March 26, 1915] 



SCIENCE 



453 



new revelation will result when next the 

 possibility is shown of so interchanging un- 

 defined with defined ideas and postulates 

 with demonstrated propositions that, de- 

 spite such interchange of basal with super- 

 structural elements, the doctrine as an au- 

 tonomous whole will remain absolutely 

 unchanged. But this is not all nor nearly 

 all. It is only the beginning of what may 

 be made a veritable apocalypse. Of great 

 interest to any intellectual man or woman, 

 of very great interest to students of logic, 

 psychology, or philosophy, should be the 

 light which it will be possible in this con- 

 nection to throw upon the economic role of 

 logic and upon the constitution of mind or 

 the world of thought. I refer especially to 

 the recently discovered fact that in inter- 

 preting a system of postulates we are not 

 restricted to a single possibility, but that, 

 on the contrary, such a system admits in 

 general of a literally endless variety of 

 interpretations; which means, for such is 

 the make-up of our Gedankenwelt, that an 

 infinitude of doctrines, widely different in 

 respect of their psychological character and 

 interest, have nevertheless a common form, 

 being isomorphic, as we say, logically one, 

 though spiritually many, reposing on a 

 single base. And how foolish the instructor 

 would be not to avail himself of the oppor- 

 tunity of showing, too, in the same connec- 

 tion, how various mathematical doctrines 

 that differ not only psychologically, but 

 logically also, are yet such that, by virtue 

 of a partial agreement in their bases, they 

 intersect one another, owning part of their 

 content jointly, whilst being, in respect of 

 the rest, mutually exclusive and incompati- 

 ble. If, for example, it be some Euclidean 

 system that he has been expounding, he 

 will be able readily to show upon how seem- 

 ingly slight changes of base there arise now 

 this or that variety of non-Euclidean geom- 

 etry, now a projective or an inversion 



geometry or some species or form of higher 

 dimensionality. I need not say that anal- 

 ogous phenomena will in like manner pre- 

 sent themselves in other mathematical fields. 

 And it is of course obvious that as various 

 doctrines are thus made to pass along in 

 deliberate panorama it will be feasible to 

 point out some of their salient and distinc- 

 tive features, to indicate their historic set- 

 tings, and to cite the more accessible por- 

 tions of their respective literatures. Nat- 

 urally in this connection and in the atmos- 

 phere of such a course the question will 

 arise as to why it is that, or wherein, the 

 hypothetico-deductive method fails of uni- 

 versal applicability. So there will be op- 

 portunity to teach the great lesson that this 

 method is not rudimentary, but is an ideal, 

 the ideal of intellect and science; to teach 

 that mathematics is but the name of its 

 occasional realization ; and that, though the 

 ideal is, relatively speaking, but seldom 

 attained, yet its lure is universal, mani- 

 festing itself in the most widely differing 

 domains, in the physical and mechanical 

 assumptions of Newton, in the ethical pos- 

 tulates of Spinoza, in our federal constitu- 

 tion, even in the ten commandments, in 

 every field where men have sought a body 

 of principles to serve them as a basis of 

 doctrine, conduct or achievement. And if 

 it shall thus appear that mathematics is 

 very high-placed as being, in respect of its 

 method and its form, the ideal and the lure 

 of thought in general, the fault must be 

 imputed, not to the instructor, but to the 

 nature of things. 



In all this study of the postulational 

 method the impression will be gained that 

 the science of mathematics consists of a 

 large and increasing number of more or 

 less independent, somewhat closely related 

 and often interpenetrating branches, con- 

 stituting, not a jungle, but rather an im- 

 mense, diversified, beautifully ordered for- 



