The cognitive technology of number

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A new paper on language and number cognition is in press: Michael C. Frank, Daniel L. Everett, Evelina Fedorenko, & Edward Gibson, "Number as a cognitive technology: Evidence from Pirahã language and cognition", Cognition (in press 2008). Michael put the paper on his web site, and so you can easily get and read the whole thing.

The abstract, like the whole paper, speaks for itself:

Does speaking a language without number words change the way speakers of that language perceive exact quantities? The Pirahã are an Amazonian tribe who have been previously studied for their limited numerical system [Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science 306, 496–499]. We show that the Pirahã have no linguistic method whatsoever for expressing exact quantity, not even ‘‘one.” Despite this lack, when retested on the matching tasks used by Gordon, Pirahã speakers were able to perform exact matches with large numbers of objects perfectly but, as previously reported, they were inaccurate on matching tasks involving memory. These results suggest that language for exact number is a cultural invention rather than a linguistic universal, and that number words do not change our underlying representations of number but instead are a cognitive technology for keeping track of the cardinality of large sets across time, space, and changes in modality.

For background on issues of language, culture and numbers among the Pirahã, you might want to read "One, two, many — or 'small size', 'large size', 'cause to come together", 8/20/2004; "Life without counting throwing", 8/12/2004; "The straight ones: Dan Everett on the Pirahã", 8/26/2004; "On counting and throwing", 8/27/2004; "Language, thought and counting in Amazonia", 10/30/2004.

I have only one additional comment for now. From a historical point of view, the first sentence of Frank et al.'s abstract asks the question backwards:

Does speaking a language without number words change the way speakers of that language perceive exact quantities?

This is the right question from the perspective of cognitive psychologists who are used to working with people who speak languages with number words, and now have the opportunity to learn about number cognition among speakers of a numberless language. But from the point of view of cultural evolution, the question goes in the other direction:  did developing languages with number words change the way that people perceive exact quantities? As I understand the answer proposed by Frank et al., it's "no, the basic representation remains the same, or at least some aspects of it do; what changes is mainly the ability to remember and compare".

But I'd like to point out that comparing exact quantities is only one aspect of the way that people "perceive exact quantities".  Once the "cognitive technology" of number is established, there are many other properties of particular exact quantities, or classes of exact quantities, that may become cognitively salient.

Once, a five-year-old of my acquaintance, stimulated by kindergarten exercises in counting and comparing, announced a discovery: there are a "fair numbers" and "unfair numbers". Fair numbers, he explained, are when if you have that many things, you can share them with a friend so that you each have the same. With unfair numbers, somebody always gets more. (This was not part of the lesson plan — in fact, I learned about it because his teacher perceived his enthusiasm for unscheduled discoveries, expressed in idiosyncratic terminology, as distracting and even disruptive.)

After you've understood something like the distinction between even and odd numbers, it seems to me that it becomes (to one degree or another) part of the way that you "perceive exact quantities". The set of such properties is indefinitely large, and most people don't become familiar with as many of them as Ramanujan famously did; but still…

[Also relevant here are some of the abstracts from a 2004 workshop on Numerals In the World's Languages: William McGregor, "Numerals in Australian languages"; Pattie Epps, "Tracing the development of numerals in the Guaviaré-Japurá (Maku) family"; Dan Everett, "On the absence of number and numerals in Pirahã"; Heike Wiese, "Language and the emergence of recursivity in the numerical domain". When I have time, I'll try to track down the full form of these.]


  1. Marc A. Pelletier said,

    July 11, 2008 @ 8:36 am

    " […] his teacher perceived his enthusiasm for unscheduled discoveries, expressed in idiosyncratic terminology, as distracting and even disruptive"

    Is it just me or should this teacher be given a severe beating? That kid displays analytical reasoning, inquisitiveness and insight and a teacher considers that disruptive? He should have been given the usual terminology, congratulated, and encouraged to continue his explorations!

    More on topic, it's obvious the Pirahã do not particularly care about cardinals; discoveries such as those would be culturally set aside as uninteresting or unimportant. This would inevitably end up as a lack of terminology for numerical properties (and with methods to cope with numbers in general).

