This is a repost of an old column that, given Congressional interest in militarily invading Mexico, may be of relevance.
Inasmuch as America has a large population of Latin Americans, it seems to me that people, or some people, might want to know about them, and what they are, and where they came from. Most Latinos of the south are either a mixture of Spanish and Indian, or sometimes pure Indian. We have some idea of the Spaniards. They were European. But what were the Indians? What is their contribution to the great numbers of–whether you like it or not–new Americans? In particular, what are their blood lines? Are they, as nativists insist, of very low IQ–83–and have they enstupidated the Spanish? Horrendously primitive?
Without thinking about it, I had the entrenched idea that they were just that. I wasn’t conscious that it was either an idea or entrenched–just a fact. It didn’t occur to me that I knew virtually nothing about these people, or that there was anything to know.
What pulled me up short was their architecture. Throughout a large region, sort of Yucatan through parts of Honduras, you find ruined cities of monumental architecture that would match most of what is found in the ancient Near East. A great deal of it is overgrown with jungle. To get to major sites like Palenque, you walk through dim trails with unexplored walls and passageways. But the existence of these ruins did not set well with the idea of primitive incapacity. The architecture was entirely Indian since they had no contact with Europe.
Aha! I thought with the brilliance of one who has been hit over the head by the obvious. Something screwy is going on here. How witless can you be and engineer these things? I started poking around. And found interesting stuff. For example:
The Maya invented a sophisticated base-20, positional-exponential number system, including zero. The invention of zero is regarded as major advance in mathematics, and occurred in India for sure and perhaps in other places, though never in Europe. Until Fibonacci brought zero back from the Hindu-Arab world, Europe used Roman numerals, a horrible system. I knew this, but had never thought about it. Well, it’s worth a little pondering.
In a positional number system, a number–7, say–has an absolute value–in this case unsurprisingly 7–as well as a different value depending on its position. For example, in the number 100,007, seven means, well, 7. In 100,070, its value is 70, and in 10,700, its value is 700.
“Exponential” means that each position in a number represents a different power of the base, in our case 10. Thus we have ten to the zero power equals one, to the first power, ten; squared, 100, cubed, 1000, and so on.
The Maya, using base twenty, had a similar progression, going 1, 20, 400, 8,000, 160,000 etc.. (Inevitably the choice of 20 as the base is attributed to our number of fingers and toes, though I have trouble imagining anyone actually counting on his toes.)
Neither of these ideas is obvious, or anywhere approaching obvious. Both eluded Archimedes, for example. They seem natural to us because were are steeped in them from the first grade and, since everyone has had high school algebra, exponents seem routine. Using a thing and inventing it are very different animals. Any bright freshman can sling definite integrals; it took a Newton to invent them.
Imagine that you are a Mesoamerican Indian somewhere in Central America trying to figure out how to deal with large numbers. The fact that you are interested in large numbers suggests that you are not stupid. You have never had high-school algebra or heard of exponentiation. I cannot imagine how you would get from here to “Eureka!” (though as a Maya you probably didn’t know Greek either).
The idea “Hey, what if I line up powers of 20, multiply them by sort of coefficients, and add them….?”–is a huge intellectual leap. So far as I can determine, it only happened twice. It never happened in Europe.
For the mathematically curious, the Maya system had a remarkable peculiarity. Number systems, or anyway all I have heard of, require a number of symbols equal to the base. For example, binary, base-2, has two symbols, 0 and 1; decimal, base-10, ten symbols 0-9; and hexadecimal, base sixteen, 0-F. So I thought, Oh help, I’m going to have to memorize twenty symbols of some weird sort. In fact, the Maya ran a base-20 system with only three symbols representing 0, 1, and 5. That is truly strange, but it works. If interested, the link above explains it nicely.
