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To the Pythagoreans belongs by common consent the distinction of having raised mathematics to the level of a science. We may well feel some astonishment when we reflect that this great achievement, which implies in the achievers the spirit of sober scientific reasoning, should nevertheless have been the work of men who were also enthusiasts and mystics. But we can find it in no way strange that the complication of these disparate tendencies should have resulted in the profession of philosophical doctrines, respecting the world-significance of numbers, which to us appear fantastic to the verge of absurdity, and which not improbably wore that appearance to some few contemporary intellects, more critical if less original and
When we consider these peculiarities of mental constitution, reflect upon the nature of the intellectual matrix which brought forth mathematics as a science, and call to mind that universally, and at all times, curious superstitions and extravagant notions appear to have been associated with ideas of number, the hypothesis not unnaturally suggests itself that even now the philosophy of mathematics may not wholly have freed itself from mystical implications. Such a supposition, though it may prove to be ill-founded, can hardly be regarded as inherently improbable. Not, indeed, that there is any obvious connexion between mathematics and mysticism: the former is a subject of thought, the latter, to speak generally, is an attitude of mind, probably the effect, or the concomitant, of temperamental disposition. But it may be that all ratiocinative processes, no matter
what the subject, in which the current and continual substitution of symbols (of any kind) for concepts is a prime condition of the effective conduct of the process, are provocative of that attitude of mind. I believe this to be in general the case, though with great differences between individuals; and I incline to think that not infrequently the more gifted is the individual for the prosecution of purely symbolic trains of thought the greater is the provocation to this mental attitude. Ability to resist this tendency to lapse into the mystical must, however, increase more and more with the growing insight into the nature of our mental processes which is afforded us by the progressive development of psychological investigation. The recent emergence in the domain of philosophy of what is known as Pragmatism is clearly a result of this progressive development, and Pragmatism may not inaptly be characterized as a methodical and determined attempt to rid philosophy of mysticism.
Of the two mathematical doctrines discussed respectively in Part II and Part III of this book, the first-that of Imaginary Quantity in Algebra and Imaginary Loci in Geometry-seems to have attracted but little attention from the philosopher pure and simple. Discussion of the principles which underlie this development of mathematical expression has been confined almost exclusively within the inner circle of mathematicians interested in the philosophy of their science. But wellnigh up to the close of last century the explanations of this development-especially with regard to Imaginary Quantity-were almost purely mystical: they really explained nothing, threw no light on the motives and processes of thought which obscurely prompted it, and thus left it as much of an enigma as ever. In recent years, however, a real endeavour has been made to unravel this long-standing difficulty in the development of algebraic symbolism; it is even averred that we have now at last
a completely adequate explanation of it. The claim is, in other words, that the doctrine has been purged of its mystical implications. But, as I hold, the claim outruns the performance: the purgation is not thorough, and Part II of the present work is in the main an attempt to make it thorough.
Very different were the circumstances which attended the growth of the doctrine discussed in Part III. Metageometry, or Pangeometry, or the theory of non-Euclidean spaces and geometries, was mainly developed in the second half of the nineteenth century. It gave rise to a controversy in which the opposition to the doctrine proceeded almost wholly from the philosophers, especially from the Kantians, the great body of mathematicians meanwhile standing aloof from the conflict and apparently taking little or no interest in it. That the philosophers were worsted in the encounter can hardly be denied. Their comparative ignorance of mathematics occasionally led them to misunderstand the arguments of their opponents, and hence to raise objections which damaged their cause instead of furthering it. The result was that opposition from the purely philosophical side was practically silenced. On the other side, while the doctrine has become much more widely known to and studied by mathematicians in general, it has not among these met with universal acceptance: there are some who regard it as having no real significance, there are others who remain merely sceptical.
It should be added that there are points upon which the protagonists of the metageometrical theory differ from one another, and that these differences are clearly relevant to the philosophy of the doctrine. Its real significance, if it has any, thus still remains a matter of doubt, and must so remain until these differences are composed. Thus persistence in criticism of this theory, even should the criticism turn out to be mistaken, provided it be intelligent, a 3
will be justified by its utility; for it must then at least indirectly aid in bringing about that complete concordance which is still to seek in the philosophical interpretation of the theory.
With respect to Part I, the general character of which is psychological, I gladly take this opportunity of acknowledging my indebtedness to Dr. W. McDougall, Wilde Reader in Mental Philosophy in the University of Oxford, who was kind enough to read it in manuscript and to give me the benefit of some valuable suggestions.