44 ILLINOIS ACADEMY OF SCIENCE 



statement of law which may be more useful in the future than 

 it is now, as it is extended to wider fields of science. 



Time does not permit more than a mention of the far-reach- 

 ing dependence upon the laws of statistical mathematics that 

 science has immediately at hand. Problems of distribution, 

 organized and unorganized ; problems of large numbers, prob- 

 lems of values that are given only as averages, problems of 

 functions that merely must keep their values within a given 

 approximation, all these suggest that the conception even that 

 a measurement actually has a unique value which could be de- 

 termined exactly is also an assumption. 



But I hear the objection raised, that the fact that the mathe- 

 matician is able to amuse himself with creations of this kind 

 is one thing, and that the laws of nature belong to their appli- 

 cations is another. I need only remind you, however, that 

 conic sections were studied many centuries before Kepler lived, 

 and that wireless telegraphy is very dependent upon the square 

 root of minus one. This order of events does not always 

 occur but it happens often enough to answer the objection. 



In the effort to explain the universe, science is driven more 

 and more toward the postulate that the universe is infinitely 

 complex, and away from the postulate that the universe is 

 comparatively simple. The intricacies of phenomena increase 

 year by year, and the scientist, like the mathematician, is com- 

 pelled to admit that to generalize merely by adding more terms 

 is a very poor way to generalize. The generalization neces- 

 sary to handle nature, like the generalization necessary to 

 handle increasing knowledge of mathematics, is plainly a gen- 

 eralization of kind, that kind of generalization which will ex- 

 hibit the simple case as a degenerate form of the usual case. 

 A more profound insight into the tangle of phenomena shows 

 that the threads are not simple and well known curves merely 

 mixed together, but that they are in reality infinitely complex 

 curves intertwined with themselves. That some of tliem from 

 certain viewpoints project into straight lines, or circles, or 

 simple helices, is purely an accident. Since this is the case it 

 becomes plainly evident that progress in science is very de- 

 pendent upon the creative power of the mathematician in 

 matching intricacy in nature with intricacy in mental con- 

 struction. It becomes very plain that such heavy assumptions 



