Philosophy of Mathematics

Edited by Øystein Linnebo (University of Oslo)
Assistant editor: Sam Roberts (Universität Konstanz)
54 found
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1 — 50 / 54
  1. added 2021-06-11
    The Barber, Russell's Paradox, Catch-22, God, Contradiction, and More.Laurence Goldstein - 2006 - In Graham Priest, J. C. Beall & Bradley Armour-Garb (eds.), The Law of Non-Contradiction: New Philosophical Essays. Clarendon Press.
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  2. added 2021-06-11
    Our Knowledge of Mathematical Objects.Kit Fine - 2005 - In Tamar Szabo Gendler & John Hawthorne (eds.), Oxford Studies in Epistemology Volume 1. Oxford University Press.
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  3. added 2021-06-09
    Virtue Theory of Mathematical Practices: An Introduction.Andrew Aberdein, Colin Jakob Rittberg & Fenner Stanley Tanswell - forthcoming - Synthese.
    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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  4. added 2021-06-04
    Coalgebra And Abstraction.Graham Leach-Krouse - 2021 - Notre Dame Journal of Formal Logic 62 (1):33-66.
    Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of second-order ZFC via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to identifying an initial F-algebra in a (...)
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  5. added 2021-06-02
    Are Infinite Explanations Self-Explanatory?Alexandre Billon - forthcoming - Erkenntnis.
    Consider an infinite series whose items are each explained by their immediate successor. Does such an infinite explanation explain the whole series or does it leave something to be explained? Hume arguably claimed that it does fully explain the whole series. Leibniz, however, designed a very telling objection against this claim, an objection involving an infinite series of book copies. In this paper, I argue that the Humean claim can, in certain cases, be saved from the Leibnizian “infinite book copies” (...)
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  6. added 2021-06-02
    Babbage's Guidelines for the Design of Mathematical Notations.Dirk Schlimm & Jonah Dutz - 2021 - Studies in History and Philosophy of Science Part A 1 (88):92–101.
    The design of good notation is a cause that was dear to Charles Babbage's heart throughout his career. He was convinced of the "immense power of signs" (1864, 364), both to rigorously express complex ideas and to facilitate the discovery of new ones. As a young man, he promoted the Leibnizian notation for the calculus in England, and later he developed a Mechanical Notation for designing his computational engines. In addition, he reflected on the principles that underlie the design of (...)
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  7. added 2021-06-02
    Dedekind et la crèation du continu arithmétique.Emmylou Haffner & Dirk Schlimm - 2021 - In Emmylou Haffner & David Rabouin (eds.), L'Épistemologie du dedans. Mélanges en l'honneur de Hourya Benis-Sinaceur. Paris, France: pp. 341–378.
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  8. added 2021-06-02
    Dedekind on Continuity.Emmylou Haffner & Dirk Schlimm - 2021 - In Stewart Shapiro & Geoffrey Hellman (eds.), The History of Continua. Philosophical and mathematical perspectives. New York, NY, USA: pp. 255–282.
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  9. added 2021-06-02
    The Correspondence Between Moritz Pasch and Felix Klein.Dirk Schlimm - 2013 - Historia Mathematica 2 (40):183-202.
    The extant correspondence, consisting of ten letters from the period from 1882 to 1902, from Moritz Pasch to Felix Klein is presented together with an English translation and a short introduction. These letters provide insights into the views of Pasch and Klein regarding the role of intuition and axioms in mathematics, and also into the hiring practices of mathematics professors in the 1880s.
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  10. added 2021-05-31
    Review of John Heil, The Universe As We Find It. [REVIEW]Alyssa Ney - 2014 - British Journal for the Philosophy of Science 65:881-886.
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  11. added 2021-05-29
    Halting Problem Undecidability and Infinitely Nested Simulation.P. Olcott - manuscript
    When halting is defined as any computation that halts without ever having its simulation aborted then it can be understood that partial halt decider H correctly decides that its input does not halt on the simplified version of the Linz Ĥ. -/- When this simplified concrete example is fully understood then the exact same reasoning is applied to the actual Linz Ĥ correctly deciding that it would never halt when applied to its own Turing machine description.
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  12. added 2021-05-28
    Categoricity by Convention.Julien Murzi & Brett Topey - forthcoming - Philosophical Studies.
