Threesology Research Journal
"One- Two- Many"
page b

(The Study of Threes)

We might want to consider that a similar primitive occurrence of the one- two- many reference might also be recognized in our modern usage of an organizational format found in some jails and prisons with respect to the housing of prisoners:

One man cell ~ Two men cell ~ Many men 'cell'

The "Many men cell" reference illustrates that a dormitory or open bay (military) style of bunking is used for those prisoners who can be housed together with minimal or at least exhibit manageable levels of inter-personal levels of conflict that may arise due to prevailing, long-term encarcerated circumstances.

If we broaden our appreciation of this three-patterned grouping into permitting look-a-like or sound-a-like qualities coupled with a reference to enumeration that is found in some primitive (early) counting systems [as well as being used by our own modern counting system with respect to the "teen" suffix found in numbers following twelve and the repetition of segmented groupings such as prefix "twenty" when saying twenty one, twenty two, twenty three, etc.. or "thirty" prefix when saying thirty one, thirty two, thirty three, etc., as well as larger number grouping prefixes such as one hundred..., one thousand..., one million..., etc., each of which acts as delineated segments of cognitive limits in and of themselves that are simply joined together with other grouping limits called successive ordering]; another language example can be found in the colloquial (euphemistic) expression referring to a person viewed as being intoxicated by alcohol. It is said that they have had:

One ~ Too ~ Many to drink

A third example which exemplifies the third item as being different from the first two enumerated "categories" to the exent it is used as an attendant complement in a three-part sequence as a unit of completion and designation of a cognitive parameter that illustrates a sort of stoppage, end point or border, though the act of grouping in threes can be alternatively repeated:

One ~ Two ~ Buckle my shoe

You will notice in this child's counting game that there are successive three-patterned groupings:

1st enumeration2nd enumeration 3rd non-enumerated
OneTwoBuckle my shoe
ThreeFourShut the door
FiveSixPick up sticks
SevenEightLay them straight
NineTenA big fat hen

...but each grouping is considered a separate item containing an internalized repetition similar to many primitive counting systems that exhibt a succession of counting by using some method of recursion (repetition), that also contains rhyming numbers such as if we were to count to three by repeating the number One:

One (+) One (+) One

If we consider the One ~ Two ~ Buckle my shoe in a strict numerical sense and that the third item, as a rhyme to the second item, has the same numerical value, the resulting product would be five, (One + Two + Two). Overall, as with many counting systems, this child's counting game has the value of Ten as a cognitive limit with respect to the "rules of the game," even though some children (and adults) have tried to extend this counting rhythm beyound the Ten, but have been hampered by the difficult (cognitive) task of finding some relatively simple linguistic formula which supports a widely known, accepted (and repeated) everyday word that rhymes with Twelve. We must wonder whether enumeration is limited by some inherent brain structure of maturational deficiency, or is enumeration limited merely by one's language capacity due to a specific culture's language limitations (as apposed to brain development involving language)? Both?

Alternatively, we can find some resemblance of the ONE two Many structure in the following expression:

If I told you once (one)- I've told you a thousand (many) times. The sound-alike quality of "you" to "two" could, in this instance, be considered a caricature.

Another example, without even the 'sound quality' of the "two" can be found in the TV advertizing slogan of an (RC Willey) furniture store:

One place, so Many possibilities

Likewise, the phrase "E Pluribus Unum" found on American money has been translated from the latin as: Out of Many, One. Such examples could be argued as examples representative of a perpetuated cognitive schema.

The following is an example of research in language which describes an instance of the One ~ Two ~ Many and "threeness" circumstances found in a primitive Amazon tribe.

Language & Cognition
University Seminar #681
Columbia University
New York, New York

What can the study of language contribute to our understanding of human nature? This question motivates research spanning many intellectual constituencies, for its range exceeds the scope of any one of the core disciplines. The technical study of language has developed across anthropology, electrical engineering, linguistics, neurology, philosophy, psychology, and sociology, and influential research of the recent era of cognitive science has occurred when disciplinary boundaries were transcended. The seminar is a forum for convening this research community of broadly differing expertise, within and beyond the University. As a meeting ground for regular discussion of current events and fundamental questions, the University Seminar on Language and Cognition will direct its focus to the latest breakthroughs and the developing concerns of the scientific community studying language.

