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Philosophy of Mathematics

Edited by Øystein Linnebo (Birkbeck College)
Assistant editor: Sam Roberts (Birkbeck College, University of Oslo)
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  1. added 2016-04-27
    Luca Incurvati (forthcoming). Maximality Principles in Set Theory. Philosophia Mathematica.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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  2. added 2016-04-23
    S. Hewitt, A Note on Gabriel Uzquiano's 'Varieties of Indefinite Extensibility'.
  3. added 2016-04-21
    Chris Daly & David Liggins (forthcoming). Dorr on the Language of Ontology. Philosophical Studies:1-15.
    In the ‘ordinary business of life’, everyone makes claims about what there is. For instance, we say things like: ‘There are some beautiful chairs in my favourite furniture shop’. Within the context of philosophical debate, some philosophers also make claims about what there is. For instance, some ontologists claim that there are chairs; other ontologists claim that there are no chairs. What is the relation between ontologists’ philosophical claims about what there is and ordinary claims about what there is? According (...)
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  4. added 2016-04-21
    Araceli Ramirez-Cardenas, Maria Moskaleva & Andreas Nieder (2016). Neuronal Representation of Numerosity Zero in the Primate Parieto-Frontal Number Network. Current Biology 26.
    Neurons in the primate parieto-frontal network represent the number of visual items in a collection, but it is unknown whether this system encodes empty sets as conveying null quantity. We recorded from the ventral intraparietal area (VIP) and the prefrontal cortex (PFC) of monkeys performing a matching task including empty sets and countable numerosities as stimuli. VIP neurons encoded empty sets predominantly as a distinct category from numerosities. In contrast, PFC neurons represented empty sets more similarly to numerosity one than (...)
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  5. added 2016-04-16
    Erhan Demircioglu (2015). Recognitional Identification and the Knowledge Argument. Croatian Journal of Philosophy 15 (3):325-340.
    Frank Jackson’s famous Knowledge Argument moves from the premise that complete physical knowledge about experiences is not complete knowledge about experiences to the falsity of physicalism. Some physicalists (e.g., John Perry) have countered by arguing that what Jackson’s Mary, the perfect scientist who acquires all physical knowledge about experiencing red while being locked in a monochromatic room, lacks before experiencing red is merely a piece of recognitional knowledge of an identity, and that since lacking a piece of recognitional knowledge (...)
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  6. added 2016-04-14
    Thomas Forster (2016). Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF. Philosophia Mathematica 24 (1):50-59.
    Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no (...)
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  7. added 2016-04-13
    Jaykov Foukzon, Relevant First-Order Logic LP# and Curry’s Paradox Resolution. Pure and Applied Mathematics Journal Volume 4, Issue 1-1, January 2015 DOI: 10.11648/J.Pamj.S.2015040101.12.
    In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗n^C.
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  8. added 2016-04-04
    Mohammad Husein Saleh, Doaa Mohammad Shokry & Saada A. Rahman Abu Shammala, A Numerical Approach for Solving Classes of Linear and Nonlinear Volterra Integral Equations by Chebyshev Polynomial.
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  9. added 2016-04-03
    Cezary Cieśliński (2010). Deflationary Truth and Pathologies. Journal of Philosophical Logic 39 (3):325-337.
    By a classical result of Kotlarski, Krajewski and Lachlan, pathological satisfaction classes can be constructed for countable, recursively saturated models of Peano arithmetic. In this paper we consider the question of whether the pathology can be eliminated; we ask in effect what generalities involving the notion of truth can be obtained in a deflationary truth theory (a theory of truth which is conservative over its base). It is shown that the answer depends on the notion of pathology we adopt. It (...)
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  10. added 2016-04-02
    Andrew Arana (2016). Imagination in Mathematics. In Amy Kind (ed.), The Routledge Handbook of Philosophy of Imagination. Routledge 463-477.
  11. added 2016-03-26
    Giuseppe Iurato & Giuseppe Ruta (2016). On the Role of Virtual Work in Levi-Civita’s Parallel Transport. Archive for History of Exact Sciences 70:1-13 (provisional).
    The current literature on history of science reports that Levi-Civita’s parallel transport was motivated by his attempt to provide the covariant derivative of the absolute differential calculus with a geometrical interpretation (For instance, see Scholz in ''The intersection of history and mathematics'', Birkhäuser, Basel, pp 203-230, 1994, Sect. 4). Levi-Civita’s memoir on the subject was explicitly aimed at simplifying the geometrical computation of the curvature of a Riemannian manifold. In the present paper, we wish to point out the possible role (...)
