71 found
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  1. Philosophy of mathematics and mathematical practice in the seventeenth century.Paolo Mancosu (ed.) - 1996 - New York: Oxford University Press.
    The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...)
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  2. The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - Oxford, England: Oxford University Press.
    There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.
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  3. (1 other version)From Brouwer to Hilbert: the debate on the foundations of mathematics in the 1920s.Paolo Mancosu (ed.) - 1998 - New York: Oxford University Press.
    From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...)
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  4.  20
    Abstraction and Infinity.Paolo Mancosu - 2016 - Oxford, England: Oxford University Press.
    Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core (...)
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  5. Explanation in Mathematics.Paolo Mancosu - 2014 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Stanford, CA: The Metaphysics Research Lab.
    The philosophical analysis of mathematical explanations concerns itself with two different, although connected, areas of investigation. The first area addresses the problem of whether mathematics can play an explanatory role in the natural and social sciences. The second deals with the problem of whether mathematical explanations occur within mathematics itself. Accordingly, this entry surveys the contributions to both areas, it shows their relevance to the history of philosophy and science, it articulates their connection, and points to the philosophical pay-offs to (...)
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  6. On the relationship between plane and solid geometry.Andrew Arana & Paolo Mancosu - 2012 - Review of Symbolic Logic 5 (2):294-353.
    Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.
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  7. The Philosophy of Mathematical Practice.Paolo Mancosu - 2009 - Studia Logica 92 (1):137-141.
     
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  8. The development of mathematical logic from Russell to Tarski, 1900-1935.Paolo Mancosu, Richard Zach & Calixto Badesa - 2009 - In Leila Haaparanta (ed.), The development of modern logic. New York: Oxford University Press.
    The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...)
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  9. Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?Paolo Mancosu - 2009 - Review of Symbolic Logic 2 (4):612-646.
    Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...)
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  10. Mathematical explanation: Why it matters.Paolo Mancosu - 2008 - In The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press. pp. 134--149.
  11.  97
    An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs.Paolo Mancosu, Sergio Galvan & Richard Zach - 2021 - Oxford: Oxford University Press. Edited by Sergio Galvan & Richard Zach.
    An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic, natural deduction and the normalization theorems, the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these (...)
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  12. Mathematical explanation: Problems and prospects.Paolo Mancosu - 2001 - Topoi 20 (1):97-117.
  13.  42
    Visualization, Explanation and Reasoning Styles in Mathematics.Paolo Mancosu, Klaus Frovin Jørgensen & S. A. Pedersen (eds.) - 2005 - Springer.
  14. Mathematics and phenomenology: The correspondence between O. Becker and H. Weyl.Paolo Mancosu & T. A. Ryckman - 2002 - Philosophia Mathematica 10 (2):130-202.
    Recently discovered correspondence from Oskar Becker to Hermann Weyl sheds new light on Weyl's engagement with Husserlian transcendental phenomenology in 1918-1927. Here the last two of these letters, dated July and August, 1926, dealing with issues in the philosophy of mathematics are presented, together with background and a detailed commentary. The letters provide an instructive context for re-assessing the connection between intuitionism and phenomenology in Weyl's foundational thought, and for understanding Weyl's term ‘symbolic construction’ as marking his own considered position (...)
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  15. Visualization in Logic and Mathematics.Paolo Mancosu - 2005 - In Paolo Mancosu, Klaus Frovin Jørgensen & S. A. Pedersen (eds.), Visualization, Explanation and Reasoning Styles in Mathematics. Springer. pp. 13-26.
    In the last two decades there has been renewed interest in visualization in logic and mathematics. Visualization is usually understood in different ways but for the purposes of this article I will take a rather broad conception of visualization to include both visualization by means of mental images as well as visualizations by means of computer generated images or images drawn on paper, e.g. diagrams etc. These different types of visualization can differ substantially but I am interested in offering a (...)
     
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  16. The Varieties of Mathematical Explanation.Hafner Johannes & Paolo Mancosu - 2005 - In Paolo Mancosu, Klaus Frovin Jørgensen & S. A. Pedersen (eds.), Visualization, Explanation and Reasoning Styles in Mathematics. Springer. pp. 215-250.
     
