Incan Counting System Decoded?

Discovery News ^ | Feb 3 2004 | By Rossella Lorenzi

[vannrox]

Incan Counting System Decoded?

By Rossella Lorenzi, Discovery News

Learn how to add 9+7 on the yupana abacus.

Jan. 29, 2004 ? The Inca invented a powerful counting system that could be used to make complex calculations without the tiniest mistake, according to an Italian engineer who claims to have cracked the mathematics of this still mysterious ancient population.

Begun in the Andean highlands in about 1200, the Inca ruled the largest empire on Earth by the time their last emperor, Atahualpa, was garroted by Spanish conquistadors in 1533.

Long been considered the only major Bronze Age civilization without a written language, they left mysterious objects that, according to the latest research, would have been used to store units of information.

Recent studies are investigating the hypothesis that elaborated knotted strings known as khipu contain a hidden written language stored following a seven-bit binary code. Nobody, however, had been able to explain the meaning of these geometrical tablets known as yupana.

Different in size and shape, the yupana had been often interpreted as a stylized fortress model. Some scholars also interpreted it as a counting board, but how the abacus would have worked remained a mystery.

"It took me about 40 minutes to solve the riddle. I am not an expert on pre-Columbian civilizations. I simply decoded a 16th century drawing from a book on mathematical enigmas I received as a Christmas present," engineer Nicolino De Pasquale said.

The drawing was found in a 1,179 page letter by the Peruvian Felipe Guaman Poma de Ayala to the King of Spain. A simple array of cells consisting of five rows and four columns, the drawing showed one circle in the right cell on the bottom row, two circles in the next cell, three circles in the other one and five circles in the last cell of the row. The same pattern applied to the above rows.

According to De Pasquale, the circles in the cells are nothing but the first numbers of the Fibonacci series, in which each number is a sum of two previous: 1, 2, 3, 5.

The abacus would then work on a base 40 numbering system.

"Instead, all scholars based their calculations according to a base 10 counting system. But calculations made to base 40 are quicker, and can be easily reconverted to base 10," Antonio Aimi, curator of the exhibition "Peru, 3,000 Years of Masterpieces" running in Florence, told Discovery News.

"Since we lack definitive archaeological evidence, we tested this claim on 16 yupana from museums across the world. De Pasquale's system works on all of them," Antonio Aimi, curator of the exhibition "Peru, 3,000 years of masterpieces" running in Florence, told Discovery News.

The Inca's calculating system (see an example of how it works in the slide show) does not take into consideration the number zero. Moreover, numbers do not exist as graphic representations.

According to Aimi, in most cases the Inca made their calculations by simply drawing rows and columns on the ground. The unusual counting way is described in an account by the Spanish priest José de Acosta, who lived among the Inca from 1571 to 1586.

"To see them use another kind of calculator, with maize kernels, is a perfect joy... . They place one kernel here, three somewhere else and eight, I do not know where. They move one kernel here and there and the fact is that they are able to complete their computation without making the smallest mistake," Acosta wrote in his book "Historia Natural Moral de las Indias."

The claim has sparked a dispute among scholars.

Gary Urton, professor of Precolumbian studies at Harvard University, an authority on khipu research, told Discovery News: "The fact that an explanation can be constructed for one or even several yupana that conforms to this theory of a base 40 numbering system amongst the Incas is of some modest interest.

"How would one explain the many statements in the Spanish chronicles, both those written by Spaniards and by literate Andeans, who stated quite straightforwardly that the Inca used a base 10 counting system? This system is also attested in a mountain of early colonial documents that describe how the Inca organized their administrative system according to a base 10 counting system."

As Aimi concedes, the claim has the limits of any interpretative system that isn't proven with definitive historical evidence.

"We would need to find a Rosetta yupana, something similar to the deciphering of Egyptian hieroglyphics from the Rosetta stone. Since we can't have it, I would consider a strong evidence the fact that the system works on all yupana examined," he said.

[vannrox]

Very cool.

[E. Pluribus Unum]

"How would one explain the many statements in the Spanish chronicles, both those written by Spaniards and by literate Andeans, who stated quite straightforwardly that the Inca used a base 10 counting system?

Well, we use a base 10 counting system in everyday life, but we use a base 2 counting system in our calculating machines.

[SpinyNorman]

I have to dust off my "Chariots of the Gods" book collection.....

[Redbob]

Where do you suppose we got our base-12 system?

And how well-developed could their civilization have been when they didn't know about wheels?

[Gothmog]

Their problem now is that all their base now belong to us, ha ha ha.

[Constitution Day]

LOL!

[E. Pluribus Unum]

Where do you suppose we got our base-12 system?

