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Polynesian people used a kind of binary number system 600 years ago (2013) (nature.com)
190 points by ghewgill on June 16, 2017 | hide | past | web | favorite | 115 comments

Of course, we all know what they would have used if they were real pros:


How about going in the opposite direction and using https://en.wikipedia.org/wiki/P-adic_number

Or could combine the two for even more joy.

Hi, Peter. Welcome to Hacker News!

If you get downvoted, it's because HN community takes a somewhat strict approach when moderating comments that contribute noise to the conversation. "Nice article!" comments are routinely downvoted. As is sarcasm, witticisms, memes, references and other styles of comments that occur frequently but do not contribute to the discussion. It's a knowingly doomed attempt to hold back the flood of noise that covers Reddit.

That last part feels a little aggressive in this case... The Wikipedia article he linked to is actually pretty interesting, relevant, and appropriate. Sure the presentation is tongue and cheek, but so are half of the comments on this site...

> but so are half of the comments on this site...

Yeah... There are a lot more jokes and sarcasm being voted up on HN these days. It's always been around, but it used to be voted down a lot more. Can't say I'm happy about that change. It seems the fate of all vote-based communities to eventually devolve into Digg 3.0...

I agree that his comment didn't add all that much, but I think the snark on your end was a bit more distracting and irrelevant than his comment, and honestly his comment was kind of on topic. This site is thankfully strictly moderated and usually the discussion in the comments is very on topic, jumping at the chance to be smug like that degrades the quality of conversation much farther then being semi-relevant with a bit of a joke. Downvote and move on

I started making comments like this after the tenth time seeing a green account make a joke, get downvoted to hell and reply "WTF! Why is Hacker News so mean to noobs!" (Peter was green an hour ago)

I certainly understand where you're coming from and what you're probably thinking of (Eternal September is the name I know it by)--but I did try to make my comment both relevant and funny. I personally think I did okay there, but I'm fine with disagreement on that part.

I don't disagree. I shoulda waited to see if you were downvoted first before commenting. Meanwhile, I thought my comment was nice. I used to get a lot of support for a slightly less polite version of the same message. But, apparently today a lot of people read it as aggressive, snarky and smug :P

That's the problem when forums have a lot of sarcasm. It becomes even more difficult for anyone to judge your intentions. When people in "witty" forums say mean/dumb things they get surprised by how many readers think they are serious. And, some of them respond with mean/dumb things in seriousness. And, in not-seriousness... Usually there's an indistinguishable mix. And, even when you say something constructive, a lot of people will assume a negative intention and get upset that you are so disparaging.

So, yeah. Every few months I get in the mood to welcome a witty noob. Doesn't always go well :P

Wait, green means new?

I believe it means "throwaway"

Pretty sure it just identifies to logged in users that a particular account is relatively new.

After all, who is to know that a "throwaway" account actually gets thrown away?

Proposed solution - a colour picker of Information quality as vote tool? Factual disputed - Factual accurat? (Red - Black) Information Rich - Information Poor? (Black - Faded Text)

That's what Slashdot did. Didn't help very much in the general case.

I think the Slashdot voting system was excellent and loved the multi-dimension approach. In what way did/do you feel it didn't work.

FWIW Steam has a second voting dimension for humour in users game reviews.

Maybee analyse the discussion following with a NN for quality indicators? Long statements, comenting part of the original post and a absence of sarcasm/name calling should allow for a good evaluation of productive posts. Maybee the bot could even Merge uniform opionion posts into a stack, so that ideology biased posts shrink from view and the interesting nuggets show up earlier.

I'm more for KISS and YAGNI.

This sounds too convoluted a process.

We have vote up and vote down buttons. If the people like and upvote a funny (but trivial) comment, in the end it's their prerogative.

Slashdot is still a better community than HN

The Polynesian history of the (european) middle ages keeps getting more interesting and mysterious.

Evidence for polynesian contact with the Americas is mounting. Regardless, the medieval coonization of all the pacific islands is evidence of seamanship and navigation skills European mariners didn't achieve until well into the age of sail. European sailing was much more trade/cargo oriented & we know a lot less about "pre-contact" polynesian maritime cultre so it's hard to do put the technologies side-by-side. That said, it looks the polynesian ability to aim for and land on small remote islands circa 300CE was not surpassed by any other culture until the 18th century.

Then there's the strange case of Easter Island with it's giant stone statues. Giant stone statues & "how-the-f%£$-did-they-do-this" megalithic monuments have existed for a long time (EG the sphinx), but these are usually produced by populous civic cultures. "High civilization" to use an out of date term. For a tiny island to produce this kind of an artistic culture without obvious predecessors or descendants is strange.

As always in prehistory (in the literal, "before written records" sense), what we know is very little. But, it wouldn't be hard for me to believe that surprising numerical or mathematical systems existed there. It's a culture that has surprised us to the point of disbelief several times.