Numerical cognition without words:
evidence from Amazonia for strong determinism

Peter Gordon
Department of Biobehavioral Studies
Teachers College, Columbia University
4 October 2001

There has been a resurgence of interest in recent years regarding the Whorfian hypothesis that the structure of language can affect how we think and perceive the world. Such a hypothesis takes two forms: weak determinism and strong determinism. The former claims that linguistic structure can affect perception and cognition through the habitual ways in which language is used. However, it does not claim that language can prohibit or restrict the range of thoughts and perceptions. This position does not question Brown and Lenneberg’s original contention that any idea can be expressed in any language. Strong determinism challenges this assumption, claiming that speakers of one language can fail to entertain concepts spoken in another language. In this talk, I will present data from the Pirahã a tribe from lowland Amazonia who lack words for numerical quantities greater than two. Experiments investigating the Pirahã ability to perceive and encode numerosities greater than three revealed that below three, performance on most tasks was close to perfect. Above this number, performance dropped off depending on the cognitive demands of the task and reliance on strategies such as chunking. However, despite poor performance on larger numerosities, Pirahã participants showed excellent estimation for quantities up to eight or nine, with performance almost perfectly on target and showing level coefficients of variation with increased set size. These results are very much in line with current research in infant, child, adult and animal numerical cognition, consistent with an innate dual system for numerical competence. In addition, they suggest that the lack of numerical words in the Pirahã language prevents bootstrapping their innate individuation abilities into a formal counting system in which mandatory numerical perceptions are possible.

There is an old joke I sometimes tell when starting this talk, which is: There are three kinds of people in this world, those who can count and those who cannot. This is a talk about the latter kind. We’re going to talk about the very old question of whether a language can determine the way that you think. This is also known as the Whorfian hypothesis, but he should not take all the credit, or all the blame, depending on your perspective. His predecessors were also in part responsible for some of these ideas.

I want to distinguish between weak determinism and strong determinism. I will go into this a bit more, but basically weak determinism refers to the way a language habitually refers to time or space. There is a preferred way of referring to reality. Evidence of language effects on people is that people may show a preference to categorize or think in a particular way compared to speakers of another language, but it does not rule out the ability to construe ideas that are expressed in another language. In other words, I am not saying we cannot think in this other way. That is what I am calling strong determinism; that there are actual incremental abilities between languages, that there are actually ideas that you can express in one language that you cannot express in another. The Whorfian hypothesis is a stronger view, which would claim that you cannot express those ideas because you cannot actually think those ideas. I have heard this referred to as a straw man argument, that this could not possibly be true and that the only interesting opinions of this were for weak determinism. I am going to try and argue that this hypothesis is actually true in the case that I am going to show you.

I am going to talk about a tribe called the Pirahã, who I worked with quite a while ago as a result of interacting with Dan Everett, who became the chair of Linguistics at the University of Pittsburgh, and who had spent about twenty years working and living with this tribe. I am just going to put a tape on in the background to show you some of the way that they live.

[Professor Gordon shows a video of the Pirahã tribe and village.]

They are hunter-gathers so basically they will stay in a village for about a year or two and then move on. Often when people start dying in a village they will think its bad Karma and they will move to a different location. They live along the Maici River. This is a typical village; it has about ten or fifteen people in it. The river they live on, the Maici, runs into the Amazon. They have the philosophy of not telling their children to avoid dangerous situations.

[In background video, child playing with a knife.]

They have a population of about sixty people total in the villages along the river. They are monolingual in their language Pirahã, so they may know one or two words in Portuguese but they are not at all fluent. They resist assimilation to Brazilian culture, primarily because they have had bad experience with local traders who generally take advantage of them. There is some limited trading, sometimes with other tribes, sometimes with traders, but no money. Generally they will exchange things for goods. They have a very limited social structure, there is no clear chief in the village but there might be a dominant male who beats you up if you do not do what he says. They do not have any external representation, they do not write, they do not have any art, or toys.

I am going to talk about the counting system in Pirahã which is a one - two - many system. The word for one is hói, two is hoí and many is baagi. One and two are distinguished by a falling tone versus a rising tone. One of the things that got me interested in this case initially was that at the time I had been reading Gelman and Gallistel’s early work on children’s counting. Quite a large part of that book was dedicated to work by Zaslavsky who was essentially trying to get rid of some of the numerical myths of new world cultures. Gelman and Gallistel at the time were putting forward a fairly strong hypothesis that the ability to count was like the ability to use language; that there was some innate disposition there. If you have cultures that do not really count in the way that we know it, it is problematic for that theory. Zaslavsky claimed that cultures that have one - two - many counting systems had alternative finger gestures; so they were able to use their fingers instead of count words. They might use non-decimal recursive systems such as the Gumulgal in Australia that essentially have a one-two system which is fully recursive so you can say, one-two, two-one, two-two-one and so on.

Zaslavsky also claimed that there are certain taboos and certain things that you cannot count for religious reasons. When I first went to the Pirahã I stayed for about ten days and I basically just asked them some questions so that I could get a sense of what was going on there and try to get to their numerical quantities. There is clearly no evidence either for numerical taboos, they were able to count anything, or recursive systems. Dan has been going for twenty years and has never seen anything like that, so that is clearly not the case. The other interesting thing, that may follow from having a fairly low counting system, is that you do not actually count in the sense of a serial count. The fact is that even though we translate these words as "one" or "two" it is really fairly loose. It is more like when someone says in English, "Give me a couple of those." If someone gives you three, you do not say, "I said a couple!" It means "two, more or less." Hói in Pirahã means, "one more or less," so it could express two, three and even quite large numbers. In this first year we did some informal experiments about the meaning and use of these number words.