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  17.  81
    (1 other version)Harvard 1940–1941: Tarski, Carnap and Quine on a finitistic language of mathematics for science.Paolo Mancosu - 2005 - History and Philosophy of Logic 26 (4):327-357.
    Tarski, Carnap and Quine spent the academic year 1940?1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnap's Nachlaß in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking (...)
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  18.  98
    Fixed- versus Variable-domain Interpretations of Tarski’s Account of Logical Consequence.Paolo Mancosu - 2010 - Philosophy Compass 5 (9):745-759.
    In this article I describe and evaluate the debate that surrounds the proper interpretation of Tarski’s account of logical consequence given in his classic 1936 article ‘On the concept of logical consequence’. In the late 1980s Etchemendy argued that the familiar model theoretic account of logical consequence is not to be found in Tarski’s original article. Whereas the contemporary account of logical consequence is a variable‐domain conception – in that it calls for a reinterpretation of the domain of variation of (...)
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  19.  94
    Totality, Regularity, and Cardinality in Probability Theory.Paolo Mancosu & Guillaume Massas - 2024 - Philosophy of Science 91 (3):721-740.
    Recent developments in generalized probability theory have renewed a debate about whether regularity (i.e., the constraint that only logical contradictions get assigned probability 0) should be a necessary feature of both chances and credences. Crucial to this debate, however, are some mathematical facts regarding the interplay between the existence of regular generalized probability measures and various cardinality assumptions. We improve on several known results in the literature regarding the existence of regular generalized probability measures. In particular, we give necessary and (...)
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  20.  21
    Tarski, neurath, and kokoszynska on the semantic conception of truth.Paolo Mancosu - 2008 - In Douglas Patterson (ed.), New essays on Tarski and philosophy. New York: Oxford University Press. pp. 192.
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  21.  22
    Beyond unification.Johannes Hafner & Paolo Mancosu - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press. pp. 151--178.
  22. Heinrich Behmann’s 1921 lecture on the decision problem and the algebra of logic.Paolo Mancosu & Richard Zach - 2015 - Bulletin of Symbolic Logic 21 (2):164-187.
    Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on (...)
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  23.  91
    Between Russell and Hilbert: Behmann on the foundations of mathematics.Paolo Mancosu - 1999 - Bulletin of Symbolic Logic 5 (3):303-330.
    After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to (...)
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  24.  37
    Bolzano and Cournot on mathematical explanation / Bolzano et Cournot à propos de l'explication mathématique.Paolo Mancosu - 1999 - Revue d'Histoire des Sciences 52 (3):429-456.
  25.  56
    Wittgenstein, Finitism, and the Foundations of Mathematics.Paolo Mancosu - 2001 - Philosophical Review 110 (2):286.
    It is reported that in reply to John Wisdom’s request in 1944 to provide a dictionary entry describing his philosophy, Wittgenstein wrote only one sentence: “He has concerned himself principally with questions about the foundations of mathematics”. However, an understanding of his philosophy of mathematics has long been a desideratum. This was the case, in particular, for the period stretching from the Tractatus Logico-Philosophicus to the so-called transitional phase. Marion’s book represents a giant leap forward in this direction. In the (...)
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  26. Between Vienna and Berlin: The Immediate Reception of Godel's Incompleteness Theorems.Paolo Mancosu - 1999 - History and Philosophy of Logic 20 (1):33-45.
    What were the earliest reactions to Gödel's incompleteness theorems? After a brief summary of previous work in this area I analyse, by means of unpublished archival material, the first reactions in Vienna and Berlin to Gödel's groundbreaking results. In particular, I look at how Carnap, Hempel, von Neumann, Kaufmann, and Chwistek, among others, dealt with the new results.
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  27.  47
    Three Letters on the Foundations of Mathematics by Frank Plumpton Ramsey†.Paolo Mancosu - forthcoming - Philosophia Mathematica.
    Summary This article presents three hitherto unpublished letters by Frank Plumpton Ramsey on the foundations of mathematics with commentary. One of the letters was sent to Abraham Fraenkel and the other two letters to Heinrich Behmann. The transcription of the letters is preceded by an account that details the extent of Ramsey's known contacts with mathematical logicians on the Continent.
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  28.  95
    The adventure of reason: interplay between philosophy of mathematics and mathematical logic, 1900-1940.Paolo Mancosu - 2010 - New York: Oxford University Press.
    At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of .
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  29.  77
    Grundlagen, Section 64: Frege's Discussion of Definitions by Abstraction in Historical Context.Paolo Mancosu - 2015 - History and Philosophy of Logic 36 (1):62-89.
    I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools . In addition, I put Frege's considerations in the context of a widespread debate in Germany (...)
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  30. In Good Company? On Hume’s Principle and the Assignment of Numbers to Infinite Concepts.Paolo Mancosu - 2015 - Review of Symbolic Logic 8 (2):370-410.
    In a recent article, I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided (...)
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  31.  85
    A Note on Choice Principles in Second-Order Logic.Benjamin Siskind, Paolo Mancosu & Stewart Shapiro - 2023 - Review of Symbolic Logic 16 (2):339-350.
    Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to (...)
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  32.  24
    (3 other versions)2. Quine and Tarski on Nominalism.Paolo Mancosu - 2008 - Oxford Studies in Metaphysics: Volume 4 4:22.
  33.  24
    Neologicist Foundations: Inconsistent Abstraction Principles and Part-Whole.Paolo Mancosu & Benjamin Siskind - 2018 - In Gabriele Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium. Berlin, Boston: De Gruyter. pp. 215-248.
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  34. Aristotelian Logic and Euclidean Mathematics: Seventeenth-Century Developments of the Quaestio de Certitudine Mathematicarum.Paolo Mancosu - 1991 - Studies in History and Philosophy of Science Part A 23 (2):241-265.
  35.  56
    The Russellian influence on Hilbert and his school.Paolo Mancosu - 2003 - Synthese 137 (1-2):59 - 101.
    The aim of the paper is to discuss the influence exercised by Russell's thought inGöttingen in the period leading to the formulation of Hilbert's program in theearly twenties. I show that after a period of intense foundational work, culminatingwith the departure from Göttingen of Zermelo and Grelling in 1910 we witnessa reemergence of interest in foundations of mathematics towards the end of 1914. Itis this second period of foundational work that is my specific interest. Through theuse of unpublished archival sources (...)
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  36.  75
    Mathemetical Explanation.Christopher Pincock & Paolo Mancosu - 2012 - Oxford Bibliographies in Philosophy.
  37.  26
    Definitions by Abstraction in the Peano School.Paolo Mancosu - 2018 - In Alessandro Giordani & Ciro de Florio (eds.), From Arithmetic to Metaphysics: A Path Through Philosophical Logic. De Gruyter. pp. 261-288.
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  38.  44
    Introduction: Interpolations—essays in honor of William Craig.Paolo Mancosu - 2008 - Synthese 164 (3):313-319.
  39. On the status of proofs by contradiction in the seventeenth century.Paolo Mancosu - 1991 - Synthese 88 (1):15 - 41.
    In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations (...)
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  40.  20
    An Essay Review of Three Books on Frank Ramsey†.Paolo Mancosu - 2021 - Philosophia Mathematica 29 (1):110-150.
    No chance of seeing her for another fortnight and it is 11 days since I saw her. Went solitary walk felt miserable but to some extent staved it off by reflecting on |$\langle$|Continuum Problem|$\rangle$|1The occasion for this review article on the life and accomplishments of Frank Ramsey is the publication in the last eight years of three important books: a biography of Frank Ramsey by his sister, Margaret Paul, a book by Steven Methven on aspects of Ramsey’s philosophy, and the (...)
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  41.  12
    Neologicist Foundations: Inconsistent Abstraction Principles and Part-Whole.Paolo Mancosu & Benjamin Siskind - 2018 - In Gabriele Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium. Berlin, Boston: De Gruyter. pp. 215-248.
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  42.  40
    Wittgenstein’s Constructivization of Euler’s Proof of the Infinity of Primes.Paolo Mancosu & Mathieu Marion - 2003 - Vienna Circle Institute Yearbook 10:171-188.
    We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during the transitional (...)
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  43.  23
    Detleff Clüver: An Early Opponent of the Leibnizian Differential Calculus.Paolo Mancosu & Ezio Vailati - 1990 - Centaurus 33 (3):325-344.
  44. Explanation.Paolo Mancosu & Johannes Hafner - 2008 - In The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press.
     