Polynesian history blows my mind. I was reading about it a few months ago, which introduced me to the concept of thalassocracy.

It blows my mind how sophisticated their navigational skills were, and how they transmitted the information without writing.


I'm having trouble imagining a society with mathematics (beyond just counting) but not writing. Are there other known examples?

Those don't seem related to me at all. Quipu was an Incan record-keeping system that we don't understand well. Abaci are very ancient calculation devices that don't use writing. The oldest Mesopotamian and Roman versions were just a table with pebbles on it. And that's where we get the word for "calculate".

They aren't required, it's certainly possible to have a number system and not a writing system. But for anything where you need to do non-trivial arithmetic or accounting you can't just do all that in your head and memorize the results. You need some way of recording the results in a persistent way, even if it's just tallies carved into a piece of wood. And I imagine any culture that figures out a recording system for numbers would pretty quickly (in historical terms, so like years or decades) develop a writing system. I could be wrong of course, I haven't really formally studied ancient history.

Aside: Is Quipu not a writing system? I thought there was some evidence indicating it was.

Pebble-tables are fine for that, but don't leave a persistant record. Mostly when I hear "writing system" I think "linguistic writing system", a writing system for language, rather than "mathematical writing system".

But IIRC a number of writing systems started off first as mathematical record systems before developing to encode linguistic information. Maybe a good example is cuneiform, which started off as tally marks in balls of soft clay stored in jars, for tax purposes, which developed symbols for encoding what kind of goods were being counted. Quipu may be similar, with more nascent development as a linguistic system.

As far as I know the invention and use of counting boards went hand-in-hand with written (cuneiform) records. Or at least, we don’t have any evidence of earlier positional counting boards, only artifacts like tally sticks and piles of (non-positional) tokens.

The Sumerian society that developed cuneiform numeration/writing was a large bureaucratic urban society built on complicated debt/taxation/social organization.

A couple of Native American tribes used octal because base 8 works for the number of things you can carry between the fingers. Would have made computing a bit simpler if that had been adopted instead of base 10.

Oh wow, I didn't know that. Reminds me of the Aborigines who looked for shapes and figures in the empty space between the stars as opposed to our own ancients who'd rather 'connect the dots'.

Negative space astronomy - interesting...

Gotta say, I'd never have thought about space between fingers as the motivation for a base. We must investigate lobsters.

That's also the explanation for base 60

"Polynesian people used binary numbers 600 years ago"

Disgrace to Nature for click bait. The number system might be clever but as described is not binary numbers.

Especially the claim in the introduction

"Binary arithmetic, the basis of all virtually digital computation today, is usually said to have been invented at the start of the eighteenth century by the German mathematician Gottfried Leibniz."

implying Leibnitz didn't invent binary and equate the existence of 4 numbers (10,20,40,80) which lack the simplicity of 2 states by representing it with KTPV with the invention of the binary number system is sensationalist.

Interesting the quote

“It’s puzzling that anybody would come up with such a solution, especially on a tiny island with a small population,” Bender and Beller say.

It's only puzzling to scientists from "Department of Psychosocial Science" but would not be puzzling to scientists from the "Department of Mathematics".

The Polynesian people used binary numbers, but not only binary numbers. The article's title is correct. What's sensationalist about that?

Is it an important finding? I think so. It helps shed light on the scientific and technical originality of other cultures. It matters a lot to acknowledge that knowledge does not always flow from west to east or north to south. It helps to rid of the notion of there being 'advanced' and 'primitive' cultures.

Are the scientists from the "Department of Psychosocial Science" trying too hard to make that case? I don't think so. Some people like me thinks this is a newsworthy discovery. Apparently mathematicians would not agree but even so, should they dictate that I should be dismissive of this article as well? The way I see it, the article does not make outrageous or unfounded claims. I'm free to appreciate that numbers have a certain universality as an abstract concept or language that isolated people from different era and backgrounds naturally converge to.

> Some people like me thinks this is a newsworthy discovery.

Because you don't understand math. Binary arithmetic is a painfully obvious development. A bored high-schooler predisposed to mathematics would figure it out in an afternoon.

And we've documented cultures using mixed base systems in the past. All this would have taken is a single person to say, "Hey, some things are easier if you do it this way" and taught all of their kids to use that system.

How do you know I don't understand math? And if a high schooler could figure this out, then Leibnitz own discussion or contribution to this topic is then meaningless?

The main point is that the finding in the article has cultural significance, rather than purely in terms of mathematics. I think you either don't understand that or are not willing to see that. The article is not trying to elevate a Polynesian people's contribution to mathematics. There is little mathematical significance there as these people from 600 years did not celebrate or promote their number system. Rather, the article explores how mathematical knowledge arises and how it was used. Could there an underlying commonality in the way humans learn and organize knowledge? That's an interesting question to ask and is not diminished - but is in fact supported - by the fact that the same knowledge gets rediscovered in multiple isolated instances.