What base 12 system?

And how well-developed could their civilization have been when they didn't know about wheels?

We are talking about their mathematical counting system.

You must have a mighty large inferiority complex to want to turn that into passing judgement on a civilization that died out hundreds of years ago.

[martin_fierro]

"It took me about 40 minutes to solve the riddle. I am not an expert on pre-Columbian civilizations. I simply decoded a 16th century drawing from a book on mathematical enigmas I received as a Christmas present," engineer Nicolino De Pasquale said.

Memo to self: Must party sometime with that guy. Wooo hooo!

[martin_fierro]

LOL! Damn, ya beat me to it. For Great Justice.

[NotQuiteCricket]

How dare this man, this person think that he figured something out, when archaeologist have beaten their heads against this problem for decades! The nerve. He should have just shut up and not even bothered to mention it. Everyone knows that the only person who can make a judgment on something like this, is someone who has gone to university to study it. The arrogance of this fool to assume that he can understand this is amazing!

[George from New England]

12 inches = 1 foot
12 times 12 = 1 gross

[azhenfud]

"And how well-developed could their civilization have been when they didn't know about wheels?"

They were fixing to get ready to get a "round" to it...

[azhenfud]

1x12=1 dozen

[singletrack]

6 pack, 12 pack, case............

[E. Pluribus Unum]

Okay, so where did we get our base-12 system?

And what difference does it make?

[ClearCase_guy]

"We would need to find a Rosetta Yupana"

I'll check my High School Yearbook -- I think she's in there.

[mhking]

Just damn.

If you want on the list, FReepmail me. This IS a high-volume PING list...

[cinFLA]

How dare this man, this person think that he figured something out, when archaeologist have beaten their heads against this problem for decades! The nerve. He should have just shut up and not even bothered to mention it. Everyone knows that the only person who can make a judgment on something like this, is someone who has gone to university to study it. The arrogance of this fool to assume that he can understand this is amazing!

Are you so closed to new ideas? He makes no claims, just presents an idea. Most of the article is devoted to other people's evaluations of his idea. Did you even bother to read the article?

[elli1]

And how well-developed could their civilization have been when they didn't know about wheels?

The wheel invention spread by diffusion, invented in Sumaria around 3500 BC; the Brits didn't know about the wheel until approximately three thousand years later. In fact, evidence points to Mexico as the only place that invented the wheel (thought to have been a toy) independent of the Sumarian version. The folks in the Americas didn't know what to do with it, in part probably because they didn't have draft animals.

OTOH, it sure as heck took a long time for ''superior cultures'' to decipher the Incan calculator.

[Bikers4Bush]

1001110011010110010101000110101010011000110

[antidisestablishment]

"How would one explain the many statements in the Spanish chronicles, both those written by Spaniards and by literate Andeans, who stated quite straightforwardly that the Inca used a base 10 counting system? This system is also attested in a mountain of early colonial documents that describe how the Inca organized their administrative system according to a base 10 counting system."

It couldn't be that the Spaniards might have understood the system incorrectly. After all, we know how interested they were in cultural understanding:

the Inca ruled the largest empire on Earth by the time their last emperor, Atahualpa, was garroted by Spanish conquistadors in 1533.

Brilliant mathematicians those conquistadors! Outsiders bring a fresh perspective to "intractable" issues. Never look to experts for novel ideas.

[NotQuiteCricket]

Nope. Actually, I think it was cool. I should have used a /sarcasm tag, huh?

[azhenfud]

Don't bother - I think she's newly illegally immigrized.

[fishtank]

36-24-36

[Oberon]

Are you so closed to new ideas?

You have to imagine the [sarcasm] tags, then read it again. The comment is really an indictment of academia.

[elli1]

Cricket was sarcastically referring to the uppity ''scholar'', Gary Urton, professor of Precolumbian studies at Harvard University, an authority on khipu research,....who was obviously miffed that some lowlife dared to solve the problem in 40 minutes from a freaking picture in a book. Not only was it obvious that Cricket had read the article, but it was apparent that Cricket has a really good ear for tone.

(Kuddoes, Cricket!)

[Restorer]

Where do you suppose we got our base-12 system?

Sumeria. Also the 60 minute hour and the 360 degree circle.

And how well-developed could their civilization have been when they didn't know about wheels?

Wheels really don't have a great deal of utility without draft animals to pull a cart. Apparently, the wheel in all cultures was first used for chariots, carts or wagons. Some of its other uses, such as the wheelbarrow that is actually useful by man-power, didn't come along till thousands of years later. In western Europe the wheelbarrow wasn't used till the middle ages.

