ABSTRACT

Schoenflies 8 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 9 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.10