“Old Bill” Suggests—
Far back in human thought, logic and mathematics became separated, perhaps because the language used in logic led through the study of Greek, while the application of mathematics led into the field of science. In more modern times, the two have drawn together until a university audience finds them merging into one.
Counting on our fingers, which is the most easily grasped mathematical concept for most of us, can lead either forward, into further complexity until we are juggling trillions and light-years by differentiation and integration, or backward by analysis to greater abstraction and logical simplicity.
Bertrand Russell, for instance, points out* that early Greek geometers, passing from empirical rules of Egyptian land surveying to general propositions by which these rules were found justifiable, and thence to Euclid’s axioms and postulates, were working backward into logic.
Taking natural numbers 2, 3, 4, 5, etc., as a middle ground, however, is a purely arbitrary estimate of our present level of comprehension. In a lower stage of civilization, the degree of abstraction involved in realizing that three days and three cows involved the same mathematical quality might have been difficult.
So far, it is too much to ask that the deepest levels of logic merge in turn into poetry and mysticism, yet we should be the gainers if poets ceased to shun commensuration and mathematicians realized that their symbols might also express, quite abstractly, the music of the spheres.
ROYAL F. MUNGER.
* Introduction to Mathematical Philosophy.