The Borders of Understanding
My molecular biologist friend, Barry Bowman, brewed up this stimulating thought: One can understand much about nature without mathematics. Can one understand quantum mechanics only through its arcane mathematics? Can one not comprehend quantum physics other than through its mathematical expression?
To explore the thought some agreement is needed on the meaning of understanding! I offer, for our data bank of shared images of nature, the following on mathematical understanding.
There exists counting because thingness is a property of nature; there are things in the world; in particular there are things that can be classified by sameness. You can count them. Eventually counting became indispensible for the pursuit of everyday affairs. Six shells for those two axe heads.
An understanding of integers derived from the needs of human interaction; of commerce. Those same needs brought about the notion of addition. Two shells plus four shells make up the six I need. The interesting thing about addition is that it always works; you add two numbers and you get another number.
The exigencies of business produced subtraction. It is the inverse of addition. If the merchant had six and two are removed he has four left. The remainder is four because adding two to four yields six. The addition is inverted. The remarkable thing about subtraction was this: It didn’t always work. Four minus six did not produce another number! It produced a meaningless thing. By ‘number’ was meant what we now call a positive integer. At the start of mathematics nothing else was known.
Imagine the brilliance of the man who gave meaning to a meaningless notion; to 4 minus 6! He said, “Let us accept ‘minus 2′ as a form of number – like 2 itself”. He invented negative numbers. So the practice of counting gave rise to a new kind of thing. Negative numbers acquired physical meaning: the number lacking. A feature of nature took mathematical expression.