Mathematics: A Beautiful Evolution

Most of the mathematical concepts we encounter every day - numbers, addition, subtraction - seem so basic, so hard to avoid in discussing reality on even the most basic level, that it's hard to imagine someone having to sit down and invent them. Who was the first person to look at two rocks and think, "Two more and I've got four?" The very idea almost seems absurd.

But mathematics is, in part, a language - not just a set of logical relationships and entailments that seems deeper than words, but a set of notations that allow us to discover those relationships. You can't see that twice two makes four, until you have a symbol for "two" that your brain can operate with. And those symbols - that language - did have to develop, strange as it may seem. (Prehistoric artifacts seem to indicate that the earliest humans had only four "numbers" at their disposal "none," "one," "two," and "many" - showing just how much our ability to talk about numbers depends on having the right words for them.)

We don't know which culture was the first to develop a number system more elaborate than "one, two, lots!" A 20,000-year-old bone found near the Nile River seems to show a sequence of prime numbers - which would indicate fairly sophisticated mathematical knowledge from fairly early on. Then there was the Harappan civilization of the Indus Valley in present-day North India and Pakistan. As far as we know, these folks were the first to use decimals, among many other important concepts.

Archaeology also seems to find evidence of a sophisticated number system during the Shang Dynasty in China, 1600 years before Christ. Archaeologists often turn up new discoveries bearing on the history of human consciousness - so it's hard to say who was the first to develop this or that idea with any certainty.

But many mathematical ideas - like many other things - begin with the Sumerians. This culture - considered by some historians the cradle of civilization - flourished near present-day southern Iraq between three and five thousand years ago, and besides contributing the world's first known work of literature (the still-impressive tale of Gilgamesh), they developed a numerical system based on sixes. If you've ever wondered why an hour has sixty minutes, or a minute sixty seconds - after all, it'd be much simpler if everything went by 100 (so that our basic unit of time was made of 100 smaller units, rather than sixty seconds, sixty minutes)- it's in part because of lingering Sumerian influence. As Sumer's culture declined, it was absorbed into the Babylonian Empire, which also seems to have produced mathematical thought, if the handful of Babylonian mathematical writings still remaining to us provides any indication.

Babylonians, Egyptians and ancient Indians all seem to have shared at least one important discovery - the so-called Pythagorean theorem, a rule having to do with how to figure the length of the sides of certain kinds of triangles. (Clearly, this discovery was of use to the culture that built the Pyramids.) The fact that this theorem was common to all three major ancient cultures suggests the degree of traffic they may have had with each other, despite some historians' suggestion that each culture was mainly closed-off to other places. And the fact that we know the theorem as the Pythagorean theorem - after the much-later Greek mathematician and philosopher Pythagoras - illustrates the well-known, and often-criticized, tendency of some historians to want to give the ancient Greeks credit for everything.

Not that the Greeks don't deserve plenty. Greek mathematics grew up alongside Greek philosophy and Greek science - indeed, the three disciplines weren't really separated; for the ancient Greeks, all the disciplines of knowledge were one thing. Thales, for example - whom you'll often find cited as the first Western philosopher - used geometry to calculate the height of the pyramids, among other things. In any case, Greek thinkers took the young art of math to a new level of sophistication. Euclid wrote a geometry textbook so percipient as to remain useful today, Aristotle defined laws of logic, and Archimedes remains near the top of some math historians - all-time greats list. The tight relationship between math and philosophy in ancient Greek is well-expressed by the inscription found on the door of Plato's Academy: "Let nobody ignorant of geometry enter here."

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