    On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity theorem (...)
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  13. added 2021-05-26
    Review of Theodore Sider's The Tools of Metaphysics and the Metaphysics of Science. [REVIEW]T. Scott Dixon - 2021 - Notre Dame Philosophical Reviews.
  14. added 2021-05-26
    Abstract Objects and Semantics: An Essay on Prospects and Problems with Abstraction Principles as a Means of Justifying Reference to Abstract Objects.Gnatek Zuzanna - 2020 - Dissertation, Trinity College, Dublin
  15. added 2021-05-25
    Knowledge, Number and Reality: Encounters with the Work of Keith Hossack.Nils Kürbis, Bahram Assadian & Jonathan Nassim (eds.) - forthcoming - London: Bloomsbury.
    Centered around our knowledge of mathematical, modal, and a priori truths, this is a collection that celebrates the work of Keith Hossack, who, throughout his career, has made outstanding contributions to the theory of knowledge, metaphysics, and the philosophy of mathematics. Starting with a focus on our knowledge of abstract entities such as mathematical objects and the source of the necessity of mathematical truths, attention moves to the notion of necessity and its interaction with a priori knowledge: Is it the (...)
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  16. added 2021-05-23
    Towards a New Philosophical Perspective on Hermann Weyl’s Turn to Intuitionism.Kati Kish Bar-On - forthcoming - Science in Context.
    The paper explores Hermann Weyl’s turn to intuitionism through a philosophical prism of normative framework transitions. It focuses on three central themes that occupied Weyl’s thought: the notion of the continuum, logical existence, and the necessity of intuitionism, constructivism, and formalism to adequately address the foundational crisis of mathematics. The analysis of these themes reveals Weyl’s continuous endeavor to deal with such fundamental problems and suggests a view that provides a different perspective concerning Weyl’s wavering foundational positions. Building on a (...)
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  17. added 2021-05-21
    Peano on Symbolization, Design Principles for Notations, and the Dot Notation.Dirk Schlimm - 2021 - Philosophia Scientae 25:95-126.
    Peano was one of the driving forces behind the development of the current mathematical formalism. In this paper, we study his particular approach to notational design and present some original features of his notations. To explain the motivations underlying Peano's approach, we first present his view of logic as a method of analysis and his desire for a rigorous and concise symbolism to represent mathematical ideas. On the basis of both his practice and his explicit reflections on notations, we discuss (...)
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  18. added 2021-05-21
    Multiple Readability in Principle and Practice: Existential Graphs and Complex Symbols.Dirk Schlimm & David Waszek - 2020 - Logique Et Analyse 251:231-260.
    Since Sun-Joo Shin's groundbreaking study (2002), Peirce's existential graphs have attracted much attention as a way of writing logic that seems profoundly different from our usual logical calculi. In particular, Shin argued that existential graphs enjoy a distinctive property that marks them out as "diagrammatic": they are "multiply readable," in the sense that there are several di erent, equally legitimate ways to translate one and the same graph into a standard logical language. Stenning (2000) and Bellucci and Pietarinen (2016) have (...)
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  19. added 2021-05-21
    Iconicity in Mathematical Notation: Commutativity and Symmetry.Theresa Wege, Sophie Batchelor, Matthew Inglis, Honali Mistry & Dirk Schlimm - 2020 - Journal of Numerical Cognition 3 (6):378-392.
    Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects—those which visually resemble in some way the concepts they represent—offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative (...)
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  20. added 2021-05-11
    On the Buck-Stopping Identification of Numbers.Dongwoo Kim - forthcoming - Philosophia Mathematica.
    Kripke observes that the decimal numerals have the buck-stopping property: when a number is given in decimal notation, there is no further question of what number it is. What makes them special in this way? According to Kripke, it is because of structural revelation: each decimal numeral represents the structure of the corresponding number. Though insightful, I argue, this account has some counterintuitive consequences. Then I sketch an alternative account of the buck-stopping property in terms of how we specify the (...)
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  21. added 2021-05-10
    Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups.Joshua Habgood-Coote & Fenner Stanley Tanswell - forthcoming - Episteme.
    In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little (...)