[Professor Gordon shows a video of experimenter Karen Edwards working with some members of the Pirahã tribe on various number naming tasks.]

This is Karen Edwards, Dan Edwards’s wife, she is laying out objects, lemons and so forth. She is speaking to them in Pirahã but just giving them the Portuguese numbers. She is asking them to make another row that is the same. You will see in this tape that the person is using his fingers but it is not an anchoring device. So when she says three he says two again.

[Video shows a Pirahã participant using hoí for two, three and four items; he uses baagi for five items.]

Dr. Inge-Marie Eigsti:

What is the task?

Professor Gordon:

She is asking him to say how many in his own language.

[Video shows the Pirahã participants cheering loudly.]

They become pretty enthusiastic when they get it correct. They do not get it wrong all the time. I will show you at the end if you look at the average they actually do pretty well.

[Resumes video. Shows tasks asking Pirahã to make identical lines composed of flashlight batteries or lemons. Next the experimenter knocks twice with a fist on the table, and the Pirahã participant, asked to match the number of knocks, knocks three times.]

This gives you a sense of how they see numerosities. There is quite a lot of research recently that suggests that the weak form of determinism is applicable in cases where people of different cultures may have a propensity to think in ways that reflect their language that may have to do with time, space, number, and so on. The question is whether there are concepts in one language can be translated into another. My hypothesis is that the Pirahã language is numerically incommensurate with languages that do count, that the Pirahã do not have concept of number or counting that can be equated with a counting culture, the numbers are not rigid designators and they are not used in serial counting as I said before. The accurate estimations of numerosity are limited to those quantities that can be directly perceived, which are those small numbers up until about three.

There is quite a history that the first three or four numbers or quantities show a discontinuity with the rest of the number line. This goes back to the idea of subitizing, which we find in Cattell. We find reaction times and error rates fairly level up to about three and then we show an increase after three. These are for brief presentations in which you have to say how many objects there were. There are other experiments in the literature that suggest the same limitation of three or four units. With pre-attentive units, you can follow moving objects in a display amongst other objects. You can keep track of those objects to about three or four and then you are not able to do it. Dehaene has talked about this notion of parallel individuation, which is the number of individuals that you can keep in mind at the same time, is about three. There is also this idea of three being the real magic number not seven. Things actually come in threes more than seven; mom saying, "I will count to three." Or "One, two, three…go!" And, remember all the religious significance of three and anthropological literature where three actually plays a big role. For large numbers, there is a lot of literature. Rather than having an accurate individuator, what you have is a sort of analog estimator of quantity. So the representation is thought to be of more continuous in nature rather than discrete. You have estimation for example in tasks that involve:

[Professor Gordon bangs on table a large number of times]

How many times did I bang? It is too fast to count but you tend to be fairly accurate. As the number gets larger your error rate gets larger. Studies with infants and with animals also show similar properties. I want to talk about where the one-two-many counting system fits into a universal possible counting system. The base of the counting system can be either perceptual or body based. In other words, you can use the three-ness as your base and it could either be nonrecursive, in which case the system becomes one - two - many or a one - two - three - many. There are not too many systems that go beyond that, so for instance you do not get a one-two-three-four-five-many system. Nor are there many binary systems like the one that we find in Australian culture, which is recursive. Obviously you can employ the ten base systems which tend to be recursive. There are different cultures that use fingers. There are also cultures that use other parts of the body and you usually get counting that goes up to thirty-three or so.

[Professor Gordon shows slides of the various body parts used as numerical markers.]

In general symbol systems borrowed from the three perceptual base. Even the words for one, two, and three tend to derive from the I to one view, you to two and three as a notion of through, so a merging into that other system. You see the three as merging in the notations for numerosity. They tend to show these repetitive elements up to three and sometimes up until four but not beyond. In Cuneiform, Etruscan, Roman, Mayan, Chinese, and Arabic we see these repetitive elements and then they go to something else after four. Chinese and Egyptian plurals are indicated by three-ness. You can see the threes emerging in these cave paintings from .... B. C.

[Professor Gordon shows slides of the cuneiform and examples of cave paintings.]

There is something fundamental about this three-ness that seems to reflect a limited individuation in a perceptual system.

I want to talk now about a second set of studies that I did when I went back the second year. I stayed there for two months and did some more systematic studies. I wanted to see if the Pirahã could perceive numerosities despite lacking linguistic labels, and I wanted to develop tasks that allowed them to show this without having to count overtly. We started with simple one-to-one matching and we added different configurations to make them harder. You then add things requiring memory and spatial transformations and so on. The studies were carried out with two villages in six weeks. As I said before, these are very small populations. In addition to the small number of people in these villages, it is impossible to test any of the women because of taboos and restrictions on interaction between females and outside males. The children just run away screaming if you try and test them. I had about seven subjects total.

