Everett J. Nelson

Mar 1, 1947

Citations

0

Citations

Journal

Journal of Symbolic Logic

Abstract

pates in a realm of Dialectic ("Dialec t ique") , constituting a higher reality of Ideas ("une rgalite' plus haute e t plus cached, qui consti tute, a mon avis, un veritable monde des Ide«s") . The task of mathematical philosophy is to erect the theory of Ideas, by describing the ideal s t ructures incarnated ("incarn^es") in mathematics, establishing a hierarchy of such s tructures , and demonstrating the reasons for their applications to the sensible universe ("les raisons de leurs applications a l 'Univers sensible") . I t is understandable that the comments of the mathematical participants in the discussion show them to be somewhat a t a loss. I doubt, however, tha t their a t t i tude of wondering respect is justified. The metaphysical conclusions presented by Cavailles and Lautman seem to the reviewer to be based upon fragmentary and incomplete evidence and to be expressed in a terminology which can only promote obfuscation. The work' of Godel and Skolem, to which so much weight is at tached in these remarks, by no means shows the impossibility of "reducing" mathematics to logic; nor is i t ever made clear why mathematics should be in the peculiar position of resisting a t t empts a t definition. In the absence of clear understanding of these relatively technical points, i t seems premature to engage in metaphysical speculation about the ontological basis of mathematics. M A X BLACK

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