> How do you know I don't understand math?

Because if you had a deep appreciation for mathematical structures you wouldn't be surprised by this development.

> And if a high schooler could figure this out, then Leibnitz own discussion or contribution to this topic is then meaningless?

Mostly, yes. Binary is useful for computers, but it's not like we use binary algebra in our daily lives. FWIW, Leibnitz did a much better job at calculus than Newton ... mathematical construct that has been "discovered" at least 3 times.

> The main point is that the finding in the article has cultural significance, rather than purely in terms of mathematics. I think you either don't understand that or are not willing to see that.

The OP was complaining about Nature screwing up the understanding of the mathematics and chiding the social scientists for ignorant remarks about the mathematics involved. I am also tired of seeing silly pop-science articles being posted HN.

> Could there an underlying commonality in the way humans learn and organize knowledge?

Yes, it does and this finding does not contribute to what we already know about mathematics and its relation to cognition [0]. I'm frankly a bit embarrassed by my social science colleagues and Nature for not doing a better job framing their findings more appropriately.

Sorry for being curt.


I am not surprised by the mathematical development described in the article. Rather, I appreciate that there is yet new evidence for what I believe about human capacity and potential across cultures. I felt a small level vindication and inspiration, not surprise.

I'm not sure why you insist on finding fault with what I understand or not, or what you wrongly imagine my reaction is to this article. It's possible to appreciate stories like that without being shallow or sensationalist. At the same time, it is possible to be unnecessarily dismissive about something when the focus is solely on technical aspects and lose the bigger picture of what a story is about.

I think we agree that truly unique discoveries or inventions are extremely rare. I did not get the sense that the article framed the islander's number system as such. It's okay to disagree if you see the framing differently.

> I'm not sure why you insist on finding fault with what I understand or not

Again, I am just trying to validate the OP's points about social scientists screwing up basic mathematics on Nature's website.

> At the same time, it is possible to be unnecessarily dismissive about something when the focus is solely on technical aspects and lose the bigger picture of what a story is about.

Agreed! I spend all my time on this stuff and it's the coolest thing ever ... and I'm really sad that Nature doesn't come right out and say what you are saying.

Again, sorry for being curt. I didn't mean to insult you, we really don't have time to understand everything.

>implying Leibniz didn't invent binary

Humanity is really old. At most, we can say that Leibniz was the last to invent it.

Leibniz recognized the concept of binary in the hexagrams of the I Ching and even published a paper on it. http://www.historyofinformation.com/expanded.php?id=454

And relevant to a classic contradiction in the startup community: idea vs (execution and capital). What you (say or think) matters a lot less than what you do.

> ...which lack the simplicity of 2 states...

If you're not working with computers, does this matter?

Also, in terms of computers, binary is only "simple" when you have the ability to create, assemble, and "warehouse" the "mechanics" necessary to represent them.

Babbage arguably didn't use base-10 for his Analytical Engine because it was more complex, but rather that it lended itself to gearing mechanisms (both in being more compact, and somewhat efficient (at least in how Babbage implemented it).

On the other hand, Zuse came up with an ingenious method of representing binary states using simple and efficient (and somewhat compact) sliding rod-like mechanisms in his first computer, the Z1.

Prior to the application of electrical and electronic circuits to computation, coming up with a mechanical system to represent binary (and be able to calculate with useful numbers, given friction and such) wasn't a trivial task. That said, it took a while once potential circuitry came into existence; I've always found it strange (to a certain point) that Babbage didn't use relays and binary (as he was contemporary with Boole, and had to know about his work, as well as relays as used in telegraphy - the only explanation I've been able to surmise is that they were still too early in their development for them to be reliable for calculation, but that's just my reasoned opinion).

Hollerith used electricity for his calculating machines, but they weren't binary-based (being essentially tallying machines they didn't have to be).

Strangely enough, the ENIAC was also a base-10 machine (it's ring counters consisted of 10 flip-flops more or less); again, this might be a reliability issue with the vacuum tubes of the period...

Thank you. This needed to be said.

If this stuff is interesting to you, you should read about Geoff Saxe's work [0] with the Oksapmin people of Papua New Guinea. This book is newish (and I haven't read it, but I've read almost all of the studies the book is built on) and synthesizes a couple of decades of research into how these people count and talk about number and how that practice changed over time (there's some fascinating conclusions about how the base changed from base-27 to base-20 when western money, in particular the 1 pound : 20 shilling ratio, came to be more widely-used).

And yeah, more or less base-27. They used body parts to refer to numbers 1-27, and often anything higher than that was just "a lot", but they could count more using the system if needed. How the Oksapmin people mingled both traditional counting and modern arithmetic is pretty fascinating (and says a lot about the underlying cognitive processes that make humans able to reason about number).