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  12. added 2016-03-24
    Burt C. Hopkins (forthcoming). Claire Ortiz Hill and Jairo José da Silva. The Road Not Taken: On Husserl's Philosophy of Logic and Mathematics. Texts in Philosophy; 21. London: College Publications, 2013. ISBN 978-1-84890-099-8 . Pp. Xiv + 436. [REVIEW] Philosophia Mathematica:nkw006.
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  13. added 2016-03-21
    Peter Smith, Category Theory: A Gentle Introduction.
    This Gentle Introduction is very much still work in progress. So far, at least in a rough and ready way, we cover the basic notions of elementary category theory -- explaining the very idea of a category, then treating limits, functors, natural transformations, representables, adjunctions. The long-term plan is to go on to have a chapter on monads, and then move on to say something about categorial logic, explore categories of sets, and edge towards some initial themes in topos theory. (...)
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  14. added 2016-03-21
    Ricardo Crespo & Fernando Tohmé (forthcoming). The Future of Mathematics in Economics: A Philosophically Grounded Proposal. Foundations of Science:1-17.
    The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.
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  15. added 2016-03-18
    Burkay Ozturk (2012). On a Perceived Inadequacy of Principia Mathematica. Florida Philosophical Review 12 (1):83-92.
    This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and (...) Mathematica. (shrink)
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  16. added 2016-03-15
    Andrea Sauchelli (2016). The Definition of Religion, Super-Empirical Realities and Mathematics. Neue Zeitschrift für Systematicsche Theologie Und Religionsphilosophie 58 (1):67-75.
    Providing a precise definition of “religion”—or an analysis in terms of sufficient and necessary conditions of the concept of religion—has proven to be a difficult task, more so in light of the diverse types of practices considered religious by scholars. Here, I discuss Kevin Schilbrack’s recent definition of “religion”, elaborate it and raise several objections, one of which is based on a specific theory in philosophy of mathematics: mathematical realism.
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  17. added 2016-03-14
    John Corcoran (1978-9). CORCORAN's THUMBNAIL REVIEWS OF OPPOSING PHILOSOPHY OF LOGIC BOOKS. MATHEMATICAL REVIEWS 56:98-9.
    PUTNAM has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is largely correct as far as it goes, or at least that it is rational to believe in it. The main thesis of the book is that consistent acceptance of contemporary (...)
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  18. added 2016-03-13
    Nigel Vinckier & Jean Paul Van Bendegem (forthcoming). Feng Ye. Strict Finitism and the Logic of Mathematical Applications. Synthese Library; 355. Springer, 2011. ISBN: 978-94-007-1346-8 ; 978-94-007-1347-5 . Pp. Xii + 272. [REVIEW] Philosophia Mathematica:nkw005.
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  19. added 2016-03-11
    Brendan Larvor (2008). What Can the Philosophy of Mathematics Learn From the History of Mathematics? Erkenntnis 68 (3):393-407.
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  20. added 2016-03-09
    Cristian Soto (2014). Mark Colyvan's An Introduction to the Philosophy of Mathematics. [REVIEW] Critica 46 (138).
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  21. added 2016-03-03
    Stefan Buijsman (forthcoming). Accessibility of Reformulated Mathematical Content. Synthese:1-18.
    I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don’t have many beliefs about (...)
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  22. added 2016-03-02
    Dan Baras (2016). Our Reliability is in Principle Explainable. Episteme.
    Non-skeptical robust realists about normativity, mathematics, or any other domain of non-causal truths are committed to a correlation between their beliefs and non-causal, mind-independent facts. Hartry Field and others have argued that if realists cannot explain this striking correlation, that is a strong reason to reject their theory. Some consider this argument, known as the Benacerraf–Field argument, as the strongest challenge to robust realism about mathematics (Field 1989, 2001), normativity (Enoch 2011), and even logic (Schechter 2010). In this (...)
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  23. added 2016-02-29
    T. Button (2006). Realistic Structuralism's Identity Crisis: A Hybrid Solution. Analysis 66 (3):216-222.
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  24. added 2016-02-28
    Raffaele Pisano (2016). The Algebra Between History and Education. [REVIEW] Metascience:1-5.
