Delineating the Gap
“Between the idea
And the reality
Falls the Shadow.”
T.S. Eliot — The Hollow Men
The ancient Greek invention of hypothesis, together with the highly developed abstractions they developed, allowed mathematics to be extended far beyond simply dealing with numbers. But the Greeks had rooted their mathematics firmly in geometry, which they considered a critical part of the way the world worked. So they were always able to come back to something real for a comparison. They never lost sight of the gap between the real world and their abstractions.
The ancient Greeks were not able to develop tools to delve into the gap, but they were able to work around it. While Euclid’s Elements is purely abstraction, Eratosthenes’ measurement of the Earth was a highly practical application of basic geometry. Eratosthenes knew that his data were imperfect, and made allowances, recognizing that he was providing a rough estimate. However, that was better than no estimate, which was the situation prior to his exercise. In fact, he was very close to modern estimates, and it would be over a millennium before his result was improved upon.
Thomas Hobbes has suggested that the life of people living without any form of organized government was ‘nasty, brutish and short,’ but human beings seem to have organized themselves in a wide variety of ways since the dawn of humanity. As civilizations grew and became more complex, the resulting economic systems allowed the development of groups of people who had time to think about things in the world around them.
Among the early civilizations, that of ancient Greece seems to have attained a situation where a reasonably large group of people had the leisure to think about and discuss the world around them. While this situation also existed in most of the other great civilizations of the same period, the ancient Greeks seem to have been able to advance further than the others.
If we consider the period up to about 500 BC, there had been many developments in mathematics that attempted to solve the problems of building structures. That these methods succeeded can be seen in the longevity of the Pyramids and the magnificence of other structures of the period. But the people of the time recognized that their connection to the world was imprecise: measurements never closed perfectly, for example. So the mathematics of the time had this imprecision built into it. It was very much developed from experience and practice, not derived from theory. Indeed, the idea of ‘theory’ would have made little sense at the time.
This lack of basic numeracy is reflected in languages, even today. Many languages lacked words that related to numbers beyond two or three. The idea of number beyond physical objects didn’t exist, so the description of the number of objects often varied with the type of objects. English has a number of words that mean ‘two’ but are applied to a limited range of objects or today in limited circumstances: couple, pair, brace, twin, twain; and borrowed from other languages: dyad, duo, duet, duplet. But there are very few words that deal with three, four or five objects that have not been borrowed from other languages, and mostly from the counting numbers of those languages. Similarly, while most modern languages have singular and plural forms of words, some languages have three number forms for one, two, or more than two objects, e.g., Ancient Greek.
So the ancient Greeks were able to have groups of people with time to think, as did other civilizations of the day. However, the ancient Greeks introduced several important thinking developments or tools, which allowed them to develop their thinking in new and highly productive directions. Among these tools were:
- Deductive Logic; and
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