THE PRINCIPLES OF SCIENCE 177 



... it is the glory of geometry that from those few principles, 

 brought from without, it is able to produce so many things. 



Does this sense of glory have another, deeper root? Beyond the 

 exemplary clarity they oflFer, do mathematical representations of the 

 world have for us another antipodal appeal rooted in an obscure 

 mysticism holding number to be the essence of things? However 

 paradoxically, this mysticism is close kin to the principle of intelligi- 

 bility—as when the Pythagorean philosopher Philolaus argues that: 



. . . Number, fitting all things into the soul through sense-percep- 

 tion, makes them recognizable and comparable with one another. . . . 

 Actually, everything that can be known has a Number; for it is im- 

 possible to grasp anything with the mind or to recognize it without 

 this. 



Even numerical laws (as distinct from theories) then take on a 

 special meaningfulness. Koestler remarks that: 



The Pythagorean discovery that the pitch of a note depends on the 

 length of the string which produces it, and that concordant intervals 

 in the scale are produced by simple numerical ratios (2:1 octave, 

 3 : 2 fifth, 4 : 3 fourth, etc.), was epoch-making: it was the first re- 

 duction of quality to quantity, . . . 



. . . The gross strings of the lyre are recognized to be of subordi- 

 nate importance; they can be made of different materials, in various 

 thicknesses and lengths, so long as the proportions are preserved: 

 what produces the music are the ratios, the numbers, the pattern of 

 the scale. Numbers are eternal while everything else is perishable; 

 they are of the nature not of matter, but of mind; they permit mental 

 operations of the most surprising and delightful kind without refer- 

 ence to the coarse external world of the senses— which is how the di- 

 vine mind must be supposed to operate. 



Pythagorean mysticism can be deleterious, for example in encour- 

 aging neglect of the "coarse external world of the senses." But it also 

 inspires an extraordinarily fruitful effort to construe phenomena of 

 nature in terms of simple mathematical "harmonies," both numerical 

 and geometric. That inspiration informs the work of Archimedes in 

 the ancient world, and plays a decisive role in the rebirth of science 

 in the modern world at the hands of such as Copernicus, Kepler, and 

 Galileo. For such men the discovery of mathematical harmony in 

 natural phenomena was in itself an explanation of those phenomena, 