    ‘‘What Is Algebra?-Why This Book?’’ This is the amazing prelude to Taming the Unknown by Victor J. Katz, emeritus professor of mathematics at the University of the District of Columbia and Karen Hunger Parshall, professor of history of mathematics at the University of Virginia. This is an excellent book; its accurate historical and pedagogical purpose offers an accessible read for historians and mathematicians. [continue...].
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  25. added 2016-02-27
    Nora Berenstain (2016). The Applicability of Mathematics to Physical Modality. Synthese (online):1-17.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
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  26. added 2016-02-26
    Benjamin Eva (2016). Category Theory and Physical Structuralism. European Journal for Philosophy of Science 6 (2):231-246.
    As a metaphysical theory, radical ontic structural realism is characterised mainly in terms of the ontological primacy it places on relations and structures, as opposed to the individual relata and objects that inhabit these relations/structures. The most popular criticism of ROSR is that its central thesis is incoherent. Bain attempts to address this criticism by arguing that the mathematical language of category theory allows for a coherent articulation of ROSR’s key thesis. Subsequently, Wüthrich and Lam and Lal and Teh have (...)
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  27. added 2016-02-26
    James Trafford (2016). Structuring Co-Constructive Logic for Proofs and Refutations. Logica Universalis 10 (1):67-97.
    This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford :22–40, 2015). It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their relation such (...)
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  28. added 2016-02-26
    Philip Ebert & Stewart Shapiro (2009). The Good, the Bad and the Ugly. Synthese 170 (3):415-441.
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  29. added 2016-02-26
    Colin McLarty (2006). Two Constructivist Aspects of Category Theory. Philosophia Scientae:95-114.
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  30. added 2016-02-25
    Lascelles G. B. James, The Relationship Between Faith and Science.
    Fruitful dialogue between faith and science occurs when Christians and scientists meaningfully answer the questions that they pose to each other in a disciplined conversation that refines their perspectives iteratively, each respecting the diligent conscientious work of the other. When questions are answered in this way the answers then form the basis for the continuation of the discussion and agreed solutions will chart principles and procedures for feedback and modification.
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  31. added 2016-02-25
    Monica Ugaglia (2016). Aristotle and the Mathematical Tradition on Diastēma and Logos: An Analysis of Physics 3 3, 202a18-21. Greek Roman and Byzantine Studies 56:49-67.
    ARISTOTLE'S PHYSICS 3.3 contains interesting evidence of an open debate in mathematics, concerning the interchangeability of the notions of diastēma and logos in the theory of harmonics. Because of the standard interpretation of the passage, however, this reference to harmonics has gone unnoticed: a slightly different understanding is proposed in this paper, which restores the relevance of the passage and its place in the contemporary debate.
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  32. added 2016-02-25
    Monica Ugaglia (2015). Knowing by Doing: The Role of Geometrical Practice in Aristotle’s Theory of Knowledge. Elenchos 36:45-88.
    Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passage we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a philosophical (...)
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  33. added 2016-02-25
    Monica Ugaglia & Fabio Acerbi (2015). Aristotle on Placing Gnomons Round. Classical Quarterly 65 (2):587-608.
    The passage has been an object of scholarly debate: the lack of independent sources on the mathematical construction described by Aristotle, the terseness of the formulation and the resulting syntactical ambiguities make the exact interpretation of the text quite difficult, as already noted by Philoponus. What does it mean that the gnomons are ‘placed round the one and without’ (περὶ τὸ ἓν καὶ χωρίς)? And in what sense is this an indication of the even being ‘cut off, enclosed (ἐναπολαμβανόμενον), and (...)
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  34. added 2016-02-25
    Eduardo Castro (2009). Uma Solução para o Problema de Benacerraf. Principia: An International Journal of Epistemology 13 (1).
    The Benacerraf’s problem is a problem about how we can attain mathematical knowledge: mathematical entities are entities not located in space-time; we exist in spacetime; so, it does not seem that we could have a causal connection with mathematical entities in order to attain mathematical knowledge. In this paper, I propose a solution to the Benacerraf’s problem supported by the Quinean doctrines of naturalism, confirmational holism and postulation. I show that we have empirical knowledge of centres of mass and of (...)
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  35. added 2016-02-25
    Stephen Yablo (2002). Abstract Objects: A Case Study. Philosophical Issues 12 (1):220-240.
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  36. added 2016-02-25
    Miro Brada (2000). Personality Model. Problem Paradise:42-43.
