Is “one, two, many” a myth?

xkcd comic

Cue letters from anthropology majors complaining that this view of numerolinguistic development perpetuates a widespread myth. — From the alternate text to xkcd comic #764

The “one, two, many” theory is that cultures developed words for “one” and “two” before anything else, and any numbers after are referred to as “many”.

Do cultures with a one, two, many system exist?

Yes. Blake’s Australian Aboriginal Languages points out Aborigines felt no need to count, and while they all had words for “one” and “two” only some made it to “three” and “four”. 1

The Walpiri, for example, only has words for “one”, “two”, and “many”, as shown in this excerpt from The Story of 1:

(The entire hour-long documentary is online, if you’re curious.)

Intriguingly, there’s a recent study that suggests that Aborigines without counting words can manage counting nonetheless:

In tests, the children were asked to put out counters that matched the number of sounds made by banging two sticks together. Thus, said Butterworth, they had to mentally link numbers in sounds and in actions, which meant they couldn’t rely on sights or sounds alone.

“They therefore had to use an abstract representation of, for example, the ‘fiveness’ of the bangs and the ‘fiveness’ of the counters,” he said. “We found that Warlpiri and Anindilyakwa children performed as well as or better than the English-speaking children on a range of tasks, and on numerosities up to nine, even though they lacked number words.”

Sometimes assuming one-two-many can be taken too far, as is the case with …

The Pirahã people

The Pirahã of the Amazon have been cited as a using a “one-two-many” system of counting.

They are truly an extraordinary case of a tribe, and if the (admittedly controversial) claims of the linguist Daniel L. Everett are true, they have no tense to describe things that are not physically present; hence they cannot talk about the future nor tell stories about the past, nor name exact abstractions (like colors).

Abstractions include ordinal and cardinal numbers, which appear to be absent from the language. What they instead have are words for

“small size or amount”: hóì (falling tone)
“large size or amount”: hòí (rising tone)
“cause to come together” (loosely “many”): /bá à gì sò/

(Source: On the absence of number and numerals in Pirahã)

The original confusion was that the words for “small”, “large”, and “many” could be in certain context mean “one”, “two”, and “many”, but they don’t genuinely stand for the numbers; a single large fish would only be called hóì (falling tone) as a joke. Hence using the Pirahã as an example is based on a misunderstanding. 2

Linguistic evidence

Consider the cardinal words versus the ordinal words in English (if you ever mix them up, ordinal numbers referring to the order things are in).

one – first
two – second
three – third
four – fourth
five – fifth
six – sixth

While the words are initially mismatched (indicating that “first” and “second” were developed separately from the abstract notions of the numbers “one” and “two”) after three the words match linguistically.

This occurs in Spanish

uno – primera
dos – segunda
tres – tercera
quatro – cuarto
cinco – quinta
seis – sexta

and in many other languages.

So there is some circumstantial support to the content that “one” and “two” have some special significance, although the same evidence could as easily be used to claim the ordering of “first” and “second” was the real first development, and the cardinal numbers were instead developed all at once (or at least up to five).

Two-counting cultures

Here’s numbering according to the Gumulgal of Australia:


That is, their counting occurs using the words for one and two. So while their counting words are limited linguistically, they nonetheless can use them to count farther. This is much more common than the case of the Walpiri who don’t bother with counting past two at all. This map indicates the two counting cultures still in existence:

[Map adapted from John Barrow’s Pi in the Sky. For a more specific look at tribes in Paupa New Guinea, there’s an extensive reference online.]

While these tribes developed words to count past 2, the linguistic evidence demonstrates they started with the words for “one” and “two” before the later numbers.

xkcd’s possible myths

So was xkcd right or not? It depends on what is meant as their myth:

1. Cultures exist with only words for one, two, and many.

This isn’t a myth, as already explained above.

2. It’s common for cultures to have only words for one, two, and many.

This one’s definitely a myth, so by this meaning xkcd is correct. A fair number of cultures use a limited base 2 system, but not developing counting at all is rare. (Note that developing a word for 2 is not the same as counting — the idea of a “pair” can be separate from what we think of as “two”.)

3. Every strand of counting development started with a one, two, many system.

This one’s a touch foggier — there’s historical evidence that one and two were special in the development of language, and arguably cultures that started with counting up to five or ten simply were using received knowledge from cultures that went through the entire development process.

