68 found
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  1.  75
    From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s.Paolo Mancosu (ed.) - 1998 - 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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  2.  62
    Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century.Paolo Mancosu - 1996 - 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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  3.  40
    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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  4.  65
    The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - 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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  5. 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 collection A is properly included in a collection B then the ‘size’ of (...)
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  6.  37
    Explanation in Mathematics.Paolo Mancosu - 2011 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.
    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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  7.  47
    Fixed- Versus Variable-Domain Interpretations of Tarski's Account of Logical Consequence.Paolo Mancosu - 2010 - Philosophy Compass 5 (9):745-759.
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  8. 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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  9. Mathematical Explanation: Problems and Prospects.Paolo Mancosu - 2001 - Topoi 20 (1):97-117.
  10.  33
    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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  11.  66
    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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  12.  35
    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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  13.  35
    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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  14. The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940.Paolo Mancosu - 2014 - Oxford University Press UK.
    Paolo Mancosu presents a series of innovative studies in the history and the philosophy of logic and mathematics in the first half of the twentieth century. The Adventure of Reason is divided into five main sections: history of logic ; foundational issues ; mathematics and phenomenology ; nominalism ; semantics. Mancosu exploits extensive untapped archival sources to make available a wealth of new material that deepens in significant ways our understanding of these fascinating areas of modern intellectual history. At the (...)
     
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  15.  26
    Mathematical Explanation: Why It Matters.Paolo Mancosu - 2008 - In The Philosophy of Mathematical Practice. Oxford University Press. pp. 134--149.
  16.  94
    The Development of Mathematical Logic From Russell to Tarski, 1900-1935.Paolo Mancosu, Richard Zach & Calixto Badesa - 2008 - In Leila Haaparanta (ed.), The Development of Modern Logic. 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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  17.  35
    The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940.Paolo Mancosu - 2010 - 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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  18.  62
    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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  19.  20
    Wittgenstein, Finitism, and the Foundations of Mathematics.Paolo Mancosu & Mathieu Marion - 2001 - Philosophical Review 110 (2):286.
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  20.  7
    Tarski, Neurath, and Kokoszynska on the Semantic Conception of Truth.Paolo Mancosu - 2008 - In Douglas Patterson (ed.), New Essays on Tarski and Philosophy. Oxford University Press. pp. 192.
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  21.  58
    Aristotelian Logic and Euclidean Mathematics: Seventeenth-Century Developments of the Quaestio de Certitudine Mathematicarum.Paolo Mancosu - 1992 - Studies in History and Philosophy of Science Part A 23 (2):241-265.
  22.  8
    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.
  23.  14
    Mathemetical Explanation.Christopher Pincock & Paolo Mancosu - 2012 - Oxford Bibliographies in Philosophy.
  24.  33
    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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  25.  60
    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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  26. The Varieties of Mathematical Explanation.Hafner Johannes & Paolo Mancosu - 2005 - In Paolo Mancosu (ed.), Visualization, Explanation and Reasoning Styles in Mathematics. Dordrecht: Springer. pp. 215-250.
     
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  27.  35
    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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  28. Beyond Unification.Johannes Hafner & Paolo Mancosu - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 151--178.
     