In one of the villages I went to, I essentially went on my own so I had to learn enough of the language to be able to do these tasks, but I am certainly not fluent. It is a difficult language to learn. A missionary living there for ten years never really learned it. He was there to translate the Bible, and he became sick with anxiety every morning because he could not deal with the language. The payment for participation was food and beads. A subject would get easily bored so as an experimenter I would just have to go with the .ow. There is a real concern that most people who have worked with them have been threatened, so you have to be nice to them.

[Professor Gordon shows slides and describes the various tasks performed.]

The first task is a matching task using the number of batteries as the sample to be matched and as the constituents used by a subject to perform the task. Accuracy in performance drops off after three. The second task is to copy a line drawn in a notebook.

Professor Paula Rubel:

Are the tasks culture bound?

Professor Gordon:

Some people might ask if it is appropriate culturally to use batteries. Yes, they actually do use batteries for hunting alligators. The third task is a cluster line match asking the Pirahã to match batteries to a line of nuts. They did well up to eight, but performance accuracy also drops off after three, although they were able to do some sort of grouping. The fourth task is an Orthogonal Line task. I lined up the batteries parallel to the bar and they had to line them up in an orthogonal manner. You can see performance dropped off quite precipitously after two. This is a very difficult task because you see you have do some sort of tagging and do a spatial transposition. The next task is an uneven line match. I had an uneven row of batteries. This is an interesting task which shows a U-shaped variability. The Pirahã subjects did perfectly in three and four and their performance dropped precipitously and then increased again, as if they were doing some sort of chunking or grouping. When you get a large enough number you can start grouping into different sets.

To introduce a memory factor, I was doing a task that you would do in a subitizing test, where you just present an array very briefly. I would have a bunch of nuts on my side of the table and I would cover it with a book and I would lift for a second and then replace it. They would just receive a short amount of time to encode their numbers and they would have to match it with batteries. Again we can see that their performance drops off precipitously after two. That is a very low proportion at six and above.

In the nuts-in-a-can task, an array of nuts is put out on the table and they can inspect that for as long as they want. Then I would put the nuts into the can one at a time and when all the nuts were in the can I would remove them out one at a time. After each one I would say, "Are there still some in there, or is it empty?" At some point the subject has to say that the can is empty. Again you see they are very bad at this task, and performance drops off quite sharply.

The candy in the box task is a task where I would have a cassette case that would have a picture with a certain number of fish on it. I put the candy into the cassette case and I would put it behind my back and I there would be another cassette case behind my back and I would shuffle them around and bring out two of them. One would have the original configuration and one would have either one less or one more and they would have to choose which case had the candy in it. You can see that performance is not much better than chance, beyond that two versus three.

In summary, these tasks show accurate performance for small numbers although when the task demands are increased you can see that even after about two their performance drops o.. Large numbers are inaccurate and I want to argue that their perception of large numbers shows the same kind of analog representation that we might have for much larger numbers like ... It has been shown that our perception of large numbers when we cannot count them shows these properties.

[Professor Gordon shows graph of average performance across tasks.]

The average response to various target items is almost perfect. The mean of their responses across tasks lies almost directly on target. A second line in the figure represents response variation increasing with set size. So, there is a gradual climb up until about three and you take the coefficient variation, where you divide the variability by the target size. The average is about six across group and tasks, it is the only way you can get a big enough number to do this. But, this finding assures the validity of the measures by opposing the arguments that the subjects were not trying, that they did not understand the instructions, or that there was a big conspiracy to make fun of us.

Professor Michele Miozzo:

They have a notion of "six or seven" but its not as precise as your notion of "six or seven."

Professor Gordon:

Right, they have an analog estimation of it. It is not a sort of precise counting system like we have.

Professor Robert Krauss:

Doesn’t it strike you that the matching tasks are conceptually very different than the memory tasks?

Professor Gordon:

I tried generating these graphs by taking out the memory tasks and the result does look that different.

Professor Paula Rubel:

Do you think that if these people became acculturated, that their performance would indicate cultural change, or perhaps an influence attributable to Portuguese coming in? The question has to do with cognition and language is separate. And you are talking about cognition as well as language in a related fashion without separating and them and saying that language and cognition are the same? Does the language make the difference?

Professor Gordon:

I am not saying they have some genetic difference where they cannot count. Is that what you are asking?

Professor Rubel:

You are saying there is a cognitive difference. That has nothing to do with genetics—it is cognition.

Professor Gordon:

Based on the fact that they do not have number words that allow them to be precise. They would be like us if they had number words. There is an interesting question is whether there is a critical period. So if you introduce those numbers to them as adults would they be able to ever understand those number words. Dan and Karen tried this in downstream villages. They set up a school and had numbers up on the wall and did number tasks. The kids got it right away. The adults came and after a couple of days they could not get it and did not want to come to school any more. That is not knock down evidence for a critical period but there certainly is a difference between how well the kids could grasp it compared to the adults.

Mr. Ezequiel Morsella:

This is an ecological validity and testing question. I do not know what animals they hunt, but let us guess that they use bow and arrow. If you showed them six monkeys would they know how many arrows to take? The matching tasks to us seems very straight forward, but just the fact that they are being tested I think is hard.

Professor Gordon:

It is sort of an unrealistic situation. I know what you are saying, can you think of a situation where they would have to deal with number in their every day life? There never is a situation where they know precisely how many monkeys there will be. The closest thing occurs when they are framing a hut, and they never build a hut with three legs instead of four, but I do not think it is because they count, I just think they do it geometrically.

Professor Abe Rosman:

Has a trader introduced Brazilian money to them at all? When money is introduced, it is reasonable to expect the Pirahã to start diferentiating amounts based on number.

Professor Gordon:

Actually, the trader is better able to steal from the Pirahã because of their incapacity to handle numbers. Even when they deal with regular trading they are not very good at it. The Pirahã might spend two months in the jungle collecting Brazil nuts to trade, and then the traders give them a couple of cans of condensed milk for all that work and say that the Pirahã are indentured to them and have to collect more. It is a very harsh system. The stores that sell these products from the Rain Forest are really getting these supplies from very disreputable people.

Karen had three guys doing some yard work at this house that she built. They had been working really hard all day and done the same amount of work. To one she gave a big full bag of farina, and the second guy she gave half and the third guy she gave a tiny amount. When the third guy’s wife saw how little he got she started laughing. The other two guys were laughing, including the second, he did not realize that he was being ripped off. The way that they deal with quantity is different. There is a lot of trading, and possession is transitory. There is a lot of moving around so everything they have they bring with them in their boat, so personal possessions are pretty much limited to their bows and arrows. There is not a lot of counting that goes on.

Professor Boris Gasparov:

Do they have an expression to ask, "How many, how much?"

Professor Gordon:

Yes, but there are thing that the language does not permit, like, "Which has more?" To say, "This has more than that," you need to use recursive structure in language, but Pirahã lacks the construction of relative clauses. In this language it is possible to say, "This is big," or "This is small," but not "This is more or less than that."

Professor Gasparov:

Did you conduct any experiments on unconscious processes, making things into larger or smaller parts?

Professor Gordon:

No, they probably could, given that they can estimate numbers in another experiment. They could quantify things, but it is not the first thing to come to mind. I do not know if anyone has read Butterworth’s book on the number sense. He talks about sitting down to breakfast and everything has to do with numbers: You pick up the paper, and the date is numbered, the pages are numbered, everything is quantified for us, but in that culture nothing is. When we see arrays of objects we obligatorily quantify them. We cannot see things as not being numerical. Their way is a very different way of seeing the world.

Professor Marco Jacquemet:

What about days and the calendar?

Professor Gordon:

They measure things by high water and low water. Time of day is noted by the position of the sun in the sky.

Mr. Jamey Hecht:

What about age, people growing older?

Professor Gordon:

No one knows how old anyone is. When you are doing studies it is impossible to find out how old anyone is, you just have to guess by looking at them. I think that this evidence shows that the Pirahã do not have integer concepts.

Interestingly one is not a privileged quantity in their system. The word hói can mean "roughly one." Without a precise notion of one there is no way to generate a recursive system of counting. One is the central basis for having a recursive number system. We also need to think about what is the difference between what I have been calling strong determinism and expertise. Is this phenomenon different from the contrast of a master chess player who looks at a chessboard and sees configurations versus a novice who looks and just sees black and white pieces? Is it experience with quantification or is it actually the language shaping the way that we think? I think there are fundamental domains of language. and that if you do not understand the word five then you have got something wrong with your English. In contrast, if you do not understand the word molecule and you do not know what a molecule is, you still speak English. If there are domains that are fundamental to the organization of a language, and if these are the candidates for strong determinism, then it becomes more than a simple matter of having words its that the language sets up these fundamental ways of thinking.


Professor Rubel:
I am not sure whether strong determinism in your mind asserts a lack of translatability. It seemed at the beginning that was what you were saying.

Professor Gordon:

I think that is one part of it. It is incremental ability. If you believe that language determines thought, then the consequence of that untranslated ability is that it makes concepts unknowable to the people of the impoverished language.

Professor Rubel:

Are you familiar with the work of Berlin and Kay? I wonder what is the color terminology for these people. Is it black and white?

