We have already[98] referred to the contempt shown by some mathematicians for exact thought, which they condemn under the name of “scholasticism.” An example of this is given by Schoenflies in the second part of his publication usually known as the Bericht Über Mengenlehre.[99] Here[100] a battle-cry in italics—

“Against all resignation, but also against all scholasticism!”—

found utterance. Later on, Schoenflies[101] became bolder and adopted a more personal battle-cry, also in italics, and with a whole line to itself:

“For Cantorism but against Russellism!”

“Cantorism” means the theory of transfinite aggregates and numbers erected for the most part by Georg Cantor. Shortly speaking, the great sin of “Russellism” is to have gone too far in the chain of logical deduction for many mathematicians, who were perhaps, like Schoenflies,[102] blinded by their rather uncritical love of mathematics. Thus it comes about that Schoenflies[103] denounces Russellism as “scholastic and unhealthy.” This queer blend of qualities would surely arouse the curiosity of the most blasÉ as to what strange thing Russellism must be.[104]

Schoenflies[105] said that some mathematicians attributed to the logical paradoxes which have given Russell so much trouble to clear up, “especially to those that are artificially constructed, a significance that they do not have.” Yet no grounds were given for this assertion, from which it might be concluded that the rigid examination of any concept was unimportant. The paradoxes are simply the necessary results of certain logical views which are currently held, which views do not, except when they are examined rather closely, appear to contain any difficulty. The contradiction is not felt, as it happens, by people who confine their attention to the first few number-classes of Cantor, and this seems to have given rise to the opinion, which it is a little surprising to find that some still hold, that cases not usually met with, though falling under the same concept as those usually met with, are of little importance. One might just as well maintain that continuous but not differentiable functions are unimportant because they are artificially constructed—a term which I suppose means that they do not present themselves when unasked for. Rather should we say that it is by the discovery and investigation of such cases that the concept in question can alone be judged, and the validity of certain theorems—if they are valid—conclusively proved. That this has been done, chiefly by the work of Russell, is simply a fact; that this work has been and is misunderstood by many[106] is regrettable for this reason, among others, that it proves that, at the present time, as in the days in which Gulliver’s Travels were written, some mathematicians are bad reasoners.[107]

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]

CHAPTER XXXVIII