1 The situation is slightly more complicated than Blake claims. For example, the Anindilyakwa mentioned in the study above usually only use words for one, two, and many, (and the children involved in the study only knew those words) but also have rarely used words for up to 19 for rituals.

2 The idea of “six apples” can be understood without fully abstracting the number “six”. In the Tsimshian language there are separate number words for flat objects, round objects, men, long objects, and canoes.


22 Responses

  1. Very interesting, it’s amazing that two so close concepts can have evolved separately.

    Small nitpick, you seem to have mixed french with spanish, in spanish four is cuatro 🙂

  2. Jason, thanks for such a well-planned and thoughtful response. I am most struck by the study of Australian aboriginals ways ways of reasoning quantitatively. It forces me to think about the way a teacher necessarily imposes a particular cultural artifact (in this case, “mathematics”, or maybe “school mathematics”) on young minds. And that it is an explicit purpose to ensure our children/students think in certain ways… So I don’t sound too over the top, I find Dewey’s Child and the Curriculum to be a great way to think through the balance of adult ways of knowing mathematics and the student’s mathematical mind.

  3. That looks like adding 1s and 2s, not base 2.

    As English (root vocab Germanic) and Spanish (romance) are both Indo-European languages, I would not want to generalize. Likewise odin, dva (1,2) in Russian don’t match pjerviy, vtoroj (first, second), but in non-IE Turkish, bir, iki match birinci, ikinci.

    I wonder how much variety there is.

  4. A little correction :
    ” This occurs in Spanish

    uno – primera
    dos – segunda
    tres – tercera
    quatro – cuarto
    cinco – quinta
    seis – sexta ”

    For male:

    uno – primero
    dos – segundo
    tres – tercero
    cuatro – cuarto
    cinco – quinto
    seis – sexto

    for female :

    uno – primera
    dos – segunda
    tres – tercera
    cuatro – cuarta
    cinco – quinta
    seis – sexta

  5. This is “one, two, many”; A Lie.
    Aboriginal people almost universally across the continent and surrounding islands counted by the base of five. Their method granted accuracy in enumerating into the hundreds or even thousands if or when necessary. It is linguistically and intuitively very different from the western system (which was inherited off ancient Indians who are ethnically linked to Aboriginals and then Arabs of the Islamic Empire).
    1= one
    2= two
    3= two and one
    4= two and two
    5= hand
    6= hand and one
    7= hand and two
    8= hand and two and one
    9= hand and two and two
    10= two hands
    In many language groups proper tag words existed for the multiples of five (5, 10, 15, 20, 25… 100 etc) leading right up to words reported for 1,000 in places like the Torres Strait Islands.
    Unfortunately the anthropology conducted in these times was racially motivated ethnography and actively sought to prove some human groups to be “less than modern”. It was this bias that saw most of the anthropologists of this time completely dismiss this method of counting for a lack of unique tag words for each number even though compounds (two and one= 3 for example) work just as well in their place and only petty ethno-semantics and old world racism could convince you otherwise.
    Saying people can’t count or enumerate because they don’t have a specific word for “3” besides “2 and 1” is like saying a deaf mute can’t think because if they’ve never heard or spoken language how can they have an inner monologue? What language are they thinking in?Aboriginal people conceptualised numbers perfectly well when tested sensibly (bringing a white man 150 spears when he demanded 10, by showing all his fingers, groups of 15, by showing 15 sticks) and only failed when they were expected to meet western standards of linguistic traditions in counting. For example;
    “How many fingers do you have on your hand?” seemed like such a stupid question to them but they answered anyway and held up their hand, “hand” being the word for 5 in most language groups across Australia. The stupidity was on the side of the ethnographer not the Aboriginals.
    Even when words for 50 and 100 and 1,000 were produced the “scientists” said boldly themselves in their own handwriting things like;
    “I do not believe these numbers”
    LOL!!! Idiots. The fact that modern desert “full bloods” grasp western mathematics now show the entire point was moot in the first place. Culture is king and if your culture has no need for the abstract science of maths it won’t discover it. As it was Europeans only took the germ of modern maths home with them as booty after the Crusades.
    And if counting based in 1’s and 2’s seems primitive to you then adding 8 more is all that separates us and the stone age. But you also might want to look into African sand divination which was used scratched in the dirt as a kind of ancient random number generator using 1’s and 2’s to predict the future of conduct religious ceremonies or whatever. The Arab mystics recorded this and eventually it was passed on to Europeans. It took a few thousand years but Gottfried Wilhelm Leibniz finally changed the 1’s and 2’s to 0’s and 1’s which led to the binary code which is used in the computer you’re reading this on.