Professor Gordon:

It is black, white, red and green. In my third year there I wanted to do something equivalent with colors, but we went to the other end of the river where they did not know me and they were suspicious of me so that I did not get much work done. They thought I was with the Indian agency and they thought I was carrying a gun.

Professor Rubel:

It would be really interesting if it were just black and white and you had only two numerical figures.

Professor Gordon:

If you look at the color literature, with Brown and Lenneberg, when they introduce memory tasks there were problems with remembering colors that they did not have words for. I think that color is so hard wired; that it is hard to make the case that you would actually not see. It is dark, light, blood and bright, which is red and green. I did do a sort of experiment; I had a towel which had lots of different colors on it. I would point to something that was yellow and they would say red, I would go to green and they would say green. I would go to blue and they could not call it green because they just called the other green, but on another occasion they might call it green. So now they might call it dark or red. The names that they give the colors are not really fixed; they shift around a lot depending on the context.

Professor Robert Krauss:

I have two questions; one simply has to with the method. I am supposing you gave them the choice between things that differ in quantity. Would they not be able to appreciate that difference?

Professor Gordon:

I think it would depend on how big the difference was. If it was the difference between eight and nine then fifty percent of the time they would get it. If it was the difference between eight and four then they would probably get it.

Professor Krauss:

The other question, in going back and reading the older literature up to the present you get the sense that in terms of these big metaphors like: language is the mold that thought is poured into, that there was not a real clear sense of the way that language affects thought. Lenneberg, in a paper that is not really well known, made the point that it is very hard to separate the concept from the culture. Is it the fact that they do not have the language, or is it that that culture does not need the concepts, that a language lacks a particular term?

Professor Gordon:

I think there are other cultures that live very similar lives to the Pirahã that do have counting systems. I do not think you can necessarily predict based on the simplicity of the lifestyle whether a culture will have a counting system or not. I have a hunch that there is a relation between the language not having recursion and not having an aggregate counting system.

Mr. Jamey Hecht:

If language does not have the term then it is unlikely that they are going to learn the concept. You can conceptually separate the notion of counting and how to determine for counting. So, is it the language, or is the language a marker for experience that that person has had?

Professor Gordon:

What Whorf says is that language sets up the categories of experience. Depending on your language you experience the world in a different way. John Lucy interprets that as a claim that language affects habitual ways of thinking. I think he is wrong about a lot of things. There are lots of concepts we have in English that other languages do not have, which I do not think count against the hypothesis, concepts like molecule. If we go into another culture that does not have science they are not going to understand what a molecule is. I do not think that they see the world in a different way because they do not have that concept. For the Whorfian hypothesis to be interesting, we have to be dealing with some core concepts of the language. Core concepts tend to be things that reflect a cognition that is not just a fundamental way of looking at the world.

Professor Miozzo: I was impressed at seeing how good they were at the tasks, when you presented the graph it showed a linear increase. This is very similar to what I have found in some brain-damaged patients. For example there are patients who cannot recognize that the number five is five and three is three. If five is presented as an Arabic digit they would say eight or nine. But if you ask them to do another task that taps into the semantic system for these numbers, not only are they perfect but you also see a linear increase, a kind of distance effect.

For example, they are faster at saying that the difference between eight and five is bigger than the difference between six and five. Perhaps there are different (forms) of representations, one that is based on integral systems, which permit an individual to know that 5 is exactly what it stands for, and another system that is reflective of size, so five is something that is bigger than four and smaller than six, but you do not have a possibility of setting very precise contrasts. It is remarkable that some of our subjects demonstrate semantic appreciation it is very similar to what you have found.

Professor Gordon: I think that is important, the idea that the primitive number system is a dual model. The one-two-three and then greater is a less precise analog estimator. I think that is what we start out with. What the language has to do is make the rest of the number system look like the beginning of it. I think you have to bootstrap from that small quantity into the large quantity system.

Professor Remez:

It would seem to me that there is no strong evidence that the Pirahã have one. I am not sure that the evidence requires a conclusion that they have one through three and then the system breaks down. It seems to break down right from the crack of the gun. To make the case for one-two-many warrants evidence that there is a conceptual notion of integer as opposed to a graded quantity distributed about one. If you could show that they had one, then I would be willing to accept that they had something like a number system. I would be very surprised if their language did not have quantifiers, in the normal linguistic sense, which include some, many, every, all, each, none. Those expressions I think are intrinsic to language, as intrinsic as prepositions are. As far as numbers go, I think as Bob says they lack the referential practice of counting, and the evidence here is just not strong for one.

Professor Gordon:

You are right. I sort of assume they have the small numbers just extrapolating from everything else.

Professor Rosman:

You mentioned that I, you, we, as another aspect of grammar, showed certain similarities to one, two, three. When you were asked about color categories, you said they do the same thing; they do not mean to say the color red, they mean the color blood; they do not mean the color green, they mean the color light. You have done a lot of analogizing into the different parts of the grammar which sound very much as though what Robert was saying is true, if indeed you could take parts of a grammar and say that it is related to other parts of a grammar. Things like shapes, pouring. In English you can pour liquids but you can also pour sugar. There are analogies everywhere.