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  45.  31
    Torricelli's Infinitely Long Solid and Its Philosophical Reception in the Seventeenth Century.Paolo Mancosu & Ezio Vailati - 1991 - Isis 82 (1):50-70.
  46.  42
    Unification and Explanation: A Case Study from Real Algebraic Geometry.Paolo Mancosu & Johannes Hafner - 2008 - In The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press. pp. 151--178.
  47.  24
    Generalizing classical and effective model theory in theories of operations and classes.Paolo Mancosu - 1991 - Annals of Pure and Applied Logic 52 (3):249-308.
    Mancosu, P., Generalizing classical and effective model theory in theories of operations and classes, Annas of Pure and Applied Logic 52 249-308 . In this paper I propose a family of theories of operations and classes with the aim of developing abstract versions of model-theoretic results. The systems are closely related to those introduced and already used by Feferman for developing his program of ‘explicit mathematics’. The theories in question are two-sorted, with one kind of variable for individuals and the (...)
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  48.  19
    Descartes and Mathematics.Paolo Mancosu - 2007 - In Janet Broughton & John Carriero (eds.), A Companion to Descartes. Wiley-Blackwell. pp. 103–123.
    This chapter contains section titled: Introduction Descartes's Early Engagement with Mathematics (up to 1623) Rules for the Direction of the Mind Discourse on the Method Geometry, Book I: The Algebra of Segments Geometry, Book I: Pappus' Problem Geometry, Book II: Descartes's Classification of Curves Conclusion Acknowledgments Note References.
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  49.  20
    Introduction à la théorie de la démonstration : Élimination des coupures, normalisation et preuves de cohérence.Paolo Mancosu, Sergio Galvan & Richard Zach - 2022 - Paris: Vrin.
    Cet ouvrage offre une introduction accessible à la théorie de la démonstration : il donne les détails des preuves et comporte de nombreux exemples et exercices pour faciliter la compréhension des lecteurs. Il est également conçu pour servir d’aide à la lecture des articles fondateurs de Gerhard Gentzen. L’ouvrage introduit également aux trois principaux formalismes en usage : l’approche axiomatique des preuves, la déduction naturelle et le calcul des séquents. Il donne une démonstration claire et détaillée des résultats fondamentaux du (...)
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  50. Plane and Solid Geometry: A Note on Purity of Methods.Andrew Arana & Paolo Mancosu - 2014 - In Giorgio Venturi, Marco Panza & Gabriele Lolli (eds.), From Logic to Practice: Italian Studies in the Philosophy of Mathematics. Cham: Springer International Publishing. pp. 23--31.
     
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