0: http://a.co/6gbRM1x

Did they use 27 as a radix when they went for bigger numbers?

For example and Oksapmin might have say "four score and seven years ago", using their 27-numeral vocabulary to supply words for "four", "seven" and "twenty", but in this case the (closest thing to) a radix is 20, not 27.

for the most part, they didn't go for bigger than 27 very often. At the time Saxe started studying them in the late 1970s, they were living very close to nature, and often didn't have cause to need to accurately count higher than that (if you have one or two fish, the distinction is important - if you have 27 or 28, less so).

But yes, the word fu (not foo, but enjoyably similar) which kind of means "a full amount", or enough, and was used as "one radix", so people do something like fu + another number word to mean 27 + that other number. I don't remember if there was evidence of treating it like a radix (like multiple fu + something), but I don't think that happened.

Again, in this particular culture and economy, there wasn't much of a need to think about large numbers precisely. Which is part of what makes all this pretty fascinating - humans have some inbuilt capacity to reason about numbers (some studies of babies show that they can distinguish between two and three of things just as readily as two hundred to three hundred, sort of a visual field scale independence - I'd have to dig to find that study, I read it about eight years ago, I think it was in nature), but beyond a certain level it becomes intrinsically intertwined with our development of language and other conceptual frames.

Again, in this particular culture and economy, there wasn't much of a need to think about large numbers precisely.

I suppose so, but there's a reason I picked 87 years. It's a time-scale that we care about when talking about people and society. We don't strictly need precise numbers for that you could say "around when granddad was born" -- but if we do have precise numbers then we can and will use them.

While not intended to be used mathematically, the I Ching hexagrams developed around 1000AD represented the binary numbers from 0-63 (2^6), as discovered by Leibniz who was quite fond of binary himself.

Came here to say this.


And as you know each hexagram was composed of two groups (trigrams) of three pairs. Similar to bytes and nibbles.

Even though the superstition is unfounded, the trigrams could combine in any way to refer to concepts and expositions that could give a lot of food for thought. Quite a nice system and very elegant because of its basis in binary numbers.

Edit: and I think you meant to say 1000 BC, not AD. In short a LONG time ago!

I was thinking of the Shao Yang (binary) arrangement, rather than the original King Wen arrangement, which would have been created during SY's lifetime (1012-1077). This is the version Leibniz examined [1]. But, yes, as you point out, if you go back to the King Wen version, even earlier. If only Leibniz was around today!

[1] https://www.jstor.org/stable/1399337?seq=1#page_scan_tab_con...

That is weird, right? Where did the precise use of broken and unbroken lines, in all the combinations necessary to represent binary 0 to 63 originate? It's a compact representation to label 64 different things using 6 "sticks"...but why did they do it like this? Were they aware of binary numbers?

It's not that hard. If you limit yourself to two symbols and try to think of ways to represent as many things as you can, you just have to keep adding additional columns/rows.

As long as you realize that you can use different combinations of the "first" spots with each additional spot you quickly come up with what is effectively binary.

I distinctly remember in early middle school trying to think of ways to enumerate things where you could never confuse one for the other based on the symbols. I was using base 10 because I wanted to store the combination of things in a variable in BASIC on a program I was trying to write on my calculator. As I was playing around on paper I came up with a great system.

I would only use one and zero because you could always tell them apart, where as if you use the other numbers then you can't tell if you meant five or if that five was a combination of two and three. When you needed to make another thing and you were out of places you would just add a digit on the left.

Then about five seconds later it hit me that what I had was just binary. All of a sudden the idea of using individual bits in masks to represent things (which I'd seen before) made a lot more sense.

> ...but why did they do it like this?

...because people are smart. People have _always_ been smart, it's one of the defining attributes of humanity.

It strikes me oddly (not specifically your comment, but the general color accompanying these kinds of articles)--there's a kind of ground assumption that by looking into the past, one sees nothing but a gradually descending IQ.

> people are smart. People have _always_ been smart

I don't disagree, but I think there's more to it than just that.

Our raw mental capabilities may have always been the same, but how smart we are also depends on our learning. Learning gives us extra leverage. If you have two smart people and one of them was trapped by themselves all their life on a desert island and the other learnt a lot about (say) maths and science, then the latter person could in practical terms have greater intellectual capabilities.

Over the centuries we've made great gains in mathematical tools, scientific knowledge, in democratizing education and in disseminating knowledge. And this means over the centuries we've (as a species) obtained more leverage that we can apply to our raw mental capabilities, giving us (overall) greater intellectual capabilities.

> Our raw mental capabilities may have always been the same, but how smart we are also depends on our learning.

FWIW, I tend to equate "smart" with "cleverness," as a separate measure distinct from "experience," for the same reason you describe--so what I usually go by is something along the lines of:

Cleverness is a measure of "what can you do with what you have," whereas experience is a measure of "what do you have?"

