Tuesday, June 23, 2009

Great zeros in history.

Now then. There is great debate and hair-splitting in math history circles about 'true zeros' and 'some kind of place holder that isn't a zero for some dumbass reason I will expound upon for an hour/ten pages'. It seems to me that the defining point for the hard core math geeks has to do with whether the zero symbol is used in a 'true' positional notation system. (The definition of 'true positional notation system' varies by who is talking/writing, of course.) So I'm going to just include the most noteworthy of the symbols that meant 'nothing' or 'empty set' or whatever from around the world. If you want to explain to me why the Mayan monumental zero wasn't a true zero because it wasn't a 'real' positional notation system, go right ahead in the comments, but know that I will roll my eyes and shake my head when I read it.

I'm trying to do this in chronological order, but there's some overlap. Also when possible I will include the date of the earliest known usage of the symbol, and a picture of said symbol.


The Babylonians inherited the unholy number/cuneiform writing system originally developed by the Sumerians. This civilization ran in a fairly unbroken line for thousands of years, with breaks to change names and move locations. So while I was talking the other day about Sumerians and base sixty, THOSE Sumerians were about four thousand years back from THESE Babylonians.

That said, the 'double wedge' shown above is considered a symbol "to indicate the absence of units of a given order of magnitude" (from Georges Ifrah). Meaning it was used just like our zero today. Their notation system DID use positions like ours does, but because of the base sixty, it was not a simple progression by tens like ours is - 1, 10, 100, 1000, etc. Their system went up in sixties (near as I can figure without the degree in advanced math I'm not ever getting): 1, 60, 3600, 216000. The exact 'steps' of the system vary a bit over the four or five thousand years the civilization shifted around, but by the time they settled on a positional notation system about 2000 BCE, that was generally how they did it. Interestingly, I think, is that they used the positional notation system for about two thousand years before finally inventing (or deciding to use) a zero symbol, under the Selucid Turks in about 300 BCE.


The Mayans seem to have had TWO types of zero, to go with their two types of writing. The first kind was a fairly straightforward text style that was used for actually doing mathematical computation; the zero from that system is the eye-shaped symbol in the upper left of the picture. The second type of writing was the flowery symbolic 'writing' they used for monuments, shown in the bottom row of the above picture - there were several versions of the zero used for monuments. Mostly they look like four-petaled flowers. They'd make a cool tattoo for a math history geek.

The Mayans are kind of entertaining because they only NEEDED the first zero. The Mayans ran on base twenty and wrote their numbers out much the same way we do in base ten, with the positional notation and all (ones, twenties, four hundreds, etc). It was an efficient system, which is probably why the Maya were some of the best mathematicians in the ancient world. However, when they did inscriptions on monuments, they wrote their numbers out formally, so - for instance - 1234 would be written out "one thousand, two hundreds, three tens, four ones". Obviously for that kind of system you don't need a zero to hold a place, you just skip the empty positions - 1000 would be "one thousand". But for religious and aesthetic purposes, the Mayans liked to spell it out. "One thousand, zero hundreds, zero tens, zero ones." So it's a useless sort of zero, though kinda pretty.

Their real zero, in their math papers and notes, dates back to at least 300 BCE. Our understanding of the Mayans is kind of sketchy thanks to Christian missionaries torching huge piles of their manuscripts, so it's possible the New World zero is older. We just don't know.


This one is ours; you can see the evolution of it above, as it traveled along the trade routes from India, where it was invented (along with the rest of the positional notation system), to the Arabs, and then to Western Civilization.

There is a text (the Ganitasarasamgraha, if you must know) dated to 850 CE that has calculations that math types agree HAD to be done with some kind of positional notation. Experts claim from analysis of complex manuscripts that the entire system - positional notation and a use of zero for empty 'slots' - was in place by about 950 CE but not much before that. There appears to be some kind of academic slug-fest over these dates, with accusations of forgery and misinterpretation flying about, so have a big grain of salt with this. (Personally, I suspect the system goes back further and we just haven't found evidence of it yet.) There is, in fact, commentary by an Arab dude (Severus Sebokt; no, I did not make that up) written in 662 CE, talking about how the Greeks didn't know jack and just inherited their civilization from the Middle East, and how the Hindus had a better number system that ran on 'only nine figures'. This is interesting for two reasons. First, of course is the implication that the base ten system is older than the Hindu manuscripts can demonstrate, and second is that even in the 600s, people were sick to death of hearing about how awesome the Greeks are (I'm in august company).

From there, the system moved over to the Arabs, who were busy dominating the trade routes between east and west and making big bucks off it. They needed a good system to count their big bucks, and adopted the Indian base-ten-with-zero method to do so. We're not sure when it happened - more academic slug-fests there - but by looking at the symbols above, anyone, even those of us with no official training, can see the evolution.

Western civ is said to have discovered the numbers while slogging about, killing people, during the crusades. I'm kinda skeptical. I think what really happened is the system oozed north from Spain (which was held by Arab peoples from 711 to 1492 CE) and west from Byzantium which, while considered European, had Asian trade routes running through it constantly, and had close ties to Western Europe until it was taken by the Ottoman Turks in 1453 CE. Just like most of the rest of Arab knowledge got into Europe (including knitting).

There you go, those are the high points anyway. I'm sure there are thousands of gory details I could have included, but I wanted a brief overview, not a dissertation.

At the moment, I feel an urge to write about the Phonecians. I'm tired of numbers.


Louiz said...

Very interesting. Himself asked, when I said you were going to do a post on zero, whether you would point out the difference between "null" and "zero". Himself would be what I would call a complete numbers geek though. Not to mention a programmer who actually has to use the concept of null in his programmes.

Alwen said...

Finally catching up on Girl Genius, what with power being out - Friday's! Pierogi wagon! And "Why, you're not Flopsy, Mopsy, and Nietzsche!"

Hold me while I fall down laughing...

Shoveling Ferret said...

Hurray, someone else who is with me on the whole Greeks = overrated!

Emily said...

I'm having to read this one over & over. My first surprise, in the previous post, was the positional thing. (Told ya I was dumb.)That idea is so obvious that I was amazed it had never occurred to me.