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  1. Evolutionary Debunking Arguments: Ethics, Philosophy of Religion, Philosophy of Mathematics, Metaphysics, and Epistemology, edited by Diego E. Machuca.Peter Königs - 2023 - International Journal for the Study of Skepticism 14 (1):73-78.
  2. Theorem proving in artificial neural networks: new frontiers in mathematical AI.Markus Pantsar - 2024 - European Journal for Philosophy of Science 14 (1):1-22.
    Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, machine learning results (...)
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  3. Why do numbers exist? A psychologist constructivist account.Markus Pantsar - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers to (...)
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  4. A Conventionalist Account of Distinctively Mathematical Explanation.Mark Povich - 2024 - Philosophical Problems in Science 74:171–223.
    Distinctively mathematical explanations (DMEs) explain natural phenomena primarily by appeal to mathematical facts. One important question is whether there can be an ontic account of DME. An ontic account of DME would treat the explananda and explanantia of DMEs as ontic structures and the explanatory relation between them as an ontic relation (e.g., Pincock 2015, Povich 2021). Here I present a conventionalist account of DME, defend it against objections, and argue that it should be considered ontic. Notably, if indeed it (...)
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  5. 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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  6. Rules to Infinity: The Normative Role of Mathematics in Scientific Explanation.Mark Povich - forthcoming - Oxford University Press.
    [EDIT: This book will be published open access. Check back around April 2024 to access the entire book.] One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which (...)
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  7. Are Large Cardinal Axioms Restrictive?Neil Barton - 2023 - Philosophia Mathematica 31 (3):372-407.
    The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of (...)
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  8. A Critique of Yablo’s If-thenism.Bradley Armour-Garb & Frederick Kroon - 2023 - Philosophia Mathematica 31 (3):360-371.
    Using ideas proposed in Aboutness and developed in ‘If-thenism’, Stephen Yablo has tried to improve on classical if-thenism in mathematics, a view initially put forward by Bertrand Russell in his Principles of Mathematics. Yablo’s stated goal is to provide a reading of a sentence like ‘The number of planets is eight’ with a sort of content on which it fails to imply ‘Numbers exist’. After presenting Yablo’s framework, our paper raises a problem with his view that has gone virtually unnoticed (...)
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  9. Justified Epistemic Exclusions in Mathematics.Colin Jakob Rittberg - 2023 - Philosophia Mathematica 31 (3):330-359.
    Who gets to contribute to knowledge production of an epistemic community? Scholarship has focussed on unjustified forms of exclusion. Here I study justified forms of exclusion by investigating the phenomenon of so-called ‘cranks’ in mathematics. I argue that workload-management concerns justify the exclusion of these outsiders from mathematical knowledge-making practices. My discussion reveals three insights. There are reasons other than incorrect mathematical argument that justify exclusions from mathematical practices. There are instances in which mathematicians are justified in rejecting even correct (...)
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  10. 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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  11. The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics.Alain Badiou, Zachary Fraser & Tzuchien Tho - 2007 - Re.press.
    In The Concept of Model Alain Badiou establishes a new logical ’concept of model’. Translated for the first time into English, the work is accompanied by an exclusive interview with Badiou in which he elaborates on the connections between his early and most recent work-for which the concept of model remains seminal.
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  12. Is math real?: how simple questions lead us to mathematics' deepest truths.Eugenia Cheng - 2023 - New York: Basic Books.
    Where does math come from? From a textbook? From rules? From deduction? From logic? Not really, Eugenia Cheng writes in Is Math Real?: it comes from curiosity, from instinctive human curiosity, "from people not being satisfied with answers and always wanting to understand more." And most importantly, she says, "it comes from questions": not from answering them, but from posing them. Nothing could seem more at odds from the way most of us were taught math: a rigid and autocratic model (...)
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  13. Kritik der mathematischen vernunft.J. E. Gerlach - 1922 - Bonn,: F. Cohen.
    Die allgemeine anzahlenlehre.--Der araum und die grössenlehre.--Die gestaltenlehre.--Besondere gestalten.--Gleich und gleich.--Plus, minus und das irgend-i.--Anhang: Zur "gemeinverständlichen" erörterund der relativitätstheorie.
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  14. On the continuum fallacy: is temperature a continuous function?Aditya Jha, Douglas Campbell, Clemency Montelle & Phillip L. Wilson - 2023 - Foundations of Physics 53 (69):1-29.
    It is often argued that the indispensability of continuum models comes from their empirical adequacy despite their decoupling from the microscopic details of the modelled physical system. There is thus a commonly held misconception that temperature varying across a region of space or time can always be accurately represented as a continuous function. We discuss three inter-related cases of temperature modelling — in phase transitions, thermal boundary resistance and slip flows — and show that the continuum view is fallacious on (...)
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  15. Towards a critical epistemology of mathematics.David Kollosche - 2023 - Prometeica - Revista De Filosofía Y Ciencias 27:825-833.