For the record, from The Story of Mathematics: “The importance of astronomy and calendar calculations in Mayan society required mathematics, and the Maya constructed quite early a very sophisticated number system, possibly more advanced than any other in the world at the time ….The pre-classic Maya and their neighbours had independently developed the concept of zero by at least as early as 36 BCE, and we have evidence of their working with sums up to the hundreds of millions, and with dates so large it took several lines just to represent them. ”
Curious from a Stone Age people, which they essentially were. I note that Europe did not invent zero.
It is often said that the Maya never invented the wheel. Actually they did. Hundreds of these wheeled pull-toys for children have been found. Several writers have commented that it is difficult to understand why the Maya were unable to make the mental leap to the idea of making full-sized carts. But of course they could. Thing is, there were no animals to pull them, such as horses or donkeys. Making a mental leap to horses does not get you a horse.
Others have asked why they didn’t make carts and push them themselves. For a couple of reasons, I suspect. With no metals to make shod wheels, they would have to have used wood which would quickly have worn out. Also because of a lack of iron or steel, a cart able to carry, say, five hundred pounds would have to have been made of heavy wooden beams, to include wheels and axles. To push a five-hundred-pound cart with a similar load for, say, a mile, would have meant moving a thousand pounds for that distance, whereas dividing the load into parts and carrying it by mean would mean moving only five hundred pounds for that difference.
The Maya in the popular mind are thought to have been murdering, torturing savages given to human sacrifice. This is probably because they were in fact murdering, torturing savages given to human sacrifice. Why this is thought especially reprehensible is a mystery. The Romans sacrificed huge numbers in the arena so that the public could enjoy watching them die, crucified large numbers, and poured molten lead down the throats of criminals. In the European witch hunts, sort of 1450-1750, some 500,000 were killed depending on whose numbers you accept, mostly by burning alive. The Tudors hanged criminals, cut them down still alive, opened their abdomens and removed their bowels while still alive, and had four horses attached to their arms and-legs put them into pieces. And of course everybody and his dog put entire cities to the sword, from Joshua to Hiroshima. Despite their best efforts the Maya could not keep up with the moderns.
The invention of writing is among the major intellectual achievements of humanity and one that occurred at most three or four times on the planet, and perhaps fewer. Specialists argue, idiotically in my view, over whether Chinese was or was not influenced by earlier writing. Specialists have to do something with their time. What is not arguable:
Wikipedia: “It is generally agreed that true writing of language (not only numbers) was invented independently in at least two places: Mesopotamia (specifically, ancient Sumer) around 3200 BC and Mesoamerica around 600 BC. Several Mesoamerican scripts are known, the oldest being from the Olmec or Zapotec of Mexico.”
The Maya script is logosyllabic and said to be functionally similar to Japanese, to which it is utterly unrelated. It is not “proto-writing,” but actual real writing. This was not immediately known because the script had not been deciphered, but now about ninety percent can be read. This doesn’t help as much as might be expected since the Spanish Christians, as destructive as the Muslims of today, burned almost all Maya books–codices actually–and so everything we know comes from inscriptions carved on buildings. Imagine how we would look to Martians with the same problem. The book to read if interested is Breaking the Maya Code.
The aesthetic is a matter of taste but these to my eye appear artistically respectable. The Maya of today do nothing in math and technology, but retain a fine sense for design and the use of color.
Maya Civilization Keeps Growing
The general public knows little of the Maya and, until recently, archaeologists were not much better. This is changing. For example, some 60,000 Mayan structures, previously unknown, were recently found in the Guatemalan rain forest. A few snippets and link:
BBC: “Results from the research using Lidar technology, which is short for “light detection and ranging,” suggest that Central America supported an advanced civilization more akin to sophisticated cultures like ancient Greece or China….The archaeologists were struck by the “incredible defensive features,”which included walls, fortresses and moats….
“With this new data it’s no longer unreasonable to think that there were 10 to 15 million people there,” said Mr Estrada-Belli,..Another discovery that surprised archaeologists was the complex network of causeways linking all the Maya cities in the area. The raised highways, allowing easy passage even during rainy seasons, were wide enough to suggest they were heavily trafficked and used for trade
Again from The Story of Mathematics: The Maya “were able to measure the length of the solar year to a far higher degree of accuracy than that used in Europe (their calculations produced 365.242 days, compared to the modern value of 365.242198), as well as the length of the lunar month (their estimate was 29.5308 days, compared to the modern value of 29.53059).” Try to imagine how they did it.