> Over the centuries we've made great gains in mathematical tools, scientific knowledge, in democratizing education and in disseminating knowledge. And this means over the centuries we've (as a species) obtained more leverage that we can apply to our raw mental capabilities, giving us (overall) greater intellectual capabilities.

I don't think we've gained greater intellectual capabilities--our intellectual capabilities are the same, we just operate in a completely different mental environment than our priors did.

Moreover, having a different view of the world allows for different connections to be made, and different potentials to be expressed (irrespective of one's individual level of cleverness).

So, for example, by placing a priority on stories & views that encourage greater investigation of the physical world, we get to where we are today. And we can teach the next generation slightly different stories that optimize for different kinds of usefulness.

To bring it to the HN contingent--if I learn a new programming language, I've gained experience in different ideas and operate in a different mental landscape. But I'm not smarter afterwards, and I wasn't dumber before.

> looking into the past, one sees nothing but a gradually descending IQ

Actually, it quite literally might be so: https://en.wikipedia.org/wiki/Flynn_effect

Though it's questionable as to how IQ is related to actual intelligence over time.

Agree with that too. Maybe IQ is out of date? Also, the huge effect that all these new technologies must be having is surely not going to be factored into that.

I think the parent was referring to the fact that the numbering does not resemble any modern definition of binary --- there are 64 ways to arrange 6 2-stated things, but the I Ching numbering doesn't have any clear 32-16-8-4-2-1 (or other set of) weights to each "bit".

I agree. People are also arrogant. We like to look back and think "how quaint" and we are better than them. It's humbling to think that we actually weren't. But it's probably not very adaptive to think that way. Since, believing we are better now, by falsely diminishing the whatever metric of the past, probably helps us keep going forward to create the bright future.

Hopefully without repeating the mistakes of the past tho!

Even that phrase "mistakes of the past" is telling, right? I mean it's not like, IMHO, you hear a similar amount of talk about, "the brilliance of the past", except it a sort of quaint, dismissive way: "oh, look, plumbing in ancient Rome, weren't they sort-of clever!"

I think that it's fair to say that, in the past, people were (overall) a lot more ignorant about the character of the world and the universe, and that this led them to (overall) believe a lot more incorrect things.

This is not to beat our chests in a "we're better" kind of fashion, but just to acknowledge that we have the benefits of the knowledge that people who came before us built.

As time goes on, you have more giants standing on the shoulders of other giants, and so on. Our feeble wetware is going to look pretty inferior to the digital cognitive systems of the next millennium, so we would do well to stay humble :)

the sticks are literal sticks, used in fortune telling. It's roughly equivalent to flipping a coin. If you had a game where you flipped six coins, and each combination had a different meaning, you would be close to inventing binary

The I Ching hexagrams don't represent numbers. They're not ordered numerically and any of the traditional sequences and they're not used to perform arithmetic. The individual lines represent (AFAIK) 2's, 6's, 3's, and 9's depending on whether the line is male or female, moving or static.

The correspondence between the hexagrams and the binary numbers nothing more than a necessary consequence of the representation chosen: the number of states of a string of 6 characters in a system of 2 symbols.

While the described system has some powers of 2, which is interesting, it is a far cry from positional binary. It most certainly does not prefigure the system described by Leibniz, nor the application of arithmetical operations to logic, which was the profound insight in question. This shares far more with things like the base-60 fractions used by Mesopotamians than anything recognizably similar to modern binary and logic systems. (Which of course derive a great deal from Aristotle, significantly more than 600 years ago).

The ability of the Polynesians to settle huge stretches of the Pacific might be one of the most impressive human achievements ever.

Odds are they reached South America even if we never find direct evidence (there is some).

There's a study that found Polynesian DNA markers in indigenous Brazilian remains. It's probably what you are referring to by the some evidence. https://www.nature.com/news/dna-study-links-indigenous-brazi...

I've visited Mangareva. It's a lovely island and a step in the journey to get to Pitcairn Island.

Did your trip include Pitcairn Island? I'd be interested in your impressions of the place (population 50, boat every 3 months or so).

Yes it did. I visited in 2012. I put some photos online here http://cd.pn/pn2012/

That covers the trip from Mangareva to Pitcairn and back.

I loved it on Pitcairn. The island is small but very hilly so there's more walking around and things to see than I expected. The people were friendly and welcoming.

I went there via a yacht from Mangareva that was chartered through an island run travel agency. I stayed for about three weeks. There's definitely a feeling of isolation once the yacht left knowing that there's no way to leave until it comes back. I also had to adjust to no mobile phone and power only on half the day. I was constantly checking my pocket for the first few days to check mail and messages only to realise I didn't have a phone!

I stayed with an island couple - accomodation fee covers food, etc. They were great. They have a shop on the island that gets stocked through the three monthly supply ship. Most of the food comes from island gardens, food growing wild and fishing. I fished a lot!

