260 LIFE: ITS NATURE AND ORIGIN 



servations, it is necessary to stress the difference between pure and 

 applied mathematics. 



Pure mathematics is comprised of arithmetic, algebra, geometry, 

 trigonometry, calculus, etc., while applied mathematics includes 

 mechanics, astronomy, navigation, and physics (including molec- 

 ular physics). Pure mathematics develops logical consequences 

 from assumed facts — indeed it has been said that a pure mathema- 

 tician is never so happy as when he does not know what he is 

 talking about. But when we come to apply mathematics to any 

 particular problem, we must be careful to start with proper facts 

 and to apply suitable mathematical forms or formulas. 



Scientific Facts and Mathematical Deductions 



An essential preliminary to the application of mathematics to 

 any problem is a statement of the facts and the assumptions in- 

 volved. The facts are determined by such careful observations as 

 we are able to make; the assumptions, either by our inability to 

 determine the facts, or by the compulsions of mathematical ex- 

 pediency. The resulting deductions are frequently given the hall- 

 mark of "rigid or rigorous mathematical analysis," notwithstanding 

 the assumptions or uncertainties on which they depend. 



Take, for example, Stokes' formula or "law," which is widely used 

 for calculating the fall of a sphere in a homogeneous fluid. While 

 weighing the electron on extremely small oil droplets, Millikan found 

 that Stokes' law failed, because to such tiny ultramicroscopic particles 

 air was no longer a homogeneous fluid, and the drops fell more 

 rapidly than the law indicated, through the voids between the mole- 

 cules of the gases comprising air. 



The often fruitful practice of mathematicians, physicists, and biolo- 

 gists of dealing with "fields" is a deliberately chosen form of escape 

 mechanism, which enables scientists to deduce and explain the beha- 

 vior of material units when complications of numbers, structures, and 

 interactions become too great for the human mind to follow. The 

 comparatively simple "three body problem," in general, is conceded 

 to be insoluble; but mathematicians deal safely with four, five, and 

 polydimensional space and with V — 1, an imaginary quantity. 



While throwing these sops to the mathematical Cerberus, we 

 attempt, as far as our mentality permits, to envisage the physical 

 mechanisms underlying phenomena. Energy may be correctly 

 expressed by a formula, but we visualize it as due to material 

 units in motion. When matter is "converted" into energy, as in 