Professor Rubel:

They must have words for child and old men. You said they did not have words for young or old.

Professor Gordon:
No I did not say they did not have words for young and old. They do not have precise ways of saying how old someone is. But Robert, they never made any mistakes with one or the quantity of one.

Professor Remez:

If the measures were sharpened you might see contrasts appear. It might require a test to see if they can use rational numbers.

Professor Michael Studdert-Kennedy:

One of the things that struck me about this curve, if you are judging seven would they be very likely to give you eight or six. So in other words they have got a good idea, an approximate idea. Where do children perform?

Professor Gordon:

It is very similar to babies. A lot of the early baby literature that looked at the ability to distinguish numerosities showed that they went up to three. When they try to three versus four they could not do it. The message early on, is that they could appreciate numerosity because we are showing that they could distinguish one versus two and two versus three. They could never get three versus four, but they can do eight versus four, but they cannot do eight versus six.

Professor Studdert-Kennedy:

It is normally the situation that an apprehension can be guided by language once they are given the words. I propose that the baby in our culture has a capacity, which seems to be about what the Pirahã have, but then when given words, they start doing it.

Professor Ann Senghas:

I am intrigued by Robert’s comment about them having one more or less. If they are twenty percent off, roughly, then any time you are presenting them with one versus two they are going to pick the right quantity. I would like to see how they were performing with mass quantities, for instance, with a cup of flour. If they were going to match these things would they also be twenty percent off all the time? If they are treating everything as analog then they would not even have one cup of something. Maybe they are really treating these things as analog that we treat discretely. My second comment is that I am struck by the lack of recursion. Do they have coordinate structures?

Professor Gordon:

No, the only thing they can do is to stack adjectives.

Professor Jim Magnuson:

Is it likely that in their culture a difference of one does not matter much because there are no precious commodities. Perhaps an operant training regimen would be successful.

Professor Gordon:

It is sort of like learning a second language. Some people can learn it really well if they have the right approach and the right kind of teaching they can learn to speak a second language without an accent. But most people cannot. What is possible and what is normal are two different things. Probably with the right training, right circumstances we could probably teach them.

Dr. Inge-Marie Eigsti:

Weren’t you saying that someone did try teaching them?

Professor Gordon:

Yes, but it wasn’t very systematic. It was more like they just stopped coming to practice because they did not like it. The kids liked it. All we know is there is a difference between the attitude of the children and the adults.

Professor Remez:

Let us thank Professor Gordon and adjourn.

--- Language and Cognition, 2001- 2002, University Seminar #681, ---
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Note: when speaking of patterns-of-three, whether in an A- B- C, *- **- ***, or 1- 2- Many configuration, it must be acknowledged that there may not be any underlying numerical order being used on a conscious level. Just because you or I use a numerical referencing tool in order to discribe a grouping, does not mean the user of the grouping necessarily has any conscious reference of the same as a number series. This lack of enumeration for a grouping that we reference as "three" may in fact illustrate that the grouping is related to something other than a numerical influence. Thus, the usage of the idea of there being three families of fundamental particles, a triplet codon system to DNA, or a triune brain, represent something more fundamental than a mere numerical symbol. In other words, does the development of a DNA strand have a blueprint that references the "three" as a number quantity, or merely functions as a molded object of sorts made by an uneducated production-line worker pouring wax into a pre-fabricated mold to make a three-wicked candle? No less, did the maker of the mold engineer its structure by way of using a blueprint which designates the numerical reference of a "three," or are they also merely acting as an uneducated production-line worker with a title of "engineer" that merely acts as a means of differentiation for purposes of placement in the overall manufacturing plant processes of assignment where numerical referencing is an attribute of affectation akin to workplace jargon that has little relevance outside a specific context?

One, two, ... er, too many

Tim Radford, science editor
Friday August 20, 2004
The Guardian

Researchers claim to have solved the mystery of the people who simply do not count. It could be because they are lost for words. The Piraha of the Amazon have almost legendary status in language research. They have no words at all for number. They use only only three words to count: one, two, many. To make things confusing, the words for one and two, in Piraha, are the same syllable, pronounced with a falling or rising inflection. And to make things really difficult, the word for one can sometimes mean "roughly one", and the word for two can sometimes mean "not many".

Peter Gordon, a behavioural scientist at Columbia University in New York, reports in Science today that the Piraha may may not be very good at counting because because they do not have the words for it. The Piraha have puzzled anthropologists for decades. Around 200 Piraha speakers live in settlements of 10 or 20 people on the banks of the Maici river in the lowland Amazon region of Brazil, using the same pronoun for "he" and "they" and being imprecise about quantities of fish and manioc. They provided a test for an old riddle: do words determine thought or does thought determine words?