There are artifacts from the Bounty and other shipwrecks around the island. Cannons, anchors, etc. There's a museum with more artifacts and island history. The church holds an original bible from the Bounty.

There were a number of island events while I was there for birthdays and a market day where islanders sell there unwanted goods to other islanders or visitors. I was surprised at the number of visitors to the island. Lots of yachties ticking off their bucket list. And a surprising number of solo sailors visiting. I met a lot of interesting people.

My perception may be coloured by the fact that I'm related to many people there through my grandmother who was born on the island in the early 1900s. I went there to learn about where she lived and my family ancestral history.

Thanks for sharing! Hopefully one day I'll be visiting Pitcairn myself. Btw, I infer from your comment that you descend from one of the Bounty crew members, am I correct in assuming that? If so, that trip must have been awesome since so many connections to your ancestry are still preserved there.

> Odds are they reached South America even if we never find direct evidence (there is some).

Isn't sweet potatoes in Polynesia (which originated from South America) pretty much a proof?

Here in Hawaii the Hokulea is a big deal (http://www.hokulea.com) because they have taken modern day people, and using the Hawaiian methods of navigation they can still get around without modern technology and figure out where to go. Showing it wasn't just Polynesians sending out random boats hoping they would hit an island.

The origin of Polynesian sweet potatoes being from an old visit to South America is the most plausible origin story, but the lack of anthropological evidence is also a powerful dissuading factor. There's no oral stories of either the Polynesians or the South Americans of such a voyage taking place, like there was for the Viking colonization of Vinland (recalled in the Icelandic sagas).

Note, however, that the Australian aboriginals' oral histories (songlines) lack any reference to coming from elsewhere, although DNA clearly demonstrates otherwise.

In such cases, if any given generation omits an element from its oral history (by choice or mistake), it is lost forever.

There are two orders of magnitude difference in time though. Australian Aboriginals have been on the continent for 40-60 thousand years, Polynesians made it to Hawaii 800-2000 years ago.

> Mangarevans combined base-10 representation with a binary system. They had number words for 1 to 10, and then for 10 multiplied by several powers of 2

Because base-10 seems to originate from two handfuls of fingers, I wonder why they didn't end up with a base-5 representation with a binary system?

> takau (K) means 10; paua (P) means 20; tataua (T) is 40; and varu (V) stands for 80

By using their numeral for five (say, F) to mean 5 in this scheme, they could have gotten rid of their numerals for 6 to 9. So 157 would be VTPKF2 instead of VTPK7.

If you think this is cool you might also find the Marshall Island stick charts, and Polynesian navigation methods particularly intriguing. See



This reminds me of the story of ancient Ethiopians using binary maths. I read about it a while back and finally managed to track it down (the original link seems dead): https://web.archive.org/web/20170609082700/http://www.uh.edu...

They had number words for 1 to 10, and then for 10 multiplied by several powers of 2. The word takau (which Bender and Beller denote as K) means 10; paua (P) means 20; tataua (T) is 40; and varu (V) stands for 80. In this notation, for example, 70 is TPK and 57 is TK7.

To my non-mathematically trained ears this doesn't sound like a binary system at all, but more like the highly inefficient Roman system. Am I missing something?

It's not at all like a Roman system except for the use of letters. You can separate a natural number n into two parts:

n = 10 * q + k, where 0 <= k < 10 and q in N

q = 1 * K? + 2 * P? + 4 * T? + 8 * V?

where K?, P?, T?, and V? are 0 if the letter is absent and 1 if the letter is present. n is a textbook base 10 decomposition, and q is a sparse binary representation.

Roman numerals didn't use quotient/remainder or a geometric expansion at all. Since the 'base' seems to switch between 2 and 5 each time, you can't cleanly decompose it. And there was that weird subtractive case.

Roger Bacon wrote about binary notation in the 13th century, although he called bits `fingers'. John Wilkins credited Bacon in his book Mercury, or the Silent and Swift Messenger published in 1641.

Wilkins' book is basically a tutorial on communications security (COMSEC) that touches on channel coding, reliability, secrecy, key management, cryptanalysis, OPSEC, and data compression.

ETA: Wilkins takes a clear position on the full disclosure debate but cautions of the hazard of experimenting with crypto technologies:

    `...the chiefe experiments are of such nature, that
    they cannot be frequently practised, without just cause
    of suspicion, when it is in the Magistrates power to
    prevent them.'

Computers use base two because their most natural unit can occupy one of a couple states: 0 or 1. One, two, base two.

Human hands, on the other foot, have ten fingers. Since our favorite mapping between things and integers is finger counting, we naturally end up with more than two states. Zero, one, two, three, four, five, six, seven, eight, nine, and the fully-extended state, ten. That's eleven states, which is why the global human standard is base-eleven.

