Nearly all mathematicians agreed that the way to solve these paradoxes was simply not to mention them; but there was some divergence of opinion as to how they were to be unmentioned. It was clearly unsatisfactory merely not to mention them. Thus Poincaré was apparently of opinion that the best way of avoiding such awkward subjects was to mention that they were not to be mentioned. But[108] “one might as well, in talking to a man with a long nose, say: ‘When I speak of noses, I except such as are inordinately long,’ which would not be a very successful effort to avoid a painful topic.”
Schoenflies, in his paper of 1911 mentioned above, adopted the convenient plan of referring these logical difficulties at the root of mathematics to a department of knowledge which he called “philosophy.” He said[109] of the theory of aggregates that though “born of the acuteness of the mathematical spirit, it has gradually fallen into philosophical ways, and has lost to some extent the compelling force which dwells in the mathematical process of conclusion.”
The majority of mathematicians have followed Schoenflies rather than Poincaré, and have thus adopted tactics rather like those of the March Hare and the Gryphon,[110] who promptly changed the subject when Alice raised awkward questions. Indeed, the process of the first of these creatures of a child’s dream is rather preferable to that of Schoenflies. The March Hare refused to discuss the subject because he was bored when difficulties arose. Schoenflies would not say that he was bored—he professed interest in philosophical matters, but simply called the logical continuation of a subject by another name when he did not wish to discuss the continuation, and thus implied that he had discussed the whole subject. Further, Schoenflies would not apparently admit that the one method of logic could be applied to the solution of both mathematical and philosophical problems, in so far as these problems are soluble at all; but the March Hare, shortly before the remark we have just quoted, rightly showed great astonishment that butter did not help to cure both hunger and watches that would not go.[111] The judgment of Schoenflies by which certain apparently mathematical questions were condemned as “philosophical,” rested on grounds as flimsy as those in the Dreyfus Case, or the Trial in Wonderland.[112]
[98] Chapters VII and XXXVI.
[99] Die Entwickelung der Lehre von den Punktmannigfaltigkeiten. Bericht, erstattet der deutschen Mathematiker-Vereinigung, Leipzig, 1908.
[100] Ibid., p. 7. The battle-cry is: “Gegen jede Resignation, aber auch gegen jede Scholastik!”
[101] “Ueber die Stellung der Definition in der Axiomatik,” Jahresber, der deutsch. Math.-Ver., vol. xx., 1911, pp. 222-5. The battle-cry is on p. 256 and is: “Für den Cantorismus aber gegen den Russellismus!”
[102] Ibid., p. 251. “Es ist also,” he exclaims with the eloquence of emotion and the emotion of eloquence, “nicht die Geringschätzung der Philosophie, die mich dabei treibt, sondern die Liebe zur Mathematik;...”
[103] “Ueber die Stellung der Definition in der Axiomatik,” Jahresber, der deutsch. Math.-Ver., vol. xx., 1911, p. 251.