Philosophia Mathematica 1 (1):24-49 (1993)

Michael Detlefsen
University of Notre Dame
In the early years of this century, Poincaré and Russell engaged in a debate concerning the nature of mathematical reasoning. Siding with Kant, Poincaré argued that mathematical reasoning is characteristically non-logical in character. Russell urged the contrary view, maintaining that (i) the plausibility originally enjoyed by Kant's view was due primarily to the underdeveloped state of logic in his (i.e., Kant's) time, and that (ii) with the aid of recent developments in logic, it is possible to demonstrate its falsity. This refutation of Kant's views consists in showing that every known theorem of mathematics can be proven by purely logical means from a basic set of axioms. In our view, Russell's alleged refutation of Poincaré's Kantian viewpoint is mistaken. Poincaré's aim (as Kant's before him) was not to deny the possibility of finding a logical ‘proof’ for each theorem. Rather, it was to point out that such purely logical derivations fail to preserve certain of the important and distinctive features of mathematical proof. Against such a view, programs such as Russell's, whose main aim was to demonstrate the existence of a logical counterpart for each mathematical proof, can have but little force. For what is at issue is not whether each mathematical theorem can be fitted with a logical ‘proof’, but rather whether the latter has the epistemic features that a genuine mathematical proof has.
Keywords Poincaré  Russell  Kant  proof  role of logical reasoning in proof  intuition  intuitionism  constructivism  non-logical character of mathematical proof
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DOI 10.1093/philmat/1.1.24
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References found in this work BETA

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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.
Towards a Theory of Mathematical Argument.Ian J. Dove - 2009 - Foundations of Science 14 (1-2):136-152.
Poincaré's Conception of the Objectivity of Mathematics.Janet Folina - 1994 - Philosophia Mathematica 2 (3):202-227.

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