Wait, what?

Maybe because by the time hands were closed into fists, people had forgotten about the negotiating and gotten down to fighting.

In other words, the "fist" symbol was a reserved word for a very different purpose than counting.

Or possibly more realistically: if the way you communicated numbers didn't rely on position to indicate magnitude, you didn't need zeros as place holders. You could just "sign", 3 * 100 + 5 * 10 (whatever the symbols for those were ). Hell, maybe the symbol for 10 was 1 x "fist" and 100 was "pump your first twice."

There's a lot more possible states than eleven. 32 if you count just whether a finger is extended. More if you allow crossing fingers, interaction between hands, etc. And this isn't theoretical either:


Maybe a coincidence (I don't know details of computer history), but another interesting reason why binary is used might be information density.

Base e (2.71...) has highest information density. Of course, computers can't use irrational numbers as their base, so base 3 would be a closest pick. However, base 3 is much harder to represent in digital logic, so base 2 is a better pick and is still relatively close to e.

EDIT: Finally found a reference. https://en.wikipedia.org/wiki/Non-integer_representation#Bas...

I wonder if the idea originated before Hayes described it in 2001. Now it looks more likely to be a coincidence :)

What's "information density"? Is there a formula for that? Googling didn't turn up anything

Probably a bad translation on my part from my native language. I meant radix economy, which is ideal with radix e.


"e has the lowest radix economy


A base with a lower average radix economy is therefore, in some senses, more efficient than a base with a higher average radix economy."

Zero was introduced well after base ten was in use.

Base 10 uses 0 to 9 digits, just like base 2 uses 0 and 1.

Roman numerals are closer to finger counting because they lack zero.

What actually happened, probably, is that humans first counted by putting things in correspondence with fingers; actual bijections between fingers and things. This naturally led to ten, the number of fingers total and thus a grouping size of understandably frequent use, becoming a very significant number (culturally, linguistically, etc.). Ten was a very significant number for a very long time before positional notation with its digits and bases was invented. (The Romans had no digits, but they loved their fives and tens)

And thus, when such notation for numerals WAS invented, it felt quite natural to take ten as base, it being such a significant number.

The Sumerians counted the individual finger bones on one hand to count to twelve, but adopted base sixty. I guess they combined it with five fingers on the other hand?

> Computers use base two because their most natural unit can occupy one of a couple states

Not really. That's just the way things worked out, mainly due to how electronics (transistors) work, and with the miniaturization of electronics.

Babbages Analytical Engine was designed as a base-10 machine, mainly because this was the most efficient way to represent things using gears (and with all the friction that entails especially with large numbers - lot's of ingenious mechanism was designed by him to work around this limitation).

Even ENIAC, which used vacuum tubes, was base-10; this had to do with a number of reasons (reliability of vacuum tubes, war rationing issues possibly - at least when it was designed; there are probably other reasons as well).

Other bases have also been tried, but base-2 is default mainly because its easy to create simple circuits (especially miniaturized) to represent such states, and replicate/connect them efficiently (essentially, once you have a NAND or NOR gate represented, with transistors or anything else, you have the basic building block for the rest of a computer).

Other than that, there isn't anything particularly unique in regards to binary representation (base-2) for computation versus other bases.

Imagine you had two fingers and not ten. Try counting in binary. That should clear up the anomaly.

I'm a Lisp programmer, damn it! I will use one of the fingers for one tag bit which indicates a pointer into the heap when zero, and from there I will boostrap bignums somehow, and the rest of the numeric tower.

You can count from zero to two on two fingers, so two-fingered aliens may invent numerals for zero, one and two. If they develop that into a place value system, it will be (fingers+1)-ary, or in their case trinary.

you can count from zero to three :)

It's been speculated that prior to base ten (or maybe alongside it) people used base twelve (using knuckles) hence anachronisms like dozens (inches, hours), 60 (minutes) and 360 (degrees), etc.

Not correct, the frequency of multiples of 12 in human counting is thought to be because of the lunar cycle and that each human hand has 12 phalanges -- aka finger bones -- in the fingers excluding the thumb. See how the Chinese count to ten on one hand, using a combination of thumb and fingers, bent for some numbers, for six to ten. There are very few languages with a duodecimal system.

Using two hands you can represent up to 1024.

What you're missing is that positional numeral systems developed relatively late--in recorded history, which is well after number systems start to be encoded in language. Earlier numeral systems are counting systems, where zero isn't a proper number but rather a signifier for a lack of stuff.

Let's look at numbers in natural languages. In English, we start with 12 basic numbers--one through twelve--and then we start counting "three-ten", "four-ten", etc. through "nine-ten." After that, we say "two tens" (the "tens" gets corrupted to -ty in Modern English), then "two tens one", etc. Note that we're not saying "two tens zero"--that's a sign that zero is not really fundamental in our counting system (etymologically, the term "zero" in English appears to date only to around 1600, contrast that to the -ty affix that dates back to at least Proto-Germanic, although many of the numbers themselves have roots back in Proto-Indo-European).

