Speaking of Numbers

Chris Chatham of Developing Intelligence reminded me, in comments to the post on categorizing fish, of this paper published a little over a year ago in Science. It's an interesting description of what can best be described as preliminary or pilot research on the relationship between number words and number concepts in a culture with only three number words: one, two, and many. Here's a description of the population from the paper (p. 469):

The Pirahã... live along the banks of the Maici River in the Lowland Amazonia region of Brazil. They maintain a hunter-gatherer existence and reject assimilation into mainstream Brazilian culture. Almost completely monolingual in their own language, they have a population of less than 200 living in small villages of 10 to 20 people. They have only limited exchanges with outsiders, using primitive pidgin systems for communicating in trading goods without monetary exchange. The Pirahã counting system consists of the words hói (falling tone = ‘one’) and hoí (rising tone = ‘two’). Larger quantities are designated as baagi or aibai (= ‘many’).

Perhaps more interesting than the lack of number terms for amounts greater than 2, the Pirahã use their word for one, "hói," to refer to any small quantity, sort of like English speakers use "a couple" to mean something in the neighborhood of two.

The author of the paper, Peter Gordon, visited the Pirahã three times, and after becoming interested in the question of how their verbal number system might affect their numerical concepts, conducted several pseudo-experiments (no, that's not a derogatory term, it just means that they weren't proper experiments from which we can infer causation) designed to explore Pirahã counting abilities. Several of the studies involved presenting the Pirahã participants with an array or cluster of familiar objects, and asking them to produce an array with the same number of objects. In another study, the participants were presented with line drawings and asked to draw the same number of lines (a task that was apparently very difficult for the Pirahã participants, because drawing is something they never do). In a third type of study, participants watched as the experimenter put nuts into a can, and then removed them, and were asked to report when they thought the can was empty. For each type of task, the Pirahã participants' performance dropped significantly for numerosities greater than 2 or 3 (up to about 10).

Gordon argues that these studies provide at least preliminary evidence for a strong version of the Sapir-Whorfy hypothesis. This strong version is generally called linguistic determinism (as opposed to the weaker version, linguistic relativity). He believes that the Pirahã are able to count small numbers (less than 3), perhaps by subitizing, and that they can also use analogue representations (e.g., lines whose length represent a quantity) for larger numbers (techniques which are inherently less accurate). These are both skills that appear to be present in all cultures, and even in some nonhuman animals. However, he believes that the evidence from his studies indicate that the Pirahã are not capable of more sophisticated counting techniques for numbers above three, and that this is due to the lack of number terms in their language.

There are several problems with these studies. The most obvious is that Gordon makes a causal inference from pseudo-experiments, which is a methodological no-no. There are no control conditions, no comparisons with similar populations, and no testing of different causal explanations. As Daniel Casasanto points out in a letter to Science in response to Gordon's paper, one could predict Gordon's data with either the linguistic determinism hypothesis or with its exact opposite. It could be that the environment and cultural practices of the Pirahã make the learning of number concepts over and above numerosities of 2 or 3 unnecessary, and that this in turn has led to the absence of terms related to these concepts in the Pirahã language. Gordon replies (in the same pdf file, just read to the end of Casasanto's letter) that he's not actually making a firm statement of cause, but when someone argues for "linguistic determinism," it's hard to interpret it any other way.

Speaking directly to these issues is another paper that appeared in the same issue of Science as Gordon's paper. That paper presents experiments (real experiments, this time, though they’re very limited) conducted with speakers of the Mundurukú language, who also live in the rainforests of Brazil. Here's a description of the speakers and their language from the paper (p. 500):

Here, we studied numerical cognition in native speakers of Mundurukú, a language that has number words only for the numbers 1 through 5. Mundurukú is a language of the Tupi family, spoken by about 7000 people living in an autonomous territory in the Pará state of Brazil.

So, the Mundurukú language has more number terms than the Pirahã, but is still much more limited in its number terms than Western languages. Furthermore, the Mundurukú do not use numbers in exact ways, or in counting, but instead use them to approximate numeroisities.

In several tasks requiring Mundurukú speakers to compare the numerosity of large arrays of dots (20 to 80), they performed well but slightly worse than a French-speaking control group. They also performed as well as the French-speaking participants on an arithmetic task that required only approximating numerosities. However, when they were required to perform an arithmetic task that required giving exact answers (a subtraction task), they performed significantly worse than the French controls, particularly as numerosities increased.

The authors of this paper argue that their results provide evidence against the strong determinism view advocated by Gordon. Instead, they write:

What the Mundurukú appear to lack, however, is a procedure for fast apprehension of exact numbers beyond 3 or 4.

In other words, the lack of a counting system, not the lack of numerical terms, appears to be the primary cause of the difference between Mundurukú and French-speakers performance on the subtraction task. Since the Pirahã don't have procedures for counting, either, this may underlie their difficulty with large numbers as well.

The picture I'm trying to paint with these two sets of studies is just how messy research on language and thought really is. Even the studies with the Mundurukú speakers, though they are proper experiments involving experimental and control groups, are at best preliminary explorations into the numerical knowledge of Mundurukú speakers, and the causes of differences between their numerical abilities and those of speakers of languages with larger sets of number terms. Figuring out whether language, as opposed to other cultural and/or environmental variables are responsible for differences in cognition is damn near impossible to do with any certainty. But the research is fun anyway.