It is interesting that Europe invented neither writing, zero, nor its number system, but the Mesoamericans did all three. Perhaps the Indians were enstupidated by the admixture of Spanish blood. While this is all good fun, it again raises the question of how and why groups pass through periods of intellectual fertility and then stop, as the Maya certainly have. Always there is some pat genetic explanation that doesn’t make sense, can’t be established, or both. But the Indians did what they did. Interesting stuff, no?
The ancient Romans had the concept of zero, tho they called it “nulla” (nothing).
All this makes me wonder how did the Maya get from understanding and measuring the lunar and solar cycles to several decimal places and building as they did to where they are now? Or the Romans/Greeks etc.?
Shirt sleeves to shirt sleeves in 3 generations on a civilization scale?
Fun to ponder and maybe watch in real time in US.
Indeed, the Maya are known to have had very advanced astronomy (and concurrently astrology)…They had determined the location of the Galactic center, not known to European or Mediterranean civilizations…Which was amazing given the poor viewing conditions in Central America, then and now…
Numbers in Modern mathematics are quite different than what we learned in K-12. For example, many of the laws of physics exhibit symmetries which are expressed in the language of mathematical group theory. But, multiplicative groups can’t have 0 as an element. Also, these groups are only logically consistent with relatively-prime elements. So, nature seems to abhor 0 in systems with multiplicative symmetry. That kind of symmetry dominates physics, but not all, as additive groups do work well with 0 and they express much symmetry in physics. More important than 0 is the ability to map to an “identity” element, both within group theory and in its homomorphisms and isomorphisms. Math is mostly about maps, not numbers. In algebras, we add a second operator, so 0 is usually a part of algebras, but the mapping of one mathematical object to another becomes more important than “numbers.” For example, energy and other numbers are just scalars in a functional or maybe a quadratic form such as a Hamiltonian. Numbers lack structure to describe physical laws. e.g. a Hamiltonian operator describing the physical mechanics of an object is expressed not with numbers, but as a differential similarity relation between a static configuration (generalized position variable) and a dynamic configuration, such that only those states that conserve total energy are related. Their coordinates are numbers, but the essential description of their mechanical properties are purely mathematical.
We tend to think of numbers as having an infinite extent, but numbers that are finite form much better mathematical structures (and in software). “Real” numbers are actually just approximations in mathematics.
The mathematics taught in K-12 is hopelessly primitive — just a small step beyond stone age. In the 1960s, the Defense Department tried to push modern math into K-12, through “New Math,” during the space race with the Soviets. It was a great idea, but the public was openly hostile to it. That was a turning point for the nation. Instead of pushing towards an advanced civilization, we retreated to a stone-age culture.
It seems to me all (great) civilizations blossom, bloom then collapse – many times from external pressures but just as often from internal decay. Look at modern Iraq and Egypt; who would think these people were the creators of all we have? A visit to Newgrange near Dublin made me realize that a civilization older than Egypt once thrived in The Irish Isles. China has gone through many such cycles, but somehow has been able to reinvent itself over and over – which is exemplary. In Fred’s lifetime more or less, Great Britain has gone from THE global power to a backwater that is collapsing from within. We in America are in the process of the same inevitable collapse so it’s not easy to see, or acknowledge. I am extremely proud of my European heritage and ancestry, they also have been able to reinvent themselves after horrendous collapses from invasions, diseases, wars and revolutions. Maybe we are in a period where the Iraqis, Egyptians, Maya and Cambodians are in a rebuilding phase? Anyway, thanks Fred, I have read you many years. Everyday I read about some outlandish new cultural affront or war danger, I remember one article you wrote in which you said (paraphrasing) if the world is collapsing there’s nothing one can do about it – so I choose to sit back, pop open a beer and enjoy the fire!