[G.Mason]

"The Inca invented a powerful counting system that could be used to make complex calculations without the tiniest mistake ....... "

Well it couldn't have been very reliable.

Either that or they never divided 200 Spaniards into the Incan population and came up with extinction.

[JohnnyZ]

Gary Urton, professor of Precolumbian studies at Harvard University, an authority on khipu research, told Discovery News: "The fact that an explanation can be constructed for one or even several yupana that conforms to this theory of a base 40 numbering system amongst the Incas is of some modest interest.

Always nice to see my boy Gary being quoted. Good guy. I gave a talk on Incan mathematics once. They did some pretty complex stuff, nobody really knows how. I'd have to read more about this theory to form an opinion.

[Don Joe]

What base 12 system?

They unearthed an Incan PDP-8. :)

(Yes, yes, I'm taking certain artistic liberties. Hey, it's a joke, man, a joke.)

[JohnnyZ]

Cricket was sarcastically referring to the uppity ''scholar'', Gary Urton, professor of Precolumbian studies at Harvard University, an authority on khipu research

Personally, I got the sarcasm, but Gary Urton is a highly intelligent, open-minded individual. He is indeed a scholar and has studied khipus, ethnomathematics, et al in great depth. He was was also awarded a MacArthur "genius" grant.

And he is most certainly NOT uppity! He's -- well, he's very laid back, let's just say.

[Don Joe]

The folks in the Americas didn't know what to do with it, in part probably because they didn't have draft animals.

I wonder why they didn't do what the scandinavians did with reindeer, and use deer for draft animals?

[Elsie]

12 times 12 = 1 gross

Madonna plus Brittany = ANOTHER gross!

[Vermont Lt]

Gotta love the Conquistadors:

"yeah, great counting system...can you lift your hair off your shoulders for a moment. Yeah, yeah....this is a Spanish necklance...."

[Elsie]

Big Indian or

Little Indian???

[Elsie]

Which came first?

Smooth roads or wheels??

---

Seems to me that BOTH have to be in place before much good can come from them.

[expatpat]

Excellent point, EPU.

[Don Joe]

Big Indian or

Little Indian???

Little-Indian. They were Intelligent, after all.

[Vermont Lt]

Perhaps this is the counting system that Liberman and Dean use that tells them they still have a chance in hell of getting the nomination.

Kind of like coutning in Dog Years.....

[Erasmus]

In western Europe the wheelbarrow wasn't used till the middle ages.

Throw out yer dead!

Throw out yer dead!

[FoxInSocks]

[Gary Urton] was was also awarded a MacArthur "genius" grant.

De Pasquale seems to be the genius in this particular case -- perhaps he should get the grant.

[r9etb]

1001110011010110010101000110101010011000110

Of course! 4E6B2A354C6!!! Of course!!!!

[Bikers4Bush]

Big, they sacrificed all the little ones.

[Bikers4Bush]

Lol. I thought about going to the trouble of making it intelligible and then thought better of it.

[Walkingfeather]

.... system amongst the Incas is of some modest interest. I was struck by the use of the word "modest" it sounds a bit like academic jealously. but will take your word that he is a nice guy.

[JohnnyZ]

I was struck by the use of the word "modest" it sounds a bit like academic jealously.

I think GU's working on the assumption that although the Italian dude's system could work, it probably wasn't the way the Inca did things. I don't know the full grounds for why he came to that assumption, but I guess it's primarily grounded in the Base10 vs Base40 debate, which he seems very confident about.

[cinFLA]

Point noted. My mind is slow this morning!

[Agnes Heep]

Everyone knows that the only person who can make a judgment on something like this, is someone who has gone to university to study it.

Well, that's logical. But since I never studied logic at the University, I suppose I'm not qualified to give an opinion on the subject!

[js1138]

There ar 10 kinds of people: those who understand binary and those who don't.

[NotQuiteCricket]

It is OK. Sometimes I get these overwhelming urges of aggravation about the "establishment" and the "establishment's" unwillingness to consider that there is someone somewhere who, because of their experience and knowledge (that is outside of the "establishment") is able to see to the bottom of an issue quickly. For example the sphinx - that is water erosion, and there is no way that you can escape from the *fact* that it is water erosion...but since that *fact* will not fall in line with any of the "establishment's" current *theories* that *fact* is ignored.

Here is an example of a mathematician, who, because he grooves on math, figured out a math problem (with a fairly good probability of being correct). Well, since the people who study archaeology don't know math inside, outside, upside, downside, they sniff and poo-poo this guy's idea.

I would say that it all comes out in the wash - but stuff like this is very heavily decided on who is doing the laundry.