BioLtoGcy AND MATHEMATICS 241 
After the questions, what are facts? what is reality? 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, reality, subsequent questions, such as what 
then is a geometric fact, a geometric reality? 
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 biology 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 7, 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 every 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 in 
dependence upon the first. Such are called dependent variables 
or. functions of the independent variable. The change of the 
variables may be continuous or discontinuous. The blind 
prejudice for the assumption of continuity is so profound as to 
be unconscious. 
