The Age of Abstraction

(-600 to -300)

China

In northern China, near the Yellow River, city-states were united under the Chou dynasty, but each was run by independent lords who taxed the people heavily and had little concern for the poor. Partly in response to social chaos, the philosopher Confucius (-551 to -479) worked toward political change. His teachings mixed respect for authority with ethics in government, and humility with concern for the poor.

Toward the end of this period, the philosophy of Daoism also developed in reaction to poor government. Daoists believed that the universe had a natural order, which could be found through simplicity, peace, accommodating opposites and benevolent government.

As far as we know, the major accomplishment of Chinese mathematicians during this period was the introduction of a symbol (actually, a space) for zero as a placeholder in the decimal number system.

Greece

We know much more about the intellectual activity during this period in the region around what is now Greece. As in China, the Greeks lived mostly in city-states with little unification. Different city-states had different strengths. Aided by a number of fortuitous circumstances, the city-state of Athens became prosperous, and for about a century (-500 to -400) it had the closest thing there was to a democracy in the ancient world.

Out of the economic prosperity (and helped by a lot of slaves) grew a leisure class of free citizen men. These citizens produced great sculpture (Phideas, Polyclidos, and others), literature (such as Sophocles, Aristophanes, and Sappho), history (Herodotus and Thucydides), and medicine (Hippocrates).

And they produced philosophy. The best-known philosopher of the age was Socrates, followed by his student Plato and Plato's student Aristotle. These philosophers created an intellectual climate that influenced the work in other fields. For example, they stressed that the senses could not be trusted, and that truth and beauty could be found not in what was observed by the senses but only in the ideal. So the subjects of sculpture, such as the spear-bearer, or Dorypheros of Polyclidos, were not actual people, but rather were idealized people. This kind of abstract thinking extended to Greek mathematics as well.

Here are some of the mathematical problems the Greeks explored:

• What is number? Is everything based on number?
  
• Are there indivisible units of space and time?
  
• How can indivisible numbers be found?
  
• What are all of the regular polyhedra?
  
• The two perfect geometrical figures are straight lines and circles, which can be drawn with straightedge and compass. What other figures can be constructed using only straightedge and compass?
  
• How can we find the common measure of any two line segments as a ratio?
  
• Although the circle is one of the perfect figures, we see circles only if we look at them head-on. Otherwise circles appear to be ovals. How can we describe ovals mathematically?

These questions are indeed abstract. They're less obviously useful than the questions raised in the Age of Computation, yet their very abstractness led to mathematics that could be used in a wide variety of areas, besides intriguing mathematicians for centuries to come.

Further Reading

Boyer, Carl B., A History of Mathematics (New York: John Wiley & Sons, 1968).

Burton, David M., The History of Mathematics (USA: William C. Brown Publishers, 1991).

Eves, Howard, An Introduction to the History of Mathematics (Chicago: Saunders College Publishing, 1990).

Katz, Victor J., A History of Mathematics (New York: HarperCollins College Publishers, 1993).

Kline, Morris, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1972).