Comment by alanctgardner2 | Hacker News Reader

Comment by alanctgardner2 | Hacker News Reader

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Correction: to run the algorithm you're required to have at least as many stones as the end result. That's like saying because MAX_INT is set, math on computers is pointless. Any physical process will have a finite upper bound on the size of the result.

I agree that this isn't a remarkably efficient algorithm, but it is novel (it's completely unintuitive to the average reader), and it has a sound basis in mathematics. For the people who employed it, it was an efficient algorithm, or else they wouldn't have continued to use it. Nobody is proposing you're not as smart because you don't do math this way (although frankly, I doubt if you would have derived this by yourself, under the same conditions).

On a side note, I strongly recommend 'Guns, Germs and Steel'. I've been knocked for mentioning it before because it's a pop-sci book, but it explains why cultures 'progressed' at different rates because of largely environmental factors. This might help you overcome your ridiculously defensive bias against African cultures.

Okay, first and foremost, the algorithm obviously was never actually used in the way this story describes, because it is totally ridiculous and that is not even the point of the article.

But let's say it was. In saying that the shaman knows how to put two sets of 7 stones in a particular pile, it is implying that the shaman also knows how to put 34 sets of 7 stones in a particular pile. He also knows how to count them, so the problem is already solved. What he is doing is a bit more "rube goldberg" than novel. It may have parallels with some real algorithm for multiplication, but it is certainly not a very direct representation of it.

Does it help if you don't call them "stones" but call them "bits"?

(https://en.wikipedia.org/wiki/Multiplication_algorithm#Peasa...)

No, the algorithm as described is very different from binary multiplication. It requires a number of stones proportional to the result, while the number of bits required when doing binary multiplication is polylogarithmic.

It's a big difference: 1000 * 1000 would require 1000000 stones if done with the "Ehiopian algorithm", but it can be done with less than 1000 stones by using normal binary multiplication.

No, it absolutely does not help. In the algorithm you are citing, the numbers never leave their binary representation; there is no counting of the units that make up the number. The step of multiplying by 2 is done by shifting the binary representation of the number, which is not the way that this supposed "Ethiopian algorithm" works. The long multiplication algorithm (in binary or not) has complexity O(log(NM)), where N and M are the numbers to be multiplied, while this algorithm has complexity O(NM), and also gets there in a roundabout way.

This algorithm is totally pointless if you're not going to work with a positional notation for the number; I am sure it was never used in the way described, with stones. Anyone capable of understanding the problem in the first place - 34 goats, 7 pieces per goat - would count out the price - here is 7 for the first goat, 7 for the next, etc. - and tell the shaman to take a hike. This story amounts to calling Ethiopians idiots for taking an algorithm suited for positionally represented numbers - or at least numbers in some kind of a concise representation - and using it in a totally unnecessary way with numbers represented as groups of stones, and not understanding the pointlessness of all the drudgery.

We know that the algorithm was used because it was specifically described in ancient Egyptian texts. The calculator would use either symbols on paper or objects representing successively larger numbers (1,5,10 etc) to perform the algorithm. He obviously wouldn't use only a single stone to count out each number.

It couldn't have been efficient since it is patently idiotic. Just spread 34 groups of 7 stones each on the ground and count them! That's all you have to do. No algorithm at all, just symbolic logic, and you need fewer stones, and probably takes less time as well. The only way to use this algorithm to portray the mathematical prowess of native Ethiopian culture in some positive way is to argue that the Shamans purposefully used this algorithm to obfuscate the process and maintain the life style and privileges of the priestly class. I dare you to claim with a straight face that this "algorithm" is even remotely comparable to what the Greeks did, e.g. Archimedes' proving, PROVING I repeat, in a mathematically rigorous way, in the 3rd century BC, that a sphere has 2/3 the volume and surface area of its circumscribing cylinder.

You're getting a bit hung up on the stones.

This algorithm allows people to multiply two numbers if all they can do is multiply and divide by 2, and add.

> It couldn't have been efficient since it is patently idiotic.

And yet is it so efficient it is how computers multiply.

>This algorithm allows people to multiply two numbers if all they can do is multiply and divide by 2, and add.

Yes, and the algorithm of making N groups of M and then counting allows people to multiply if all they can do is count. And they will do it far faster than the shaman every time.

>And yet is it so efficient it is how computers multiply.

No, it isn't.

> Yes, and the algorithm of making N groups of M and then counting allows people to multiply if all they can do is count. And they will do it far faster than the shaman every time.

So why didn't they?

I don't know. I notice that I am confused, so I know that between my existing beliefs and the details of this story, something important is fictional.

My strongest guess is that it's the stones. In any base > 1 the algorithm is efficient. In unary (counting stones), it inefficient to the point of being nonsensical.

So my guess is that this was not used by counting stones, but with some form of positional number system, and that in the retelling, stones have been added as a way to make it sound more "tribal".

Edit: Alternatively, it may be the idea that they're doing this exactly. If the doubling side is done by rough estimation (eyeballing the size of the piles), it might be faster.

I don't get why there has to be a 'positive' or 'negative' angle on the story. You're trying to compare two different cultures in terms of which is 'better', which is a completely pointless goal. Nobody is saying native Ethiopians developed abstract mathematics as complicated as those of Greek, Arab, Chinese, etc. cultures.

It's not pointless if your goal was to justify racism and discrimination. Then the argument makes perfect sense while still being wrong.

You are comparing a later development of mathematics to an ancient one and then calling the ancient one stupid.It is basically like studying string theory now and then calling Galileo's speculations idiotic because he used his pulse to measure time instead of a quantum-logic clock.

No, he is comparing an ancient system (ethiopian multiplication) with an even more ancient one (just fucking counting) and calling it stupid. As far as I can tell, he's got a point.

That's a fine argument with 7 times 34, because you only need 7 holes of 34 stones.

But it breaks down when multiplying much larger numbers, because the number of holes you need increases by N. With this addition system, you only need log(N) holes.

The system exists to remove cumbersome aspects of multiplying large numbers by counting. Consider 34x34. While one fellow is out digging 34 holes, making sure not to make a mistake, or finding a piece of wood that has 34 holes marked in it, the other guy never needs more than 10 holes if his numbers are less than 1024. This keeps his working area small, and he can see all of it at the same time. It's less cumbersome this way.

I'm not sure how you think "counting" works...

As described, the Ethiopian Method would take far more stones and far more time. The larger the numbers you are working with, the larger the discrepancy.

The last time I checked Combinatorics was all "just fucking counting" and I am sure you will agree that it is not stupid.

There's simply no way that multiplication by iterated addition is a newer development than the shamanistic stones and holes ritual.

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