Other ancient number systems: Babylonians used base 60. The abacus counting device, used by Chinese, is based on the biquinary number system.
Ancient writing: Egyptian hieroglyphics were developed around 3100 B.C. Ancient Phoenicians developed a writing system around 1,00p B.C. that had about 20 letters in it, each of which represented a certain vowel or consonant sound. Using their writing system spread all around their merchant empire in the Mediterranean Sea area and gave rise to both the Greek, Roman, and Hebrew alphabets, inter alia.
There are scores, or perhaps hundreds, of writing systems used around the world for thousands of years, all based on a small number of differently shaped symbols, each representing a different vowel or consonant sound.
About 150 years ago an American Cherokee invented a writing system for all the different sounds in the Cherokee language. New languages are still being invented by authors; e.g., Tolkien for Middle earth novels, Enya for some songs, and Ewok and Klingon for Star Wars movies.
And then there is Esperanto.
Nice article. What are your thoughts on a topic called the cocaine mummies. It’s on YouTube. I highly suggest it.
How were the roofs of the temples on the pyramids made? Were they waterproof?
Thanks Fred…. You enlighten us again — Very Interesting. History does indeed repeat itself with !only the names, faces and locations changing. The Sun doesn’t shine on the same dog all the time regardless of what too many US citizens believe.
I’ve long had my doubts about whether the indigenous peoples of the Andean and Caribbean actually did murder on a huge scale, and commit sacrifices.
After all, It was reported by the Spaniards who attacked the Inca – killing them for their gold, a typical European activity – that “they cried out in puzzlement as we drew knives and attacked, stabbing them. They raised their hands, and asked why the Gods were hurting them, and seemed to have no idea how to defend themselves. they were clearly amazed and stunned by our actions”.
This does not sound like a people in the habit of killing each other. More, we should have learned by now not to trust anything an invading, looting, killing European has to say on the subject. They lie to cover their evil, and try to shape perceptions in the minds of their listeners. We do know many native peoples died from measles and other European diseases, to which they had no natural immunity.
To accuse them of being murderous sacrificing savages when it was the invaders who brought death sounds very standard and has been done over and over and over, and in fact is still being done to this day, by the invading murderous Europeans. In England, we were taught how the Chinese were nothing but ignorant savages, who lived in hovels like dogs, and killed nice kind English just trying to help them. Took a while for the truth of the Opium Wars and the desecrations and destruction of the marvellous architecture like the Summer Palace, to reach a wider audience.
No, I’m not sure at all that the Maya were anything like their European invaders and certainly wont be taking the European word for it any time soon.
Well, OK Fred, the Maya were geniuses 1000 years ago.
Maybe los hermanos de Columbia hiked up the Darien with baskets of nose candy, and wrecked their civilization?
Maybe the Spanish estupidated the Maya rather than the opposite?… Could be, but I don’t think the timelines work.
Maybe los hermanos de las estrellas came down, gave them all this STEM stuff, and then left them to their own devices?
[…] The Maya: Who Would Have Thought It? by Fred Reed […]
All true, but most of who you speak, having come to the US; are going to reference themselves to the Aztecs rather than the Mayans.
The Maya were smart and resourceful. Like all great civilizations, they moved their smartest and most intelligent to the front of the pack. As such they were able to subdue and use other tribes around them. In Belize I toured the pyramids at Xinantinech and will never forget an insight the curator shared: Architecturally the Pyramids were quite simple. A box filled with dirt and another box built on top of that. Virtually no interior rooms. No flying buttresses, or other arches. What was amazing, he said, was the ability to feed and engage the amount of labor required.
Yeah ok. My ancestors were vikings and they were a vastly misunderstood and beautiful people. Oh yeah and they enslaved and slaughtered people sometimes. Just because your wife is a beautiful woman and you have a beautiful child together doesn’t mean that her ancestors didn’t do horrible shit. Believe whatever helps you sleep better man, just lock up the knives (smart ass emoji)