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  22. added 2021-05-10
    Can We Have Physical Understanding of Mathematical Facts?Gabriel Tȃrziu - forthcoming - Acta Analytica:1-24.
    A lot of philosophical energy has been devoted recently in trying to determine if mathematics can contribute to our understanding of physical phenomena. Not many philosophers are interested, though, if the converse makes sense, i.e., if our cognitive interaction (scientific or otherwise) with the physical world can be helpful (in an explanatory or non-explanatory way) in our efforts to make sense of mathematical facts. My aim in this paper is to try to fill this important lacuna in the recent literature. (...)
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  23. added 2021-05-07
    How Do We Semantically Individuate Natural Numbers?†.Stefan Buijsman - forthcoming - Philosophia Mathematica.
    ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...)
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  24. added 2021-05-07
    William Boos. Metamathematics and the Philosophical Tradition.Brendan Larvor - forthcoming - Philosophia Mathematica.
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  25. added 2021-05-07
    Rainer Stuhlmann-Laeisz.* Gottlob Freges Grundgesetze der Arithmetik: Ein Kommentar des Vorworts, des Nachworts Und der Einleitenden Paragraphen. [Gottlob Frege’s Basic Laws of Arithmetic: A Commentary on the Foreword, the Afterword and the Introductory Paragraphs].Matthias Wille - forthcoming - Philosophia Mathematica.
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  26. added 2021-05-03
    Quine, New Foundations, and the Philosophy of Set Theory by Sean Morris. [REVIEW]Gregory Lavers - 2021 - Journal of the History of Philosophy 59 (2):342-343.
    This book has two main goals: first, to show that Quine's New Foundations set theory is better motivated than often assumed; and second, to defend Quine's philosophy of set theory. It is divided into three parts. The first concerns the history of set theory and argues against readings that see the iterative conception of set being the dominant notion of set from the very beginning. The second part concerns Quine's philosophy of set theory. Part 3 is a contemporary assessment of (...)
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  27. added 2021-05-03
    Definition in Frege's Foundations of Arithmetic.David Hunter - 1996 - Pacific Philosophical Quarterly 77:88-107.
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  28. added 2021-05-01
    Philosophie de la simulation et finitude.Franck Varenne - 2021 - Revue Philosophique de la France Et de l'Etranger 2 (146):183-201.
    This study shows firstly that it is necessary to characterize a computer simulation at a finer level than that of formal models: that of symbols and their various modes of reference. This is particularly true for those that integrate models and formalisms of a heterogeneous nature. This study then examines the ontological causes that, consequently, could explain their epistemic success. It is argued that they can be conveniently explained if one adopts a conception of nature that is both discontinuous and (...)
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  29. added 2021-04-30
    Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I).Jaykov Foukzon - 2021 - Journal of Advances in Mathematics and Computer Science 36 (2):73-88.
    In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.
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  30. added 2021-04-29
    Takeuti's well-ordering proofs revisited.Andrew Arana & Ryota Akiyoshi - 2021 - Mita Philosophy Society 3 (146):83-110.
    Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...)
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  31. added 2021-04-28
    ‘Let No-One Ignorant of Geometry…’: Mathematical Parallels for Understanding the Objectivity of Ethics.James Franklin - 2021 - Journal of Value Inquiry 55:1-20.
    It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so on). That (...)
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  32. added 2021-04-27
    Øystein Vs Archimedes: A Note on Linnebo’s Infinite Balance.Daniel Hoek - forthcoming - Erkenntnis:1-6.
    Using Riemann’s Rearrangement Theorem, Øystein Linnebo (2020) argues that, if it were possible to apply an infinite positive weight and an infinite negative weight to a working scale, the resulting net weight could end up being any real number, depending on the procedure by which these weights are applied. Appealing to the First Postulate of Archimedes’ treatise on balance, I argue instead that the scale would always read 0 kg. Along the way, we stop to consider an infinitely jittery flea, (...)
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  33. added 2021-04-26
    Core Knowledge of Geometry Can Develop Independently of Visual Experience.Benedetta Heimler, Tomer Behor, Stanislas Dehaene, Véronique Izard & Amir Amedi - 2021 - Cognition 212:104716.
