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:

urapon
ukasar
ukasar-urapon
ukasar-ukasar
ukasar-ukasar-urapon
ukasar-ukasar-ukasar

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.

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Venn diagrams explained in comic form

After reading The Manga Guide to Calculus

and Logicomix

I wondered if any and all mathematical concepts could be explained in graphic novel form.

With my classes I have experimented with having them make a comic book page on concepts where they had only a surface understanding (implementing a rule mechanically). I wanted to see if making a comic might make them look deeper.

In one case I was trying to teach how statements with “all” and “none” affect a Venn Diagram. Here’s a student example (the student gave permission for me to post but desires to remain anonymous):

[Full-size version.]

If you want to try this with your class emphasize drawing skill is unnecessary. Characters and/or plot are also optional; a succession of panels showing the math is enough, although it helps to utilize the dynamism of the comic format: