Jean Cavailles and Albert Lautman. La pensee mathematique. Separate papers by the two authors, with discussion by Cartan, Chabauty, Dubreil, Ehresmann, Freche, Hyppolite, Paul Levy, Shrecker, and the authors. Bulletin de la Societe Francaise de Philosophie, vol. 40, no. 1 (1946), pp. 1-39.
This is a report of a public discussion, held February 4, 1939, on views about the nature of mathematics expressed in two doctoral theses (III 167 and V 20) by Cavailles and Lautman, respectively. The brief opening statements made by the authors re-state their published doctrines; there follow critical comments by mathematicians and philosophers, and replies by the fist speakers.
Cavailles held that the attempt to reduce mathematics to logic ("ce fameux espoir de reduire les Mathematiques a la Logique") has failed - mathematics uses notions that are not purely logical, and Godel's work shows the impossibility of representing mathematics in a unique formal system. Accordingly, a philosopher of mathematics is justified in reaching the following positive conclusiions. Mathematics cannot be defined; constitutes a ,becoming ("un devenir"), a reality irreducible to anything except itself ("une realite irreductible a autre chose qu'elle-meme"); and is autonomous ("autonome") in the sense of developing unpredictably; but nevertheless subject to an inner necessity ("autonomie, donc necessite"); and mathematicians are engaged in an experiential activity ("une activite experiementale"); concerned with an object momentarily created by mathematical operations ("toujours correlatif de gestes effectivement accomplis par le mathematicien dans une situation donnee").
Lautman differs in holding mathematical truths to be objective: mathematics participates in a realm of Dialectic ("Dialectique"), constituting a higher reality of Ideas ("une realite plus haute et plus cachee, qui constitute, a mon avis, un veritable monde des Idees"). The task of mathematical philosophy is to erect the theory of Ideas, by describing the ideal structures incarnated ("incarnees") in mathematics, establishing a hierarchy of such structures, and demonstrating the reasons for their applications to the sensible universe ("les raisons de leurs applications a l'Univers sensible").
It is understandable that the comments of the mathematical participants in the discussion show them to be somewhat at a loss. I doubt, however, that their attitude 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 ofuscation. The work of Godel and Skolem, to which so much weight is attached in these remarks, by no means shows the impossibility of "reducing" mathematics to logic' nor is it ever made clear why mathematics should be in the peculiar position of resisting attempts at definition. In the absence of clear understanding of these relatively technical points, it seems premature to engage in metaphysical speculation about the ontological basis of mathematics.
Max Black, The Journal of Symbolic Logic, Vol. 12, No. 1, (Mar., 1947), pp. 21-22