    In 1995, as a student of psychology inspired by natural science, I defined a logical model of personality explaining psychosis. I created (for my MA thesis, 1998 and grant research, 1999) new kind of tests assessing intelligence, creativity, prejudices, expectations to show more exact methods in psychology. During my Phd study in economics, I developed 'Maximization of Uniqueness (Originality)' model enhancing the classic utility to explain irrational motivations linking economics and psychology. Later I became computer programmer developing functional programming. According (...)
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  37. added 2016-02-24
    Benjamin G. Rin & Sean Walsh (forthcoming). Realizability Semantics for Quantified Modal Logic: Generalizing Flagg's 1985 Construction. Review of Symbolic Logic.
    A semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church's Thesis and a variant of (...)
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  38. added 2016-02-21
    Russell Marcus & Mark McEvoy (eds.) (2016). An Historical Introduction to the Philosophy of Mathematics. Bloomsbury.
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  39. added 2016-02-14
    Jessica Carter (forthcoming). John W. Dawson, Jr. Why Prove It Again: Alternative Proofs in Mathematical Practice. Basel: Birkhäuser, 2015. ISBN: 978-3-319-17367-2 ; 978-3-319-17368-9 . Pp. Xii + 204. [REVIEW] Philosophia Mathematica:nkw003.
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  40. added 2016-02-10
    Paolo Rossi (ed.) (1988). Storia della Scienza. Utet.
  41. added 2016-02-10
    Paolo Rossi (ed.) (1988). Storia della Scienza. Utet.
  42. added 2016-02-08
    Michele Ginammi (2016). Avoiding Reification: Heuristic Effectiveness of Mathematics and the Prediction of the Omega Minus Particle. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 53:20-27.
    According to Steiner (1998), in contemporary physics new important discoveries are often obtained by means of strategies which rely on purely formal mathematical considerations. In such discoveries, mathematics seems to have a peculiar and controversial role, which apparently cannot be accounted for by means of standard methodological criteria. M. Gell-Mann and Y. Ne׳eman׳s prediction of the Ω− particle is usually considered a typical example of application of this kind of strategy. According to Bangu (2008), this prediction is apparently based (...)
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  43. added 2016-02-08
    Paolo Rossi, Fernardo Abbri, Enrico Bellone, Walter Bernardi, Umberto Bottazzini, Bernardino Fantini, Antonello La Vergata, Stefania Nicasi, Stefano Poggi, Mario Rossi Monti & Eugenio Torracca (eds.) (1988). Storia della Scienza - voll. 1-8. Utet.
  44. added 2016-02-08
    Paolo Rossi (ed.) (1988). Storia della Scienza - voll. 1-8. Utet.
  45. added 2016-02-08
    Paolo Rossi, Fernardo Abbri, Enrico Bellone, Walter Bernardi, Umberto Bottazzini, Bernardino Fantini, Antonello La Vergata, Stefania Nicasi, Stefano Poggi, Mario Rossi Monti & Eugenio Torracca (eds.) (1988). Storia della Scienza - voll. 1-8. Utet.
  46. added 2016-02-08
    Paolo Rossi (ed.) (1988). Storia della Scienza - voll.8. Utet.
  47. added 2016-02-08
    Paolo Rossi, Fernardo Abbri, Enrico Bellone, Walter Bernardi, Umberto Bottazzini, Bernardino Fantini, Antonello La Vergata, Stefania Nicasi, Stefano Poggi, Mario Rossi Monti & Eugenio Torracca (eds.) (1988). Storia della Scienza - voll. 1-8. Utet.
  48. added 2016-02-08
    John Corcoran (1983). 1983 Review in Mathematical Reviews 83e:03005 Of: Cocchiarella, Nino “The Development of the Theory of Logical Types and the Notion of a Logical Subject in Russell's Early Philosophy: Bertrand Russell's Early Philosophy, Part I”. Synthese 45 (1980), No. 1, 71-115. MATHEMATICAL REVIEWS 83:03005.
    CORCORAN RECOMMENDS COCCHIARELLA ON TYPE THEORY. The 1983 review in Mathematical Reviews 83e:03005 of: Cocchiarella, Nino “The development of the theory of logical types and the notion of a logical subject in Russell's early philosophy: Bertrand Russell's early philosophy, Part I”. Synthese 45 (1980), no. 1, 71-115 .
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  49. added 2016-02-08
    Ludovico Geymonat (1971). Federigo Enriques e la storia della scienza. Physis 3.
  50. added 2016-02-08
    Ludovico Geymonat (1962). Storia della matematica. In Storia delle scienza - coordinata da Nicola Abbagnano - Torino.
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