Biology and Mathematics 241 



After the questions, what are facts? what is reahty? ques- 

 tions not to be answered either by biology or mathematics, there 

 come, if we decide to retain as rough working hypotheses the 

 expressions fact, reahty, subsequent questions, such as what 

 then is a geometric fact, a geometric reahty? 



These latter questions involve a wrestling with primitive 

 •origins in physiological psychology, now entangled with meta- 

 physical constructions, all being studied at present with help of 

 the biologically given hypothesis of evolution. 



To note the essential, inter-relation of biolog}^ and mathe- 

 matics it is only needful to recall that evolution postulates a 

 world independent of man, preceding man, and teaches the 

 production of man from lower biologic forms by wholly natural 

 causes. 



If this be so, then skipping the fundamental puzzle as to 

 how a living thing gets any conscious knowledge, any subjective 

 representation of that independent world, it remains of the very 

 essence of the doctrine of evolution that man's knowledge of 

 this independent world, having come by gradual betterment, 

 trial, experiment, adaptation, and through imperfect instru- 

 ments, for example the eye, cannot be metrically exact. 



In the easiest measurements it is said we cannot even with 

 the best microscopes go beyond one-millionth of a meter; 

 that is, we are limited to seven significant figures at most. What 

 is the meaning then of the mathematics which, as in case of the 

 evaluation of tt, has gone to seven hundred places of significant 

 figures ? 



If then we are to hold to evolution, science must be a con- 

 struction of the animal and human mind; for example, geometry 

 is a system of theorems deduced in pure logical way from certain 

 unprovable assumptions precreated by auto-active animal and 

 human minds. 



So also is biology. But here the assumptions are more 

 fluctuating, and many of them are still on trial. 



Since everv science strives to characterize as to size, num- 

 ber, and, where possible, spatial relations the phenomena of its 

 domain, each has need of the ideas and methods of mathematics. 

 One of the fundamental ideas of mathematics is the idea of 

 variation, the variable, qualitative and quantitative variability. 



When related quantities vary, one may vary arbitrarily, 

 this is called the independent variable. Others may vary m 

 dependence upon the first. Such are called dependent variables 

 or functions of the independent variable. The change of the 

 variables mav be continuous or discontinuous. The blind 

 prejudice for the assumption of continuity is so profound as to 

 be unconscious. 