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  29.  2
    Torricelli's Infinitely Long Solid and Its Philosophical Reception in the Seventeenth Century.Paolo Mancosu & Ezio Vailati - 1991 - Isis: A Journal of the History of Science 82:50-70.
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  30.  12
    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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  31.  5
    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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  32.  52
    Descartes and the Cylindrical Helix.Paolo Mancosu & Andrew Arana - 2010 - Historia Mathematica 37 (3):403-427.
  33.  15
    William Ewald and Wilfried Sieg, Eds, David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917–1933. Heidelberg: Springer, 2013. ISBN: 978-3-540-69444-1 ; 978-3-540-20578-4 . Pp. Xxv + 1062. [REVIEW]Paolo Mancosu - 2015 - Philosophia Mathematica 23 (1):126-135.
  34.  1
    Torricelli's Infinitely Long Solid and Its Philosophical Reception in the Seventeenth Century.Paolo Mancosu & Ezio Vailati - 1991 - Isis 82 (1):50-70.
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  35.  27
    Introduction: Interpolations—Essays in Honor of William Craig.Paolo Mancosu - 2008 - Synthese 164 (3):313-319.
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  36.  14
    Kurt Gödel.Paolo Mancosu - 2011 - Journal of Philosophy 108 (11):642-646.
  37.  11
    2. Quine and Tarski on Nominalism.Paolo Mancosu - 2008 - Oxford Studies in Metaphysics: Volume 4 4:22.
  38.  1
    Algunas observaciones sobre la filosofía de la práctica matemática.Paolo Mancosu - 2016 - Disputatio. Philosophical Research Bulletin 5 (6):131--156.
    [ES] El artículo es una introducción a un nuevo enfoque para la filosofía de las matemáticas, conocido con el nombre de «la filosofía de la práctica matemática». El nuevo enfoque encontró su expresión en la colección La Filosofía de la Práctica Matemática editada por el autor. El artículo sitúa la «filosofía de la práctica matemática» dentro del contexto de otras tradiciones en la filosofía de las matemáticas y traza las similitudes y diferencias con estas tradiciones. [EN] The article is an (...)
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  39.  24
    Volker Peckhaus, Logik, Mathesis Universalis Und Allgemeine Wissenschaft. Leibniz Und Die Wiederentdeckung der Formalen Logik Im 19. Jahrhundert.Paolo Mancosu - 1999 - Erkenntnis 50 (1):129-132.
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  40.  7
    Literature Survey: Recent Publications in the History and Philosophy of Mathematics From the Renaissance to Berkeley. [REVIEW]Paolo Mancosu - 1999 - Metascience 8 (1):102-124.
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  41.  8
    Harvard 1940-41: Tarski, Carnap and Quine on a Finitistic Language of Mathematics for Science.Paolo Mancosu - unknown
    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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  42.  7
    Unification and Explanation: A Case Study From Real Algebraic Geometry.Paolo Mancosu & Johannes Hafner - 2008 - In The Philosophy of Mathematical Practice. Oxford University Press. pp. 151--178.
  43.  4
    Paolo Mancosu, Klaus Frovin JØrgensen, and Stig Andur Pedersen, Eds. Visualization, Explanation and Reasoning Stryles in Mathematics. Synthese Library, Vol. 327. Dordrecht: Springer, 2005. ISBN 1-4020-3334-6 ; 1-4020-3335-4 . Pp. X + 300. [REVIEW]Paolo Mancosu & Klaus JØrgensen - 2006 - Philosophia Mathematica 14 (2):265.
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  44.  6
    Michel Blay, Reasoning with the Infinite. From the Closed World to the Mathematical Universe 216 Pp. $ 30.00 ISBN 0 226 05834 4; $ 15.00 0 226 05835 2. [REVIEW]Paolo Mancosu - 1999 - Early Science and Medicine 4 (4):365-366.
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  45.  2
    The Westin Seattle, Seattle, Washington March 28–29, 2002.Geoffrey Hellman, Jeremy Avigad & Paolo Mancosu - 2002 - Bulletin of Symbolic Logic 8 (3).
  46.  2
    Acoustics and Optics in the Early Modern Period.Paolo Mancosu - unknown
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  47.  2
    2002 Spring Meeting of the Association for Symbolic Logic.Paolo Mancosu - 2002 - Bulletin of Symbolic Logic 8 (3):446-451.
  48.  4
    Book Review: Kurt Gödel. Collected Works, Volumes IV and V. [REVIEW]Paolo Mancosu - 2004 - Notre Dame Journal of Formal Logic 45 (12):109-125.
  49. Solomon Feferman, Charles Parsons, and Steven G. Simpson, Eds.: Kurt Gödel: Essays for His Centennial.Paolo Mancosu - 2011 - Journal of Philosophy 108 (11).
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  50.  1
    Answers to 5 Questions.Paolo Mancosu - 2007 - In V. F. Hendricks & Hannes Leitgeb (eds.), Philosophy of Mathematics: Five Questions. Automatic Press/VIP.
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