So, numbers. Number theory is kind of interesting because numbers are something everyone knows, but almost no one really thinks about. They're the foundation of civilization as much as peanut butter toast, and are just as ancient.
Numbers are said to be the oldest, most abstract idea the human race has ever had. They may be the FIRST abstract idea we ever had. We've used them for so long, and learn them at such a young age, that I bet everyone is thinking "abstract? What's abstract about three?" well, quite a lot, actually.
It took the human race quite a long time to understand that this
(those are herd animals of early neolithic tribes; really) is the same as this.
And from there, it was really quite a huge leap to decide that those things are both the same as this:
Think about it. What does a 3 on a piece of paper REALLY have to do with three herd animals? Nothing, except in your mind. You could make any squiggle, and assign it the value of those three herd animals and call it three or Fred or George or cement, and it would be just as accurate as 3 is. VERY abstract. You can use the counting words for all kinds of wild stuff: three hours (time), three meters (distance), three goats (concrete stuff), three planets (astronomy), you name it. Invent numbers, and you can control (or pretend to control) the world.
What's really interesting is, numbers have been invented more than once. You'd think this would be a major big deal like the wheel and only get invented once. But no. People all over the world have developed different systems of varying complexity. Tally sticks - bits of wood, ivory, clay, whatever, with little hatch marks scratched into them - date back to the stone age.
This one is from Africa and is twenty five thousand years old. Sometimes, with these really old sticks, we can figure out what they were counting. Usually days, phases of the moon (of course there are piles of sticks we've no idea what they were counting). There are ancient caves with drawings of animals that have hatch marks next to them, for all the world like the 'kills' graphic on the sides of planes:
Except the ones in caves are thousands of years old and about bears and stuff.
The more things change, the more they stay the same.
Before you decide this is all hopelessly primitive and we've left it behind in our new, high-tech, computerized world, guess again. Anyone do this, counting rows while knitting?
I do it all the time. Guess what? Goes back to those twenty-five thousand year old tally sticks. You may be counting something different, but the method is exactly the same.
To go further with this, we now have to get technical. Unfortunately.
BASES (also known as radix)
There are almost as many number systems as there are languages - the two seem to go hand in hand. One of the ways they are classified is by base. It's the number of unique digits (including zero, if they have it), before the higher numbers are expressed as combinations of those unique numbers. For instance, the number fourteen is ten-and-four, linguistically; four-ten, fourteen. The technical, modern world runs - mostly - on base ten (I am not starting on binary and hexidecimal, at least not today). We've got the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. All other numbers are expressed as variations of those ten. Get it? (I hope so, 'cause I'm trying to keep this from getting complicated.)
Right. Well. Obviously base ten - the decimal system, deci from Latin or Greek or whatever meaning ten - comes from counting on our fingers. Simple, easy, count your herd of goats on your fingers and from there gradually move forward to the Grand Unification Theory, counting on your fingers the whole way. The vast majority of the planet ran on base ten, through human history. Chinese, Egyptians both ancient and modern, Greeks, Hebrews, Romans, Indian and Indus civilizations, they ran on base ten.
Occasionally there were civilizations who ran on base five - the fingers of one hand. The Carib and Arawak tribes in the Caribbean, some Oceania islanders, the Khmer of SE Asia, and assorted African tribes used base five, which makes sense and works just fine as long as you're counting goats and not moving into really complex math.
Then there were the civs that ran on base twenty - all your fingers and toes. The thought of their multiplication tables makes my head hurt, but using your fingers and toes to count on still makes sense. From linguistic studies, it's thought the Celts used base twenty (from whom we have the holdover word, score, as a special word for twenty). The Basques still use base twenty. Some parts of Scandinavia, Inuits, and the Ainu of Japan all ran on twenty. But the most notable folks to use base twenty were the Maya, who had an extremely complex mathematical and callendrical system they worked out using their numerical system.
Then it gets weird.
The Sumerians used base sixty. And because we ultimately inherited their writing and numbering systems, we have vestiges of that damned base sixty system floating around us, all day every day. Buy a dozen donuts? Special words for eleven and twelve? Twenty-four hours in a day, sixty minutes, sixty seconds, 360 degrees in a circle... all the Sumerians' fault.
Why? Well, that's for tomorrow.
Did I mention they were the world's first accountants? This is a bill of sale.