McLARTY objection to Cantor's set theory, and was among the first to use it. His ob- jection to Zennelo's axioms (apart from a late ambiguous one to the axiom.
Poincare": Mathematics &; Logic &; Intuitiont COLIN MCLARTY*
But how have we attained rigor? It is by restraining the part of intuition in science, and increasing the part of formal logic... we have attained perfect rigor. (Poincare' 1899, 129) What does the word exist mean in mathematics? It means, I say, to be free from contradiction. (Poincare' 1905-06, 297/474)
Recent work on Poincare''s philosophy of mathematics illuminates his controversy with Russell and such affinity as he has with Brouwer. But there is another side. Poincar6's view of logic is very like Russell's, and often taken from Russell with acknowledgement. A broader reading revises the view that 'his papers on foundations are disconnected from his positive work in mathematics' (Goldfarb 1988, 62) or 'His philosophic comments [on mathematics] are almost exclusively concerned with basic number theory, set theory, and logic' (Folina 1992, xi). And we can relate Detlefsen's 1992 and 1993 account of Poincare on intuition to Poincar^'s intuitive mathematical practice. Contrast three kinds of intuitionism: banal, expansive, and restrictive. The banal merely says research and teaching require something beyond formal rigor—perhaps 'motivation'. Expansive intuitionism claims the actual content of mathematics goes beyond any formalization. The restrictive rejects some standard mathematics as inaccessible to intuition. Poincar6 was an expansive intuitionist. His problem, which we inherit here, was to avoid lapsing into the banal. He was not restrictive. He objected to none of classical mathematics. He considered standard formal logic the guarantor of rigor in mathematics. He found Peano, Russell, and Couturat's new logistic cumbersome and prey to fundamental confusion—agreeing closely with Russell on this except for the prospect of future reform. He had little t Research supported by a grant from the National Endowment for the Humanities. I thank Michael Detlefsen, Janet Folina, Michael Friedman, Warren Goldfarb, Gregory Moore, and Michael Resnik for comments improving the paper. * Department of Philosophy, Case Western Reserve University, Cleveland, Ohio 44106, U. S. A. cxm70pop.cwru.edu PHILOSOPHIA MATHEMATICA (3) Vol. 5 (1997), pp. 97-115.
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objection to Cantor's set theory, and was among the first to use it. His objection to Zennelo's axioms (apart from a late ambiguous one to the axiom of choice) was only that he was unwilhng to trust formal axioms without either a consistency proof or a principled account of why just these axioms were chosen, especially when a key technical term remained unclear. 1. Poincar^ Champ,' >ns Logic Brouwer got Poincare right and Russell wrong when he wrote: How far Poincare' is from taking the intuitive construction as the only basis of his criticism, appears from his words: 'Mathematics is independent of the existence of material objects; in mathematics the word exist can have only one meaning, it means free from contradiction'. It might almost have been written by his opponent Russell. (Brouwer 1907, 96, quoting Poincare' 190506, 819/454) Poincare understood the logicists better, and chided Couturat for not taking consistency as the proof of existence (1905-06, 297/474). After defining the arithmetized continuum Poincare dismisses the demand for a constructive definition, saying 'no one will doubt the possibility of the operation, unless from forgetting that possible, in the language of geometers, simply means free from contradiction' (1893, 27/44). He also defends non-Euclidean geometries this way: 'A mathematical entity exists, provided its definition implies no contradiction' (1921, 61). He says such things repeatedly. For his one possible reversal of position see section 7 below. He valued logic as a counterpoise to intuition: We see we have progressed towards rigor; I would add that we have attained it and our reasonings [as opposed to many older proofs—CM] will not appear ridiculous to our descendents; I refer, of course, to those of our reasonings that satisfy us. But how have we attained rigor? It is by restraining the part of intuition in science, and increasing the part of formal logic. Before, one began with a large number of concepts regarded as primitive irreducible and intuitive; such were the concepts of whole number, fraction, continuous magnitude, space, point, line, surface, etc. Today only one remains, that of whole number; all the others are only combinations, and at this price we have attained perfect rigor. (1899, 129) Poincare well knew the accuracy and productivity of Weierstrassian analysis. He is often quoted saying: 'Heretofore when a new function was invented it was for some practical end; today they are invented expressly to put at fault the reasoning of our fathers; and one will never get more from them than that' (1899, 130, repeated in 1904, 264/435). But critique is something too, and Poincar6 less famously affirmed the need for it: 'Our fathers thought they knew what a fraction was, or continuity, or the area of a curved surface. We have found they did not know it' (1904, 265/437).
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To see how serious he was, see how he clarifies the fundamentals of curved surfaces in all his mathematics cited below. One more of many passages on this implies a comment on Poincar6's own work: Now in the analysis of today, when one cares to take the trouble to be rigorous, there can be nothing but syllogisms or appeals to this intuition of pure number, the only intuition which cannot deceive us. It may be said today that absolute rigor is attained. (1900, 122/216)
2. Poincar
