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Poincaré against the logicians

Synthese 90 (3):349 - 378 (1992)

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  1. Poincaréan Intuition Revisited: What Can We Learn From Kant and Parsons?Margaret MacDougall - 2010 - Studies in History and Philosophy of Science Part A 41 (2):138-147.
    This paper provides a comprehensive critique of Poincaré’s usage of the term intuition in his defence of the foundations of pure mathematics and science. Kant’s notions of sensibility and a priori form and Parsons’s theory of quasi-concrete objects are used to impute rigour into Poincaré’s interpretation of intuition. In turn, Poincaré’s portrayal of sensible intuition as a special kind of intuition that tolerates the senses and imagination is rejected. In its place, a more harmonized account of how we perceive concrete (...)
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  • The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today.Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.) - 2006 - Dordrecht, Netherland: Springer.
    This book explores the interplay between logic and science, describing new trends, new issues and potential research developments.
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  • Logic, Logicism, and Intuitions in Mathematics.Besim Karakadılar - 2001 - Dissertation, Middle East Technical University
    In this work I study the main tenets of the logicist philosophy of mathematics. I deal, basically, with two problems: (1) To what extent can one dispense with intuition in mathematics? (2) What is the appropriate logic for the purposes of logicism? By means of my considerations I try to determine the pros and cons of logicism. My standpoint favors the logicist line of thought. -/- .
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  • Intuition in Mathematics: from Racism to Pluralism.Miriam Franchella - forthcoming - Philosophia:1-37.
    In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians who interacted with Luitzen Egbertus Jan Brouwer in order to compare their conceptions of intuition. We will see how to the same word “intuition” very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty of considering concepts (...)
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  • Towards a Theory of Mathematical Argument.Ian J. Dove - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), Foundations of Science. Springer. pp. 291--308.
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  • Non-Deductive Logic in Mathematics: The Probability of Conjectures.James Franklin - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Springer. pp. 11--29.
    Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...)
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  • Searching for Pragmatism in the Philosophy of Mathematics: Critical Studies / Book Reviews.Steven J. Wagner - 2001 - Philosophia Mathematica 9 (3):355-376.
  • Una reevaluación del convencionalismo geométrico de Poincaré.Pablo Melogno - 2018 - Dianoia 63 (81):37-59.
    Resumen: Janet Folina ha propuesto una interpretación del convencionalismo de Poincaré contraria a la que ofrecen Michael Friedman y Robert DiSalle. Ambos afirman que la propuesta de Poincaré queda refutada por la relati-vidad general pues supone una noción restrictiva de los principios a priori. Folina sostiene que el convencionalismo de Poincaré no es contradictorio con la relatividad general porque permite una noción relativizada de los princi-pios a priori. Intento mostrar que la estrategia de Folina es ineficaz porque Poincaré no puede (...)
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  • What is Categorical Structuralism?Geoffrey Hellman - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 151--161.
  • Why Do We Prove Theorems?Yehuda Rav - 1999 - Philosophia Mathematica 7 (1):5-41.
    Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...)
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  • Definitions And Contradictions. Russell, Poincaré, And Lesniewski.François Lepage - 2008 - The Baltic International Yearbook of Cognition, Logic and Communication 4.
    This paper is composed of two independent parts. The first is concerned with Russell’s early philosophy of mathematics and his quarrel with Poincaré about the nature of their opposition. I argue that the main divergence between the two philosophers was about the nature of definitions. In the second part, I briefly present Le!niewski’s Ontology and suggest that Le!niewski’s original treatment of definitions in the foundations of mathematics is the natural solution to the problem that divided Russell and Poincaré.
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  • Constructive Type Theory and the Dialogical Approach to Meaning.Shahid Rahman & Nicolas Clerbout - 2013 - The Baltic International Yearbook of Cognition, Logic and Communication 8 (1).
    In its origins Dialogical logic constituted one part of a new movement called the Erlangen School or Erlangen Constructivism. Its goal was to provide a new start to a general theory of language and of science. According to the Erlangen-School, language is not just a fact that we discover, but a human cultural accomplishment whose construction reason can and should control. The resulting project of intentionally constructing a scientific language was called the Orthosprache-project. Unfortunately, the Orthosprache-project was not further developed (...)
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  • Poincaré: Mathematics & Logic & Intuition.Colin Mclarty - 1997 - Philosophia Mathematica 5 (2):97-115.
    often insisted existence in mathematics means logical consistency, and formal logic is the sole guarantor of rigor. The paper joins this to his view of intuition and his own mathematics. It looks at predicativity and the infinite, Poincaré's early endorsement of the axiom of choice, and Cantor's set theory versus Zermelo's axioms. Poincaré discussed constructivism sympathetically only once, a few months before his death, and conspicuously avoided committing himself. We end with Poincaré on Couturat, Russell, and Hilbert.
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  • On Dialogues, Predication and Elementary Sentences.Shahid Rahman & Nicolas Clerbout - 2013 - Revista de Humanidades de Valparaíso 2:7-46.
    En sus orígenes la lógica dialógica constituyó la fundamentación lógica de un nuevo movimiento llamado Escuela de Erlangen o Constructivismo de Erlangen, el que debía proporcionar un nuevo comienzo a una teoría general del lenguaje y de la ciencia. En lo referente a teoría general del lenguaje, la Escuela de Erlangen afirma que el lenguaje no es un hecho que descubrimos, sino una realización cultural humana cuya construcción puede y debe ser controlada. El proyecto de un desarrollo constructivo de un (...)
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  • An Inferential Community: Poincaré’s Mathematicians.Michel Dufour & John Woods - 2011 - In Frank Zenker (ed.), Proceedings of the 9th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 18-21, 2011. Windsor, Canada: pp. 156-166.
    Inferential communities are communities using specific substantial argumentative schemes. The religious or scientific communities are examples. I discuss the status of the mathematical community as it appears through the position held by the French mathematician Henri Poincaré during his famous ar-guments with Russell, Hilbert, Peano and Cantor. The paper focuses on the status of complete induction and how logic and psychology shape the community of mathematicians and the teaching of mathematics.
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  • Towards a Theory of Mathematical Argument.Ian J. Dove - 2009 - Foundations of Science 14 (1-2):136-152.
    In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent (...)
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  • Carroll’s Infinite Regress and the Act of Diagramming.John Mumma - 2019 - Topoi 38 (3):619-626.
    The infinite regress of Carroll’s ‘What the Tortoise said to Achilles’ is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry.
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  • Henri Poincaré.Gerhard Heinzmann - forthcoming - Stanford Encyclopedia of Philosophy.
  • Louis Joly as a Platonist Painter?Roger Pouivet - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 337--341.
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  • Mathematical Inference and Logical Inference.Yacin Hamami - 2018 - Review of Symbolic Logic 11 (4):665-704.
    The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a (...)
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