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  1. Russell's Unknown Logicism: A Study in the History and Philosophy of Mathematics.Sebastien Gandon - 2012 - Houndmills, England and New York: Palgrave-Macmillan.
    In this excellent book Sebastien Gandon focuses mainly on Russell's two major texts, Principa Mathematica and Principle of Mathematics, meticulously unpicking the details of these texts and bringing a new interpretation of both the mathematical and the philosophical content. Winner of The Bertrand Russell Society Book Award 2013.
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  • Wittgenstein et le logicisme de Russell : Remarques critiques sur « A Mathematical Proof Must be Surveyable » de F. Mühlhölzer.Sébastien Gandon - 2012 - Philosophiques 39 (1):163-187.
    Ce texte discute certaines conclusions d’un article récent de F. Mülhölzer et vise à montrer que le logicisme russellien a les moyens de résister à la critique que Wittgenstein lui adresse dans la partie III des Remarques sur les fondements desmathématiques.This paper discusses some conclusions of a recent article from F.Mülhölzer. It aims at showing that Russell’s logicism has the means to overcome the criticisms Wittgenstein expounded in Remarks on the Foundations of Mathematics, part III.
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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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  • ‘As If’ Reasoning in Vaihinger and Pasch.Stephen Pollard - 2010 - Erkenntnis 73 (1):83 - 95.
    Hans Vaihinger tried to explain how mathematical theories can be useful without being true or even coherent, arguing that mathematicians employ a special kind of fictional or "as if" reasoning that reliably extracts truths from absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence of core mathematical theories, but right about the utility of fictional discourse in mathematics. This essay explores this area of agreement between Pasch and Vaihinger. Pasch's position raises questions about structuralist interpretations of mathematics.
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  • Poincaré's Conception of the Objectivity of Mathematics.Janet Folina - 1994 - Philosophia Mathematica 2 (3):202-227.
    There is a basic division in the philosophy of mathematics between realist, ‘platonist’ theories and anti-realist ‘constructivist’ theories. Platonism explains how mathematical truth is strongly objective, but it does this at the cost of invoking mind-independent mathematical objects. In contrast, constructivism avoids mind-independent mathematical objects, but the cost tends to be a weakened conception of mathematical truth. Neither alternative seems ideal. The purpose of this paper is to show that in the philosophical writings of Henri Poincaré there is a coherent (...)
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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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  • Toward a Topic-Specific Logicism? Russell's Theory of Geometry in the Principles of Mathematics.Sébastien Gandon - 2009 - Philosophia Mathematica 17 (1):35-72.
    Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of geometry was sustained (...)
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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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  • ‘As If’ Reasoning in Vaihinger and Pasch.Stephen Pollard - 2010 - Erkenntnis 73 (1):83-95.
    Hans Vaihinger tried to explain how mathematical theories can be useful without being true or even coherent, arguing that mathematicians employ a special kind of fictional or “as if” reasoning that reliably extracts truths from absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence of core mathematical theories, but right about the utility of fictional discourse in mathematics. This essay explores this area of agreement between Pasch and Vaihinger. Pasch’s position raises questions about structuralist interpretations of mathematics.
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  • On Formalism Freeness: Implementing Gödel's 1946 Princeton Bicentennial Lecture.Juliette Kennedy - 2013 - Bulletin of Symbolic Logic 19 (3):351-393.
    In this paper we isolate a notion that we call “formalism freeness” from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability. We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic in the (...)
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