Lapointe - 2022 - Philosophia Mathematica 30 (1):138-140.
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  40. Carnap's philosophy of mathematics.Benjamin Marschall - 2022 - Philosophy Compass 17 (11):e12884.
    For several decades, Carnap's philosophy of mathematics used to be either dismissed or ignored. It was perceived as a form of linguistic conventionalism and thus taken to rely on the bankrupt notion of truth by convention. However, recent scholarship has revealed a more subtle picture. It has been forcefully argued that Carnap is not a linguistic conventionalist in any straightforward sense, and that supposedly decisive objections against his position target a straw man. This raises two questions. First, how exactly should (...)
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  41. Lakatos' Undone Work: The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science_ - Introduction to the Special Issue on _Lakatos’ Undone Work.Sophie Nagler, Hannah Pillin & Deniz Sarikaya - 2022 - Kriterion - Journal of Philosophy 36:1-10.
    We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, which gave rise to this special issue. Lastly, (...)
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  42. Conceptual Frameworks on the Relationship Between Physics–Mathematics in the Newton Principia Geneva Edition (1822).Raffaele Pisano & Paolo Bussotti - 2022 - Foundations of Science 27 (3).
    The aim of this paper is twofold: (1) to show the principal aspects of the way in which Newton conceived his mathematical concepts and methods and applied them to rational mechanics in his Principia; (2) to explain how the editors of the Geneva Edition interpreted, clarified, and made accessible to a broader public Newton’s perfect but often elliptic proofs. Following this line of inquiry, we will explain the successes of Newton’s mechanics, but also the problematic aspects of his perfect geometrical (...)
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  43. 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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  44. Paul Weingartner and Hans-Peter Leeb, eds, Kreisel’s Interests: On the Foundations of Logic and Mathematics.Dag Prawitz - 2022 - Philosophia Mathematica 30 (1):121-126.
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  45. Kuhn, Lakatos, and the Historical Turn in the Philosophy of Mathematics.Vladislav A. Shaposhnikov - 2022 - Epistemology and Philosophy of Science 59 (4):144-162.
    The paper deals with Kuhn’s and Lakatos’s ideas related to the so-called “historical turn” and its application to the philosophy of mathematics. In the first part the meaning of the term “postpositivism” is specified. If we lack such a specification we can hardly discuss the philosophy of science that comes “after postpositivism”. With this end in view, the metaphor of “generations” in the philosophy of science is used. It is proposed that we restrict the use of the term “post-positivism” to (...)
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  46. Resolving Frege’s Other Puzzle.Eric Snyder, Richard Samuels & Stewart Shapiro - 2022 - Philosophica Mathematica 30 (1):59-87.
    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 instead argue that (...)
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  47. The story of proof: logic and the history of mathematics.John Stillwell - 2022 - Princeton, New Jersey: Princeton University Press.
    How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on the (...)
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  48. Two Lost Operations of Arithmetic: Duplation and Mediation.Lloyd Strickland - 2022 - Mathematics Today 65:212-213.
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  49. On V.A. Yankov’s Hypothesis of the Rise of Greek Mathematics.Ioannis M. Vandoulakis - 2022 - In V.A. Yankov on Non-Classical Logics, History and Philosophy of Mathematics. Springer, Outstanding Contributions to Logic (Volume 24). pp. 295-310.
    The paper examines the main points of Yankov’s hypothesis on the rise of Greek mathematics. The novelty of Yankov’s interpretation is that the rise of mathematics is examined within the context of the rise of ontological theories of the early Greek philosophers, which mark the beginning of rational thinking, as understood in the Western tradition.
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  50. Are Euclid's Diagrams Representations? On an Argument by Ken Manders.David Waszek - 2022 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics. The CSHPM 2019-2020 Volume. Birkhäuser. pp. 115-127.
    In his well-known paper on Euclid’s geometry, Ken Manders sketches an argument against conceiving the diagrams of the Elements in ‘semantic’ terms, that is, against treating them as representations—resting his case on Euclid’s striking use of ‘impossible’ diagrams in some proofs by contradiction. This paper spells out, clarifies and assesses Manders’s argument, showing that it only succeeds against a particular semantic view of diagrams and can be evaded by adopting others, but arguing that Manders nevertheless makes a compelling case that (...)
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