    Geometrical intuitions spontaneously drive visuo-spatial reasoning in human adults, children and animals. Is their emergence intrinsically linked to visual experience, or does it reflect a core property of cognition shared across sensory modalities? To address this question, we tested the sensitivity of blind-from-birth adults to geometrical-invariants using a haptic deviant-figure detection task. Blind participants spontaneously used many geometric concepts such as parallelism, right angles and geometrical shapes to detect intruders in haptic displays, but experienced difficulties with symmetry and complex spatial (...)
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  34. added 2021-04-23
    The Number Sense Represents (Rational) Numbers.Sam Clarke & Jacob Beck - forthcoming - Behavioral and Brain Sciences:1-32.
    On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique—the arguments from congruency, confounds, and imprecision—and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as (...)
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  35. added 2021-04-20
    Counterexample Search in Diagram‐Based Geometric Reasoning.Yacin Hamami, John Mumma & Marie Amalric - 2021 - Cognitive Science 45 (4):e12959.
    Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an (...)
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  36. added 2021-04-18
    Comparing Mathematical Explanations.Isaac Wilhelm - forthcoming - British Journal for the Philosophy of Science.
    Philosophers have developed several detailed accounts of what makes some mathematical proofs explanatory. Significantly less attention has been paid, however, to what makes some proofs more explanatory than other proofs. That is problematic, since the reasons for thinking that some proofs explain are also reasons for thinking that some proofs are more explanatory than others. So in this paper, I develop an account of comparative explanation in mathematics. I propose a theory of the `at least as explanatory as' relation among (...)
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  37. added 2021-04-16
    A Validation of Knowledge: A New, Objective Theory of Axioms, Causality, Meaning, Propositions, Mathematics, and Induction.Ronald Pisaturo - 2020 - Norwalk, Connecticut: Prime Mover Press.
    This book seeks to offer original answers to all the major open questions in epistemology—as indicated by the book’s title. These questions and answers arise organically in the course of a validation of the entire corpus of human knowledge. The book explains how we know what we know, and how well we know it. The author presents a positive theory, motivated and directed at every step not by a need to reply to skeptics or subjectivists, but by the need of (...)
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  38. added 2021-04-12
    Studies in No-Self Physicalism.Feng Ye - manuscript
    This book develops and defends a version of physicalism in contemporary philosophy of mind, called ‘No-Self Physicalism’. No-Self Physicalism emphasizes that a subject of cognition is itself a physical entity, a human brain (and body). -/- The book first argues (in Chapters 1 and 2) that many contemporary philosophers who openly accept physicalism in fact (though perhaps unconsciously and/or implicitly) take the stance of a non-physical Subject in understanding and using core philosophical notions, such as conceptual representation, truth, analyticity, modality, (...)
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  39. added 2021-04-12
    A Neglected Chapter in the History of Philosophy of Mathematical Thought Experiments: Insights From Jean Piaget’s Reception of Edmond Goblot.Marco Buzzoni - 2021 - Hopos: The Journal of the International Society for the History of Philosophy of Science 11 (1):282-304.
  40. added 2021-04-07
    Intrinsic Justifications for Large-Cardinal Axioms.Rupert McCallum - forthcoming - Philosophia Mathematica:nkaa038.
    ABSTRACT The aim of the paper is to answer some arguments raised against mathematical structuralism developed by Michael Resnik. These arguments stress the abstractness of mathematical objects, especially their causal inertness, and conclude that mathematical objects, the structures posited by Resnik included, are inaccessible to human cognition. In the paper I introduce a distinction between abstract and ideal objects and argue that mathematical objects are primarily ideal. I reconstruct some aspects of the instrumental practice of mathematics, such as symbolic manipulations (...)
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  41. added 2021-04-06
    Kreisel's Interests. On the Foundations of Logic and Mathematics.Paul Weingartner & Hans-Peter Leeb (eds.) - 2020 - London, Vereinigtes Königreich: College Publications.
    The contributions to this volume are from participants of the international conference "Kreisel's Interests - On the Foundations of Logic and Mathematics", which took place from 13 to 14 2018 at the University of Salzburg in Salzburg, Austria. The contributions have been revised and partially extended. Among the contributors are Akihiro Kanamori, Göran Sundholm, Ulrich Kohlenbach, Charles Parsons, Daniel Isaacson, and Kenneth Derus. The contributions cover the discussions between Kreisel and Wittgenstein on philosophy of mathematics, Kreisel's Dictum, proof theory, the (...)