Dr Gordon set them a series of simple numerical challenges. He asked the Piraha people to match small sets of objects by number. The adults performed accurately with sets of two or three items, but the accuracy declined when tribespeople were asked to match sets of eight or 10 items. Their skills, he reports "were similar to those in pre-linguistic infants, monkeys, birds and rodents".

Lions have a sense of numbers. Chimpanzees and even macaque monkeys can count up to nine. But the Piraha can be inaccurate even when they use their fingers to show numbers lower than five. "You can't get beyond the concept of three, unless you have the word for it," said Brian Butterworth, a neuroscientist at University College London. "Children of three to four can easily do the tasks the Piraha adults were unable to do. With training even some non-human species can do these tasks. Chimps can. Monkeys can in some circumstances. "It has been known for 50 years that birds can match sets of up to about seven. So I find it very strange that these Piraha adults are unable to do these tasks. "Maybe there is more to it than just having a language short of number words."

"The Piraha tribe in the Amazon has only three words used in counting, that mean one, two, and many. A psychologist testing them has found that they are unable to accurately perform tasks involving quantities as few as four or five. He says that this shows that, at least for numbers, language shapes and limits how people can think." I can't help but be reminded of the gully dwarves from Dragonlance when reading this.

"We have it...on the authority of African explorers that many Hottentot tribes do not have in their vocabulary the names for numbers larger than three. Ask a native down there how many sons he has or how many enemies he has slain, and if the number is more than three, he will answer 'many.'"

[ George Gamow, "One, Two, Three...Infinity" 1953 ]

Above two items from: --- Slashdot ---

Language may shape human thought
19:00 19 August 2004 news service
Celeste Biever

Language may shape human thought – suggests a counting study in a Brazilian tribe whose language does not define numbers above two.

Hunter-gatherers from the Pirahã tribe, whose language only contains words for the numbers one and two, were unable to reliably tell the difference between four objects placed in a row and five in the same configuration, revealed the study.

Experts agree that the startling result provides the strongest support yet for the controversial hypothesis that the language available to humans defines our thoughts. So-called “linguistic determinism” was first proposed in 1950 but has been hotly debated ever since. “It is a very surprising and very important result,” says Lisa Feigenson, a developmental psychologist at Johns Hopkins University in Baltimore, Maryland, US, who has tested babies’ abilities to distinguish between different numerical quantities. “Whether language actually allows you to have new thoughts is a very controversial issue.”

Peter Gordon, the psychologist at Columbia University in New York City who carried out the experiment, does not claim that his finding holds for all kinds of thought. “There are certainly things that we can think about that we cannot talk about. But for numbers I have shown that a limitation in language affects cognition,” he says.

“One, two, many”

The language, Pirahã, is known as a “one, two, many” language because it only contains words for “one” and “two”—for all other numbers, a single word for “many” is used. “There are not really occasions in their daily lives where the Pirahã need to count,” explains Gordon. In order to test if this prevented members of the tribe from perceiving higher numbers, Gordon set seven Pirahã a variety of tasks. In the simplest, he sat opposite an individual and laid out a random number of familiar objects, including batteries, sticks and nuts, in a row. The Pirahã were supposed to respond by laying out the same number of objects from their own pile.

For one, two and three objects, members of the tribe consistently matched Gordon’s pile correctly. But for four and five and up to ten, they could only match it approximately, deviating more from the correct number as the row got longer. The Pirahã also failed to remember whether a box they had been shown seconds ago had four or five fish drawn on the top. When Gordon’s colleagues tapped on the floor three times, the Pirahã were able to imitate this precisely, but failed to mimic strings of four of five taps.

Babies and animals

Gordon says this is the first convincing evidence that a language lacking words for certain concepts could actually prevent speakers of the language from understanding those concepts. Previous experiments show that while babies and intelligent animals, such as rats, pigeons and monkeys, are capable of precisely counting small quantities, they can only approximately distinguish between clusters consisting of larger numbers. However, in these studies it was unclear whether an inability to articulate numbers was the reason for this.

The Pirahã results provide a much stronger case for linguistic determinism, says Gordon, because, aside from their language, they are otherwise similar to other adult humans, whereas there are many more factors that separate babies and animals from adult humans. However, scientists are far from a consensus. Feigenson points out that there could be other reasons, aside from pure language, why the Pirahã could not distinguish accurately for higher numbers including not being used to dealing with large numbers or set such tasks.

“The question remains highly controversial,” says psychologist Randy Gallistel of Rutgers University in Piscataway, New Jersey. “But this work will spark a great deal of discussion.”

Journal reference: Science Express (19 August 2004/ Page 1/ 10.1126/science.1094492)

Selection from: --- New Scientist ---

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Herb O. Buckland