A brief history of numbers and counting, Part 1: Mathematics advanced with civilization
The origins of numbers are cloaked in mystery. But, I think it’s safe to say that as civilization advanced numbers advanced with it; and it is equally safe to say that civilization could not have advanced without it.
Common intuition, and recently discovered evidence, indicates that numbers and counting began with the number one. (Even though in the beginning, they likely didn’t have a name for it.) The first solid evidence of the existence of the number one, and that someone was using it to count, appears about 20,000 years ago. It was just a unified series of unified lines cut into a bone. It’s called the Ishango Bone.
The Ishango Bone (it’s a fibula of a baboon) was found in the Congo region of Africa in 1960. The lines cut into the bone are too uniform to be accidental. Archaeologists believe the lines were tally marks to keep track of something, but what that was isn’t clear.
But numbers, and counting, didn’t truly come into being until the rise of cities. Indeed numbers and counting weren’t really needed until then. Numbers, and counting, began about 4,000 BC in Sumeria, one of the earliest civilizations. With so many people, livestock, crops and artisan goods located in the same place, cities needed a way to organize and keep track of it all, as it was used up, added to or traded.
Their method of counting began as a series of tokens. Each token a man held represented something tangible, say five chickens. If a man had five chickens he was given five tokens. When he traded or killed one of his chickens, one of his tokens was removed. This was a big step in the history of numbers and counting because with that step subtraction — and thus the invention of arithmetic — was invented.
In the beginning Sumerians kept a group of clay cones inside clay pouches. The pouches were then sealed up and secured. Then the number of cones that were inside the clay pouch was stamped on the outside of the pouch, one stamp for each cone inside. Someone soon hit upon the idea that cones weren’t needed at all. Instead of having a pouch filled with five cones with five marks written on the outside of the pouch, why not just write those five marks on a clay tablet and do away with the cones altogether? This is exactly what happened.
This development of keeping track on clay tablets had ramifications beyond arithmetic, for with it, the idea of writing was also born.
But, if you’re keeping track of your wealth with marks made on a clay tablet what’s to stop you from making your own clay tablet and stamping in 50 marks, and trading those 50 marks on a clay tablet for grain?
To prevent this from happening, the Sumerians needed an official method of keeping track, and an official group of people who kept track. A select few were allowed to enter this group. They essentially became the world’s first accountants. So a farmer may have made his own clay tablet with 50 marks on it and claimed that this proved that he was the owner of 50 chickens, but if that tablet didn’t have an official seal from the accountants it was worthless.
It was the Egyptians who transformed the number one from a unit of counting things to a unit of measuring things. In Egypt, around 3,000 BC, the number one became used as a unit of measurement to measure length. If you’re going to build pyramids, temples, canals and obelisks you’re going to need a standard unit of measurement — and an accurate method of applying it to real objects. What they invented was the cubit, which they considered to be a sacred measurement. A cubit is the length of a man’s forearm, from elbow to fingertips, plus the width of his palm. Considered sacred as they were, they had officially ordained sticks which they kept in the temples. If copy cubits were needed they were made from one of the original cubits kept in the temple. Thanks to this very official, very guarded and very precise unit of measurement the Egyptians were able to create colossal buildings and monuments with wondrous accuracy.
The Egyptians were the first civilization to invent different symbols for different numbers. They had a symbol for one, which was just a line. The symbol for ten was a rope. The symbol for a hundred was a coil of rope. They also had numbers for a thousand and ten thousand. The Egyptians were the first to dream up the number one million, and its symbol was a prisoner begging for forgiveness, which was a person on its knees, hands upraised in the air, in a posture of humility.
Greece made further contributions to the world of numbers and counting, much of it under the guidance of Pythagoras. He studied in Egypt and upon returning to Greece established a school of math, introducing Greece to mathematical concepts already prevalent in Egypt. Pythagoras was the first man to come up with the idea of odd and even numbers. To him, the odd numbers were male; the evens were female. He is most famous for his Pythagorean theorem, but perhaps his greatest contribution to math was laying the groundwork for Greek mathematicians who would follow him.
Pythagoras was one of the world’s first theoretical mathematicians, but it was another famous Greek mathematician, Archimedes, who took theoretical mathematics to a level no one had ever taken it to before. Archimedes is considered to the greatest mathematician of antiquity and one of the greatest of all time. Archimedes enjoyed doing experiments with numbers and playing games with numbers.
But as trivial as his math games may have seemed to outsiders they often led to results that proved practical in the real world, some of which we still benefit from today. One example: Archimedes wondered if you could turn the surface of a sphere into a cylinder, and if you did, what would be the difference in area covered? Archimedes successfully worked this problem out, and to him that was the end of it. But thanks to the formulas he left behind, later mapmakers were able to turn the surface of the globe into a flat map.
Archimedes is also famous for his Archimede’s screw, which is a circular inclined plane (a screw) inside a tube that pumps water from one level to a higher level. He is equally famous for inventing a method of determining the volume of an object with an irregular shape. The answer came to him while he was bathing. He was so excited he leapt from his tub and ran naked through the streets screaming “Eureka!,” which is Greek for “I have found it.”
Archimedes made many, many other mathematical contributions, but they are too numerous to mention here during a brief history of numbers.
The Greek’s role in mathematics ended, quite literally, with Archimedes. He was killed by a Roman soldier during the Siege of Syracuse in 212 BC. And thus ended the golden age of mathematics in the classical world. Under the rule of Rome, mathematics entered a dark age, and for a couple different reasons.
In Part 2 we’ll look at numbers from the mathematical dark ages of the Romans to the modern digital age.
If you'd like to read Steven Law's previous science articles go to www.curiosity101.com.
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