    This essay addresses a critical epistemology of mathematics as an investigation into the epistemic limitations of mathematical thinking. After arguing for the relevance of a critical epistemology of mathematics, I discuss assumptions underlying standard arithmetic and assumptions underlying standard logic as examples for such epistemic limitations of mathematical thinking. Looking into the work of philosophically inte­res­ted scholars in mathematics education such as Alan Bishop and Ole Skovsmose, I discuss some early insights for a critical epistemology of mathematics. I conclude that (...)
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  16. Mario Bunge's Philosophy of Mathematics: An Appraisal.Marquis Jean-Pierre - 2012 - Science & Education 21:1567-1594.
    In this paper, I present and discuss critically the main elements of Mario Bunge’s philosophy of mathematics. In particular, I explore how mathematical knowledge is accounted for in Bunge’s systemic emergent materialism.
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  17. Solutions to the Knower Paradox in the Light of Haack’s Criteria.Mirjam de Vos, Rineke Verbrugge & Barteld Kooi - 2023 - Journal of Philosophical Logic 52 (4):1101-1132.
    The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge on (...)
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  18. 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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  19. L'imagination du réel.Rolin Wavre - 1948 - Neuchâtel,: Baconnière.
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  20. La recherche scientifique en mathématiques.Paul Antonin Montel - 1953 - [Alençon,: Impr. alençonnaise.
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  21. Vom Denken in Begriffen.Alexander Israel Wittenberg - 1957 - Basel,: Birkhäuser.
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  22. Les limitations internes des formalismes.Jean Ladrière - 1957 - Louvain,: E. Nauwelaerts.
  23. Structure et objet de l'analyse mathématique.Eloi Lefebvre - 1958 - Paris,: Gauthier-Villars.
  24. Sur la clarté des démonstrations mathématiques.François Rostand - 1962 - Paris,: J. Vrin.
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  25. Matthew Handelman.* The Mathematical Imagination: On the Origins and Promise of Critical Theory.Mirna Džamonja - 2023 - Philosophia Mathematica 31 (2):283-285.
    This book, published in 2019 as an open-access edition of the Fordham University Press, attracts by its title. Imagination, as we mathematicians know only too w.
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  26. Discorso epistemologico sull'insiemistica, conoscenza e matematica.Armando Brissoni - 1974 - Firenze: OS.
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  27. 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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  28. The Devil in the Data: Machine Learning & the Theory-Free Ideal.Mel Andrews - unknown
    Machine learning (ML) refers to a class of computer-facilitated methods of statistical modelling. ML modelling techniques are now being widely adopted across the sciences. A number of outspoken representatives from the general public, computer science, various scientific fields, and philosophy of science alike seem to share in the belief that ML will radically disrupt scientific practice or the variety of epistemic outputs science is capable of producing. Such a belief is held, at least in part, because its adherents take ML (...)
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  29. Die erkenntnistheoretischen Grundlagen der Mathematik.Gustav Kruck - 1981 - Zürich: Schulthess.
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  30. Rigour and Proof.Oliver Tatton-Brown - 2023 - Review of Symbolic Logic 16 (2):480-508.
    This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.
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  31. L'analyse et la synthèse selon Ibn al-Haytham.Roshdi Rashed - 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.
  32. Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919-1920 in Göttingen.David Hilbert - 1992 - Boston: Birkhäuser. Edited by Paul Bernays & David E. Rowe.
    Erster Teil Die übliche Auffassung von der Mathematik und ihre Widerlegung.- 1 Die Rolle von Anschauung und Erfahrung.- 2 Die Rolle der Voraussetzungen.- 3 Die Nichtuntrüglichkeit des mathematischen Schliessens.- Zweiter Teil Die landläufige Auffassung von der Physik und ihre Berichtigung.- 4 Physikalische Begriffsbildungen.- 5 Die Gesetze der Physik und ewige Naturgesetze.- 6 Die Beziehung zwischen Theorie und Experiment.- Dritter Teil Fragen philosophischen Charakters.- 7 Physikalische Gesetzlichkeit und Kausalität.- 8 Naturgeschehen und Wahrscheinlichkeit.- 9 Die Rolle von idealen Gebilden.
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  33. Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups.Joshua Habgood-Coote & Fenner Stanley Tanswell - 2023 - Episteme 20 (2):281-307.
    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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  34. Mathematik und Erkenntnis: Eine Möglichkeit, die "Laws of Form" zu lesen.Fritz B. Simon - 1993 - In Dirk Baecker (ed.), Kalkül der Form. Frankfurt am Main: Suhrkamp.
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  35. Standardized PUEBI EYD V’s Regularity in Formal Writing of Mathematical Existential Statement Consequences on Scientific Ontological Theorization to Indonesian Scientific Community.Raisa Rahima - 2023 - Proceeding of 10Th International Conference on Nusantara Philosophy (Icnp) 9.
    This paper discusses the consequences of the latest PUEBI EYD V regulations for scientific ontological theorization through analyzing the semantical metaphysical commitment it reflects when we write formal mathematical statements using purely mathematical symbol (e.g., “there are 22 aardvarks”). This paper shows that PUEBI EYD V commits to mathematical Platonism metaphysically. This commitment brings harm to observable entities ontological nature in scientific theorization as shown in nominalism projects of philosophy of mathematics. Scientific theories - and even mathematical theories - should (...)
