I have sent you our article on Exact and Approximate Arithmetic in Mundurucú, and the press communique related to it.
The reason is that as, you will notice, our study which involved various groups -- illustrating the social diversity of the Mundurucú -- and a French control group -- clashes with Gordon's 'incommensurability' hypothesis.
More precisely :
We personally favour a performance limitation based on the lack of a routinized counting routine.
We find the term 'incommensurability' highly inappropriate, because even in the Pirahã, there is a linear relation (and thus a "common measure") between the true numerosity and the approximate response given.
It thus seems much more likely that similar concepts of number are present in all cultures --- but that cultures vary in the invention of tools to measure number.
This is a clear and simple statement of the difference between Peter Gordon's interpretation of his research among the Pirahã -- which, he believes, calls into question Sapir's hypothesis of "formal completeness" -- and Pica et al.'s point of view. (See here, here, and here for my original post on the Pirahã article, and notes from Dan Everett and Peter Gordon on the subject.)
Here is a .pdf of the CNRS press release on the research of Pica et al., and a .pdf of their Science paper (Pierre Pica, Cathy Lemer, Véronique Izard, and Stanislas Dehaene, "Exact and Approximate Arithmetic in an Amazonian Indigene Group", Science, Vol 306, Issue 5695, 499-503, 15 October 2004). (Peter Gordon's paper from the same issue, "Numerical Cognition without Words", is here, and the discussion by Gelman and Gallistel, "Language and the Origin of Numerical Concepts", is here.)
Here is an excellent page discussing research on arithmetic and the brain at Stan Dehaene's Unité de Neuroimagerie Cognitive (Cognitive Neuroimaging Unit) in Paris, with many interesting links.
In his email, Pierre went on to discuss my silly throwing-vocabulary example, treating it with more seriousness than it deserves:
With respect to your hypothesis about the Nerdahã, (a population that you imagine) who aren't interested in throwing things, I agree with your story. The Nerdahã have no words for pitch, fling, chuck, toss, sidearm, slider, curveball, bouncepass, and so on, for trivial reasons right ? It is not an interesting fact, and the scientist is misled.
As far as I can tell the fact that the lexicon is limited in this case is not linked to any interesting properties in the case of the Nerdahã. Their case seems rather involve their lack of interest for a kind of sport (an activity which is usually not related to deep innate properties – the opposite in fact).
I do not agree with you however, that your story has anything to do with the the facts that we are talking about.
It is not even clear to me that a 'dormant knowledge' is involved in your hypothesis.
I believe that Pierre and I largely agree about the throwing business. In my imaginary example, what the Nerdahã lack is not primarily throwing vocabulary, but rather "a routinized throwing routine", i.e. a set of well-practiced motor skills and associated cognitive systems. The case is different in many ways from counting and arithmetic -- though surely it's plausible that throwing also has some "deep innate properties"? But the purpose of my example was to displace the discussion from an area that most people think of as happening mostly in the brain (counting) into an area that most people think of as happening mostly in the body (throwing). And everyone understands that the development of motor skills is mostly non-linguistic, even though socially-important skills generally have an associated vocabulary.
The example thus made the point that skills and vocabulary might co-vary, without vocabulary necessarily playing any crucial causal role in the development of the skills. The Mundurucú example demonstrates the same sort of dissociation, even more strongly, by showing that it's possible to have (some of) the vocabulary without the skills. It remains unclear whether some others could have the skills without the vocabulary -- but I would predict that the answer should be "yes". This certainly can be true for some complex skills, including some that are mainly "in the brain"; and it seems to me that it might be true for counting and associated exact-arithmetic skills, if the "routinized counting routine" were (for example) mediated entirely by finger-gestures, say of the kind used in chisenbop. Though on the other side, the author of this page on finger multiplication says that
Now that I have told you all that, I will also throw in my two cents' worth about number sense. This "trick" will not teach your students number sense. They would really be better off making arrays, and sets of numbers to "see" the different products. They can build upon those experiences later. The finger trick can be time-consuming and does not transfer to conceptual understanding.
As for the incomensurability issue, I perversely claim to agree with both Gordon and Pica, as I'll try to explain in a later post.
Posted by Mark Liberman at November 1, 2004 06:36 AM