Or could combine the two for even more joy.
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.
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...
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
After all, who is to know that a "throwaway" account actually gets thrown away?
FWIW Steam has a second voting dimension for humour in users game reviews.
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.
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.
It blows my mind how sophisticated their navigational skills were, and how they transmitted the information without writing.
Aside: Is Quipu not a writing system? I thought there was some evidence indicating it was.
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.
The Sumerian society that developed cuneiform numeration/writing was a large bureaucratic urban society built on complicated debt/taxation/social organization.
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".
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.
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.
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.
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 . 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'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.
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.
Humanity is really old. At most, we can say that Leibniz was the last to invent it.
If you're not working with computers, does this matter?
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...
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).
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.
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.
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.
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!
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.
...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.
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.
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.
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.
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!"
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.
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.
Odds are they reached South America even if we never find direct evidence (there is some).
I've visited Mangareva. It's a lovely island and a step in the journey to get to Pitcairn Island.
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.
Isn't sweet potatoes in Polynesia (which originated from South America) pretty much a proof?
In such cases, if any given generation omits an element from its oral history (by choice or mistake), it is lost forever.
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.
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?
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.
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
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.
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."
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.
Finally found a reference.
I wonder if the idea originated before Hayes described it in 2001. Now it looks more likely to be a coincidence :)
"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."
Roman numerals are closer to finger counting because they lack zero.
And thus, when such notation for numerals WAS invented, it felt quite natural to take ten as base, it being such a significant number.
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.
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.
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.
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.
Interesting. Perhaps this is the very first use of binary-coded-decimal?
This just seems like tribalism to me, yet the same people want open borders and freedom of movement, but not of ideas and culture.
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