    I'm not sure why some linguists are so attached to the Worf-Sapir hypothesis — this whole Pirahã thing is considerably easier to explain by positing that the lack of interest or ability with numerical tasks causes the lack of vocabulary and syntax to describe them than vice-versa!

  2. [links] Belated link salad for a Friday | said,

    July 11, 2008 @ 9:59 am

    […] The cognitive technology of numbers — "Cognitive technology." Wow. What a concept. […]

  3. John Cowan said,

    July 11, 2008 @ 11:32 am

    I'm not sure the question is so backwards, historically considered, as all that. Granted that the Australian semi-numerates might have split from the rest of the human stock before the invention of robust numbering systems, this surely cannot be true of the Pirahã. Their remote ancestors must have had a numbering system, which raises two questions: Why did they lose it? Why did they lose all interest in reacquiring one (as the Australians did when the English arrived)?

  4. Martin said,

    July 11, 2008 @ 11:46 am

    Is it really so different for a language to not encode number when there exist languages that don't encode gender? Or what about the "exactness" of Chinese measure words that is not found in other languages? Can this be an example of Principles and Parameters — that there's a switch for whether a language will encode for exact number?

    Regarding "fair" and "unfair" numbers, I think the teacher missed a prime opportunity to have the student learn about numbers that are "unfair" for sharing equally among any number of friends.

  5. john riemann soong said,

    July 11, 2008 @ 12:59 pm

    I remember when I was four years old when I finally decided to count all my family members — four of them, including myself, I counted. It was kind of strange — up to that point I had never really thought of my family having a distinct, definite size. Pinning down the swirling mass of people you always saw in your living room into just four definite people suddenly made me aware that the family was kind of small, even though you had close ties with each individual (naturally). It was when I was in the elevator going up to our flat. "But why do we always press 11, mommy? Why not 7?" It hadn't struck me why our floor number would remain constant. Age for a four-year-old was another thing — from how it was exposed to me, until sometime later I had not really conceived it as the time I had spent on Earth — it always struck me as some element associated with you, like your name, your favourite colour, etc. Changing your age was therefore a big thing, because it was like switching your favourite colours, and so forth.

    My teacher once recounted how (also around the age of four), he was surprised how the number of squares of a chessboard always added up to 64, no matter which way he counted them, or how he broke up the counting.

    Now, if we had no number system, it might be more difficult for us four-year-olds to have counted, but then of course you have the whole cause/effect thing that you might have had no cultural pressure to count. It strikes me that not having a number system and an affected ability to perceive exact quantities could be both effects of cultural determinism, so not having a number system and an affected ability to perceive might be /correlated/. But why should one cause the other? Isn't cultural determinism the causer of both?

  6. john riemann soong said,

    July 11, 2008 @ 1:00 pm

    "I think the teacher missed a prime opportunity"

    Prime, heh heh.

  7. Kate G said,

    July 11, 2008 @ 1:20 pm

    What about languages that use a variety of counting words? I have been told that Japanese has a set of numbers for counting flat things, and you would use entirely different ones to count people, or to number off the steps in a procedure. If this is true, has anyone looked into whether tasks like "are there more envelopes in this picture than people in this picture?" are harder for such speakers than for those who use the same number words for everything?

  8. Joe said,

    July 11, 2008 @ 1:30 pm

    > If this is true, has anyone looked into whether tasks like "are there more envelopes in this picture than people in this picture?" are harder for such speakers than for those who use the same number words for everything?

    As a student of Japanese, I can tell you that it's true that they have tons of counter words. I do not know of any such studies, however. Though I have noticed an aversion to exaggeration, which may be a cultural thing.

    Incidentally, there are also counters for birds (which are separate from other animals), small round things, generations (distinct from the counter for people in general), etc.

  9. John Lawler said,

    July 11, 2008 @ 2:31 pm

    @ Martin: There are no human languages that don't "encode gender" in the sense of distinguishing between men and women in some way — like different roots for mother and father. Granted, there are languages like Finnish that don't have grammatical gender, but everybody knows about vive la différence. Whereas the Pirahã don't have any terms for, or concept of, number, and no interest in one. If this were true of gender, there wouldn't be any Pirahã.

