Poincaré against the logicians

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

Michael Detlefsen
University of Notre Dame
Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type.
Keywords Poincaré  Brouwer  constructivism  intuitionism  mathematical proof  the role of logic in mathematical proof  logicism  Russell  rigor  mathematical architecture
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DOI 10.1007/BF00500033
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Prolegomena to Any Future Metaphysics.Immanuel Kant - 2007 - In Elizabeth Schmidt Radcliffe, Richard McCarty, Fritz Allhoff & Anand Vaidya (eds.), Journal of Philosophy. New York: Blackwell. pp. 507-508.
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The Foundations of Arithmetic.Gottlob Frege - 1953 - Evanston: Ill., Northwestern University Press.
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Mathematical Inference and Logical Inference.Yacin Hamami - 2018 - Review of Symbolic Logic 11 (4):665-704.
Towards a Theory of Mathematical Argument.Ian J. Dove - 2009 - Foundations of Science 14 (1-2):136-152.
Towards a Theory of Mathematical Argument.Ian J. Dove - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), Foundations of Science. Springer. pp. 291--308.
Definitions And Contradictions. Russell, Poincaré, And Lesniewski.François Lepage - 2008 - The Baltic International Yearbook of Cognition, Logic and Communication 4.

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