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  42. added 2021-04-04
    Strict Finitism and the Logic of Mathematical Applications, Synthese Library, Vol. 355.Feng Ye - 2011 - Springer.
    This book intends to show that, in philosophy of mathematics, radical naturalism (or physicalism), nominalism and strict finitism (which does not assume the reality of infinity in any format, not even potential infinity) can account for the applications of classical mathematics in current scientific theories about the finite physical world above the Planck scale. For that purpose, the book develops some significant applied mathematics in strict finitism, which is essentially quantifier-free elementary recursive arithmetic (with real numbers encoded as elementary recursive (...)
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  43. added 2021-04-03
    Homotopy Type Theory and Structuralism.Teruji Thomas - 2014 - Dissertation, University of Oxford
    I explore the possibility of a structuralist interpretation of homotopy type theory (HoTT) as a foundation for mathematics. There are two main aspects to HoTT's structuralist credentials. First, it builds on categorical set theory (CST), of which the best-known variant is Lawvere's ETCS. I argue that CST has merit as a structuralist foundation, in that it ascribes only structural properties to typical mathematical objects. However, I also argue that this success depends on the adoption of a strict typing system which (...)
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  44. added 2021-04-02
    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.
  45. added 2021-03-31
    Cognitive Structuralism: Explaining the Regularity of the Natural Numbers Progression.Paula Quinon - forthcoming - Review of Philosophy and Psychology:1-23.
    According to one of the most powerful paradigms explaining the meaning of the concept of natural number, natural numbers get a large part of their conceptual content from core cognitive abilities. Carey’s bootstrapping provides a model of the role of core cognition in the creation of mature mathematical concepts. In this paper, I conduct conceptual analyses of various theories within this paradigm, concluding that the theories based on the ability to subitize, or on the ability to approximate quantities, or both, (...)
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  46. added 2021-03-31
    Mary Shepherd on the Role of Proofs in Our Knowledge of First Principles.M. Folescu - forthcoming - Noûs.
    This paper examines the role of reason in Shepherd's account of acquiring knowledge of the external world via first principles. Reason is important, but does not have a foundational role. Certain principles enable us to draw the required inferences for acquiring knowledge of the external world. These principles are basic, foundational and, more importantly, self‐evident and thus justified in other ways than by demonstration. Justificatory demonstrations of these principles are neither required, nor possible. By drawing on textual and contextual evidence, (...)
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  47. added 2021-03-28
    Unification and mathematical explanation in science.Sam Baron - forthcoming - Synthese:1-25.
    Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical facts are explanatorily unified. I argue (...)
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  48. added 2021-03-28
    Impurity in Contemporary Mathematics.Ellen Lehet - 2021 - Notre Dame Journal of Formal Logic 62 (1):67-82.
    Purity has been recognized as an ideal of proof. In this paper, I consider whether purity continues to have value in contemporary mathematics. The topics (e.g., algebraic topology, algebraic geometry, category theory) and methods of contemporary mathematics often favor unification and generality, values that are more often associated with impurity rather than purity. I will demonstrate this by discussing several examples of methods and proofs that highlight the epistemic significance of unification and generality. First, I discuss the examples of algebraic (...)
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  49. added 2021-03-28
    How Nature ‘Tokenizes’ Reality.Ilexa Yardley - 2021 - Https://Medium.Com/the-Circular-Theory/.
    Pi in mathematics is mind in Nature, explaining the tokenization of 'reality.'.
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  50. added 2021-03-25
    Euclid's Error: The Mathematics Behind Foucault, Deleuze, and Nietzsche.Ilexa Yardley - 2021 - Intelligent Design Center.
    We have to go all the way back to Euclid, and, actually, before, to figure out the basis for representation, and therefore, interpretation. Which is, pure and simple, the conservation of a circle. As articulated by Foucault, Deleuze, and Nietzsche. 'Pi' (in mathematics) is the background state for everything (a.k.a. 'mind').Providing the explanation for (and the current popularity, and, thus, the 'genius' behind) NFT (non fungible tokens). 'Reality' has, finally, caught up with the 'truth.'.
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