    @Kate: The "counter words" that Joe refers to (called classifiersin the trade) are not numbers, but markers that go with numbers to identify the things being counted.

    Thus in Japanese ni-hon no nasu means "two-roundthing of eggplant", while ni-ko no nasu means "two-longthing of eggplant", referring to two different varieties of eggplant — one round and the other long, natch. The number ni and the noun nasu are the same in both cases; only the classifiers hon (also used when counting apples) and ko (also used when counting pencils) vary.

  10. Axel Theorin said,

    July 11, 2008 @ 3:02 pm

    @Kate G, Joe
    You are conflating "counting/number words" and "counter words".
    The explication below doesn't do justice to neither English nor Japanese but it might clear up matters somewhat.

    Japanese does indeed have two words (number words) for every number up to ten (and historicaly for larger numbers as well). This is because they alternate between using originally Japanese words and words borrowed from Chinese.

    Counter words (or counters for short), which is what Joe is referring to, are a different matter. These are words attached to a quantity expression in order to estavlish what is being counted – e.g, in English, "three heads of cattle" for three cows, "head(s)" being the counter.
    The difference between for example English and Japanese here is that in Japanese the counters are
    1. Numerous! I don't know how many there are (I don't even know how many I know) but there are, as Joe says, tons.
    2. Obligatory. If asked how many cows are grazing on the pasture you must answer (the equivalent of) "three heads". Of course, just "three" would also be understood but it would be ungrammatical.

    I don't know if any studies of the kind Kate G asks about have been performed, but in my experience of using, and interacting, in Japanese I have never experienced or noticed that such tasks would be harder in Japanese or for native Japanese speakers.
    It is however notoriously hard (for me) to use the right counter, or the right number word for numbers up to ten :)

  11. Andy Hollandbeck said,

    July 11, 2008 @ 3:06 pm

    @John Cowan: "Their remote ancestors must have had a numbering system…"

    This is an awfully big premise to throw in there without any underlying explanation or proof. Why must have their remote ancestors had a numbering system?

    This whole argument piques my curiosity about what the Piraha "mathematical" system is like. If they don't have labels for whole things, how do they deal with parts of things? Do they have any type of architecture? And if so, is each new structure a hit-or-miss trial-and-error?

  12. Ryan Rosso said,

    July 11, 2008 @ 5:10 pm

    @Andy: The Piraha lead relatively simple lives. They're architecture doesn't go much farther than building simple huts to live under, as far as I know. If you go to Everett's webpage at ISU, you can see a few pictures.

  13. John Cowan said,

    July 11, 2008 @ 6:09 pm

    All languages have classifiers required on some nouns; the classifier languages are those which require classifiers for every noun. To put it another way, in a classifier language every noun is a mass noun.

    Andy: I just find it unlikely in the extreme that the Piraha, who must be related to the people living near them, can trace an unbroken line of descent from innumerate ancestors all the way back. More likely they are like the Moriori, who reverted to hunter-gatherer technology on their small and marginally habitable islands, but whose Maori ancestors were gardeners and farmers, or like the indwellers in remote Appalachian valleys who until very recent times had not left the valley in living memory, but whose ancestors had crossed the Atlantic.

  14. john riemann soong said,

    July 11, 2008 @ 6:27 pm

    It's interesting that some languages make further distinctions between singular and plural; why stop at dual? Why not making separate cases for triple, quadruple, or even making nth-number agreement productive? Perhaps some numeral systems might have come from a previously productive number-agreement system? I do wonder whether the first full-fledged languages would have had numeral systems.

  15. dr pepper said,

    July 11, 2008 @ 8:01 pm

    Not wanting to go off topic, i'm putting in a request that someone do an article on certain experiments done in the late 70's with AI systems designed to explore mathmatics. I've heard different things about them, enough to suspect that there's been some mythologizing. I'd especially like to know, in light of this article, did these systems always discover discrete numbers?