You can also see this effect in early numeral systems. Note that Roman numerals--the most common numeral system in Europe until the Early Modern--has distinct letters for 5 (V), 50 (L), and 500 (D), which is the usual case in most of its contemporary numeral systems. The Greek numerals for, say, 666, would be χξϛ--same general principal as Roman numerals, even though it has distinct numerals for every digit rather than just ones and fives.

The actual development of a true zero and true positional numeral system appears to have only independently happened very few times. The Mesoamericans probably developed it around the same time as the Long Count calendar (exact date uncertain, but roughly contemporary with the Roman Empire). Hindu-Arabic numerals developed probably slightly later (thought to be around 400 AD or so)--and it's from this system that pretty much every modern numeral system comes. The quipu could definitely represent numbers in true positional fashion, although the dating of this is unknown to me.

Base 10 predominates in modern numeral systems primarily because of the primacy of Hindu-Arabic numerals. The derivation of number terms in natural languages shows a rather confusing panoply of numbering bases. The vigesimal and sexagesimal number systems of Mesoamerica and Mesopotamia do show residual base-5 and base-10 in their construction, and the terminology in relevant native languages tends to indicate a base 10 strata (so the number in "78" in Mayan and Nahuatl boils down to "three twenty ten eight"), which strongly suggests that these systems are chosen for accounting purposes, not for things like "counting on fingers and toes." It's also worth pointing out that the human visual system subitizes small numbers--basically, you don't need to count three objects, you just take a glance and immediately know "there are three"--and this process tends to break down around 4-6 objects. It's not hard to imagine that number systems like duodecimal or vigesimal are based on counting subitized groups.

Yep!! That's what's commonly misunderstood about the "invention of zero". Positional numbering is the part that's actually innovative. This hit me when I was a math teacher and trying to consider our algorithms for arithmetic from first principles. I was teaching high school students how to work with exponential expressions with variables, which seems very esoteric, unless you realize that you've been doing it for years implicitly.

> What you're missing is that positional numeral systems developed relatively late--in recorded history,

Mesoamericans were carving base-20 long count dates at least as early as they were writing words, actually. The oldest complete date is 36 BCE. This was a true number system, including a glyph for zero.

We're still talking ~2,000 years ago, which I consider recorded history (basically, the distinction between history and prehistory is roughly ~6,000 years ago). Certainly much, much newer than language.

It sounds like pedantry, but it's important: history can't be recorded if there isn't any recording being done. Again, current evidence has the long count preceeding written language in Mesoamerica. That time frame only works for the mesopotamian and nile river civilizations.

While we’re at it, the (positional) Mesopotamian sexagesimal system evolved around 2000 BC, and had a placeholder “zero” symbol for empty places sometime before 0 AD.

Fully extended is one set over none extended: base ten.

It makes sense if you think of all of them extended as a set, rather than one less than a set. And so by the time positional notation came around, it was cemented that all of them extended was a set. (It took us a while to get 0 and positional notation.)

Just look at Roman numerals:

A finger. A hand. Both hands. A hand representing both hands. Both hands representing both hands. Etc.

They find that the former Mangarevans combined base-10 representation with a binary system. They had number words for 1 to 10, and then for 10 multiplied by several powers of 2.

Interesting. Perhaps this is the very first use of binary-coded-decimal?

0b1001011000 years ago? Anybody?

I keep forgetting the New Year and write 0b1001010111 on all of my checks.

A guy in an English suit asking for cultural appropriation to be made illegal. Oh the irony.

I find that funny, but remember everyone else is allowed to appropriate white culture because reasons.

This just seems like tribalism to me, yet the same people want open borders and freedom of movement, but not of ideas and culture.

Binary numbers have been in use since the balance scale was developed.

This is not a bad point really. But taking that perspective, binary logic has existed since all forms of classification, ie. any form of recognition whatsoever, which all babies achieve innately (what is pain/hunger/warm/wet, who is mum, is that a voice I hear, etc.) In short, all structured thought requires binary logic, because otherwise classification wouldn't work, and thus neither communication, memory, etc.

Full disclosure: I had a revelatory acid trip on this subject when a research mathematician first explained to me set theory as essentially a derivative matrix resulting from a boolean test.... the world as we perceive it is merely sets! This explains children learning! Mind. Blown.

Bonus anecdote: If you like admiring early human achievements that have been unfairly obscured from popular history, check out the Polynesian crab claw sails. https://en.wikipedia.org/wiki/Crab_claw_sail#Performance

Meh. Ancient babylonians used a base-16 number system. Does it really matter what number system they used?

Um... apparently not to you but to many of the readers here (as we can roughly guess by the upvotes) it does matter.

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