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  36. Epistemologia della matematica: ontologia, verità, valutazioni.Nicola Grana - 2001 - Napoli: L'orientale.
  37. Working Toward Solutions in Fluid Dynamics and Astrophysics: What the Equations Don’t Say.Lydia Patton & Erik Curiel (eds.) - 2023 - Springer Verlag.
    Systems of differential equations are used to describe, model, explain, and predict states of physical systems. Experimental and theoretical branches of physics including general relativity, climate science, and particle physics have differential equations at their center. Direct solutions to differential equations are not available in many domains, which spurs on the use of creative mathematics and simulated solutions.
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  38. Bridge to abstract mathematics.Ralph W. Oberste-Vorth - 2012 - [Washington, DC]: Mathematical Association of America. Edited by Aristides Mouzakitis & Bonita A. Lawrence.
    Statements in mathematics -- Proofs in mathematics -- Basic set operations -- Functions -- Relations on a set -- Cardinality -- Algebra of number systems -- The natural numbers -- The integers -- The rational numbers -- The real numbers -- Cantor's reals -- The complex numbers -- Time scales -- The delta derivative -- Hints for (and comments on) the exercises.
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  39. Role of Imagination and Anticipation in the Acceptance of Computability Proofs: A Challenge to the Standard Account of Rigor.Keith Weber - 2022 - Philosophia Mathematica 30 (3):343-368.
    In a 2022 paper, Hamami claimed that the orthodox view in mathematics is that a proof is rigorous if it can be translated into a derivation. Hamami then developed a descriptive account that explains how mathematicians check proofs for rigor in this sense and how they develop the capacity to do so. By exploring introductory texts in computability theory, we demonstrate that Hamami’s descriptive account does not accord with actual mathematical practice with respect to computability theory. We argue instead for (...)
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  40. Mathematics: Method Without Metaphysics.Elaine Landry - 2023 - Philosophia Mathematica 31 (1):56-80.
    I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of (...)
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  41. Jens Erik Fenstad.*Structures and Algorithms: Mathematics and the Nature of Knowledge.Julian C. Cole - 2023 - Philosophia Mathematica 31 (1):125-131.
    This book collects eight essays — written over multiple decades, for a general audience — that address Fenstad’s thoughts on the topics of what there is and how.
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  42. A novel Holistic Risk Assessment Concept: The Epistemological Positioning and the Methodology.Carrodano Tarantino Cinzia - unknown
    Risk is an intrinsic part of our lives. In the future, the development and growth of the Internet of things allows getting a huge amount of data. Considering this evolution, our research focuses on developing a novel concept, namely Holistic Risk Assessment (HRA), that takes into consideration elements outside the direct influence of the individual to provide a highly personalized risk assessment. The HRA implies developing a methodology and a model. This paper is related to the epistemological positioning of this (...)
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  43. Knowledge and the Philosophy of Number. [REVIEW]Richard Lawrence - 2022 - History and Philosophy of Logic 43 (4):404-406.
    Hossack’s project in this book is to provide a new foundation for the philosophy of number inspired by the traditional idea that numbers are magnitudes.
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  44. Experimental mathematics.V. I. Arnold - 2015 - Providence. Rhode Island: American Mathematical Society. Edited by D. B. Fuks & Mark E. Saul.
    One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. (...)
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  45. Die dialogische Form der Mathematik.Sven Matthiscyk - 2016 - Würzburg: Königshausen & Neumann.
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  46. Benedetto Croce: la scienza, la matematica.Antonio Nigrelli & Francesco Saverio Tortoriello (eds.) - 2007 - Soveria Mannelli: Rubbettino.
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  47. 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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  48. The Quasi-Empirical Epistemology of Mathematics.Ellen Yunjie Shi - 2022 - Kriterion – Journal of Philosophy 36 (2):207-226.
    This paper clarifies and discusses Imre Lakatos’ claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that Lakatos’ motivation for classifying mathematics as a quasi-empirical theory is epistemological; what can be called the quasi-empirical epistemology of mathematics is not correct; analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics.
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  49. Mathematical Explanations in Evolutionary Biology or Naturalism? A Challenge for the Statisticalist.Fabio Sterpetti - 2021 - Foundations of Science 27 (3):1073-1105.
    This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...)
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  50. Grundlagen §64: An Alternative Strategy to Account for Second-Order Abstraction.Vincenzo Ciccarelli - 2022 - Principia: An International Journal of Epistemology 26 (2):183-204.
    A famous passage in Section 64 of Frege’s Grundlagen may be seen as a justification for the truth of abstraction principles. The justification is grounded in the procedureofcontent recarvingwhich Frege describes in the passage. In this paper I argue that Frege’sprocedure of content recarving while possibly correct in the case of first-order equivalencerelations is insufficient to grant the truth of second-order abstractions. Moreover, I propose apossible way of justifying second-order abstractions by referring to the operation of contentrecarving and I show (...)
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