Nature is ever like to herself—­the very essence of the modern scientific spirit, had yet to be born of years of unwearied labour and unceasing delving into Nature’s innermost secrets.  Only in Mathematics—­in the properties of geometrical figures, and of numbers—­was the reign of law, the principle of harmony, perceivable.  Even at this present day when the marvellous has become commonplace, that property of right-angled triangles . . . already discussed . . . comes to the mind as a remarkable and notable fact:  it must have seemed a stupendous marvel to its discoverer, to whom, it appears, the regular alternation of the odd and even numbers, a fact so obvious to us that we are inclined to attach no importance to it, seemed, itself, to be something wonderful.  Here in Geometry and Arithmetic, here was order and harmony unsurpassed and unsurpassable.  What wonder then that Pythagoras concluded that the solution of the mighty riddle of the Universe was contained in the mysteries of Geometry?  What wonder that he read mystic meanings into the laws of Arithmetic, and believed Number to be the explanation and origin of all that is?"[1]

[1] A Mathematical Theory of Spirit (1912), pp. 64-65.

No doubt the Pythagorean theory suffers from a defect similar to that of the Kabalistic doctrine, which, starting from the fact that all words are composed of letters, representing the primary sounds of language, maintained that all the things represented by these words were created by God by means of the twenty-two letters of the Hebrew alphabet.  But at the same time the Pythagorean theory certainly embodies a considerable element of truth.  Modern science demonstrates nothing more clearly than the importance of numerical relationships.  Indeed, “the history of science shows us the gradual transformation of crude facts of experience into increasingly exact generalisations by the application to them of mathematics.  The enormous advances that have been made in recent years in physics and chemistry are very largely due to mathematical methods of interpreting and co-ordinating facts experimentally revealed, whereby further experiments have been suggested, the results of which have themselves been mathematically interpreted.  Both physics and chemistry, especially the former, are now highly mathematical.  In the biological sciences and especially in psychology it is true that mathematical methods are, as yet, not so largely employed.  But these sciences are far less highly developed, far less exact and systematic, that is to say, far less scientific, at present, than is either physics or chemistry.  However, the application of statistical methods promises good results, and there are not wanting generalisations already arrived at which are expressible mathematically; Weber’s Law in psychology, and the law concerning the arrangement of the leaves about the stems of plants in biology, may be instanced as cases in point."[1]