  16. Nathan Myers said,

    July 12, 2008 @ 4:04 am

    Everything plausible that I've read of the Pirahã suggests that what is different about them is not at basis linguistic, but ideological. In other words, their peculiarities are fundamentally no different in kind from those of, e.g., American "conservative" extremists. At some point in the past, for reasons that must have seemed to made sense at the time, each group deliberately abandoned a big swath of their rich cultural heritage, and succeeded in propagating the choice through to the present.

    This differs from, e.g., Tierra del Fuegans, whose ancestors must have forgotten about clothing on the trek through the Amazon, where it would have been mostly a burden, and then just failed to re-invent it when they got further south; or Tasmanians, who lost bits of culture (IIRC, ultimately including fire!) piecemeal through population attrition.

    To me this means that the interest inherent in their linguistic and cultural details is not in their provenance, but rather in practicalities of how they are able to get along without the linguistic and other technologies they have chosen to abandon. Why they chose to abandon them (to the extent there can be said to be reasons) is probably both unrecoverable and not really interesting.

  17. Kate Gladstone said,

    July 12, 2008 @ 11:04 am

    The teacher who regarded it "as distracting and even disruptive" for a child to come to the idea of odd and even numbers would need only to extend this attitude to fit in well among the Pirahã, who perhaps find it "distracting and even disruptive" to have the idea of any sort of number at all!

  18. Kate Gladstone said,

    July 12, 2008 @ 11:05 am

    Why does this blog persistently use "URI" where the rest of the Internet uses "URL"?

  19. Zachary Spector said,

    July 12, 2008 @ 12:37 pm

    @Kate Gladstone:

    There is a technical distinction between URI and URL, explained here. All URLs are URIs, but not all URIs are URLs.

  20. James Wimberley said,

    July 12, 2008 @ 1:05 pm

    I looked them up on the W3C site and found this:
    "A Universal Resource Identifier (URI) is a member of this universal set of names in registered name spaces and addresses referring to registered protocols or name spaces. A Uniform Resource Locator (URL), defined elsewhere, is a form of URI which expresses an address which maps onto an access algorithm using network protocols."
    So far as I understand this, a URI corresponds to the address "Professor Mark Liberman, Department of Linguistics, University of Pennsylvania, USA" which is unique but doesn't allow you to contact him. A URL would be the same with a zipcode, email or phone number, which make the unique identifier into a usable contact label. There may be alternate universes in Language Log phase space in which the URI is implemented in a different URL.

    Since everybody uses URL, writing URI is IMHO just showing off.

  21. James Wimberley said,

    July 12, 2008 @ 5:47 pm

    @John Cowan on the origins of this curious lack. How do linguists think languages speciate? Do they have a Darwinian gradualist theory, in which speciation is driven by some random clock mutation process, and the languages that do remain stable are beating the system artificially with devices like written literature and schools? Or are they Gould/Mayr punctuationists, seeing new languages as evolving rapidly in geographically isolated populations under strong selective pressure, and then tending to stay put even if they spread? The analogy to species is imperfect as unlike DNA languages can mix, by borrowing (a gradualist process) and creolisation (a punctuated one).

    I find it hard to envisage a gradualist explanation of the Pirahas' loss of number words. But I can imagine violent Hobbesian disruptions – quasi-genocides and enslavements – in which children between say 2 (onset of speech) and ?4 or 5 (acquisition of number concepts) were cut off from their parents and other language mentors and forced to fend for themselves linguistically. Remember that Latin stayed SFIK pretty stable for the thousand years of the Roman state, then mutated in a third of the time into a gaggle of Romance languages in the chaos and violence of the Dark Ages. If I'm right, the Pirahas' ancestors must have gone through something worse – the very edge of extinction.

  22. dr pepper said,

    July 13, 2008 @ 11:16 pm

    I believe that by asking for a URI, the owners of this forum are asking for any contact information you want to give. If they just put url, then only websites, ftp sites, and other data retrieval sites would be expected, whereas asking for URI means you can reply by giving a file site, an email address, a mailing address, a phone number, or the coordinates of a dead drop.

  23. James Wimberley said,

    July 14, 2008 @ 6:36 am

    .."or the coordinates of a dead drop"
    Perhaps we should continue this discussion under Moscow Rules.

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