    • I agree a great deal of research on the topic has been confused in exactly the manner you state, although I am unsure you actually read the article above, or watched the video, or read the article regarding the study I linked to with the Warlpiri and Anindilyakwa.

      I would say the case with the Pirahã is still open.

  6. […] ONE, TWO, MANY Some cultures are illiterate. So, too, some cultures are virtually innumerate, and numbers more than two are too many. […]

  7. […] have long found it fascinating that there are cultures do not have words for numbers other than “one”, “two” and “ma….  I fondly imagine that if Transition had its own language it might add one more: “one”, […]

  8. […] have long found it fascinating that there are cultures do not have words for numbers other than "one", "two" and "many&qu….  I fondly imagine that if Transition had its own language it might add one more: […]

  9. […] P.S. An interesting post that uses the comic above as a jumping off point: Is “one, two, many” a myth […]

  10. Never, meaning not ever have I read a more jaded account of the mathematical, not to mention geometry and science physics of any race of people on earth.
    Just a bit of trivia, the Australian Aboriginals are the originators of all mathematics and applied sciences. The entire subject bridges time and without doubt walked with two legs and was imparted from one to the other.
    For these people this knowledge is intertwined into their daily lives. Its like eating and breathing, so familiar are they with the practice, it takes little thought.
    They cannot translate for you. On this subject matter the Europeans have missed the boat.
    If the writer of this blog has the rest of his or her lifetime to devote to this subject, for it would take that long, I would suggest the writer either give up or go the hard yards to get the facts correct.
    The saying, ignorance is bliss is not funny, its fueling a debate of woefull misconception.

  11. […] [In some ways it makes me think we are not too far removed from cultures that only have words for &#8…] So I started thinking about ways around this scale issue. And I stumbled on the one part of daily life where we are used to dealing in orders of magnitude from 10^1 to 10^9.Computer storage. […]

  12. […] books by Terry Pratchett – simplified a bit for the current purpose and resonating with the myth of indiginous tribes counting “one, two, many” (see also […]

  13. […] As a brief aside, the dimension of a dynamical systems provides some hard limits on the possible dynamics. As we’ve seen, one dimensional dynamics is more-or-less trivial (at least if we consider only the large time asymptotics). The Poincaré –Bendixson Theorem guarantees that dynamical systems of the form (1) in the plane (e.g. ) can only have steady states and oscillations in terms of bounded solutions. Unbounded solutions, or solutions that blow up in finite time, can of course exist but these are not interesting in terms of asymptotic dynamics. Three-dimensional systems (equivalently, two-dimensional nonautonomous systems) can exhibit chaotic behaviour in addition to numerous other phenomena not possible in planar dynamics. For this reason, dynamics is often broken into the trivial (1D), the planar (2D), and everything else (3+D). So you can think of this as Mathematicians’ One-Two-Many Trichotomy. […]

  14. […] one of the most revealing attributes of “3” in ancient traditions is the counting system (article and video). Shamanism is an adaptable art, but you can still perceive the reminiscence of […]

  15. […] are stories of tribal and aboriginal languages with a counting system of little more than “one, two, many”, where words for quantities like […]

  16. […] Adam Rogers summarizes new research by Ted Gibson and Bevil Conway on names for colors across cultures. If I understood it right, they have built on Berlin and Kay’s earlier finding that each culture divides up and names the color spectrum in roughly the same way. They show there is a possible link with how foreground objects tend to be warmer colors (which are more readily communicable) versus background which tend to be “cool”. We have less need to label background objects, like the forest or the sky, than to label foreground objects, which tend to be warm. Our names for colors, they suggest, developed this way because we develop color words inasmuch as they are useful: “as cultures get more industrialized, they get more color words. First comes black and white (or light and dark), then red. Then a bunch of others.” There is an interesting analogy here with the development of language for numbers. […]

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