Story

This story was given to me by my brother-in-law Tony Payne, who cannot remember where he got it from. However, a very similar story, but involving crows rather than baboons, is given in The Number Sense: How the Mind Creates Mathematics (Dehaene, 2011), From One to Zero: A Universal History of Numbers  (Ifrah, 1994) and Number: The Language of Science (Dantzig, 1941)

If a man with a gun goes to shoot baboons near the edge of a forest, the baboons will see him coming, hide in the forest and not come out until he is seen to go away.

If the first man hides and a second man with a gun joins him, and then one of them walks away, the baboons will stay hidden and not come out of the forest. They know that there is still a man hiding with a gun .

The same is true if two, three, four or possibly five men join the first man and the same, or a smaller, number of men go away. The baboons stay hidden, they know that at least one man is still hiding.

However, if six men with guns join the first one and then six of them ostentatiously walk away, after a while the baboons will come out of the forest and can be shot by the man they have failed to accounted for.

Interpretation

When a second man joins the first man and then one of them walks away, the baboons can calculate that one man remains hidden, they can subtract 1 from 2 and get the right answer that there is 1 left.

The baboons can also get the right answer with 1 plus (2, 3, 4 and possibly 5) minus (2, 3, 4 or 5 respectively).

It is therefore thought that baboons can do small number arithmetic with numbers up to about six, so have a number system something like none, one, two, three, four, five, six and many (more than six).

But when six men join the first man and then six  men walk away the baboon’s number system lets them down. This is because one plus six results in many, and many minus many gives none, the wrong answer. So it seems to the baboons that all the hunters have gone away and it is safe to come out of the forest.

As shown in the post Small Number Arithmetic, with a number system of none, one, two and many, the system works well for addition, exhibiting closure (each operation results in an unambiguous instance of the number system itself). So one plus two gives three, two plus three gives many and many plus many gives many etc.

As baboons can apparently find out to their cost, problems arise with subtraction. Subtracting anything except none (one, two, three or many) from many is always problematic. In particular, taking many from many seems most likely to result in none but in reality could also result in one, two, three or many depending on the unknowable “real” values of many.

It seems that the baboons are applying the simple rule “same minus same always results in none”. So one minus one, two minus two, three minus three and many minus many all give a result of none.

Discussion

There is no evidence of anyone hunting baboons like this. The story was probably just made up to illustrate the limitations of languages that do not have sufficient number words or a number system that can represent all numerosities.

There are human languages that use small number arithmetic and have a word like many to represent large numerosoties. (Butterworth et al., 2008) Isolated hunter gatherer cultures seem to have little need for an elaborate number system in their languages but tend to acquire them quite quickly upon contact and trade with the outside world.

The story also suggests an experimental technique for establishing the upper numerosity discrimination limits of animals and pre-verbal children, using an ‘expectancy violation technique’. See What Counts and  Otto Koehler.

Unnecessary Expansion

The problem caused by a lack of necessary number words is superficially similar to the apparent order in which languages “acquire” colour words. Some languages only have two colour words, cold and warm, corresponding to monochrome, black and white. Others have black and white plus red. Yet others add yellow then green or green then yellow, then blue, brown and finally purple, pink, orange or grey. (Berlin and Kay, 1969)

Colour Hierarchy Diagram (after Berlin and Kay 1969)

The diagram above works from left to right (following the arrows and plus signs). If a language has a particular colour word then it will also have all the colour words to the left of that word. So if a language has a word for blue, then it will also have words for yellow, green, red, black and white. The diagram also indicates that if a language has a word for say pink, then it may, or may not, have a word for purple, but it will have colour words for brown, blue etc.

The colour words in a language provide foci, or prototypes, for the colour experience but say nothing about the boundaries between these colour foci.

This sequence corresponds fairly closely to the order in which children acquire colour words and is therefore a cultural example of the discredited evolutionary theory of recapitulation. This is summarised in Ernst Haekel’s phrase “ontogeny recapitulates phylogeny”, suggesting that as embryos develop into adults, they go through stages that resemble the evolution of their species.

Acknowledgement

Thanks again to my brother-in-law Tony Payne who told me the baboon story. See also Beau Geste Hypothesis and Cafetières, Disorder, Chaos and Anarchy

Bibliography

Berlin, B. and Kay, P. (1969) Basic Color Terms: Their Universality and Evolution. Berkeley: University of California Press.

Butterworth, B., Reeve, R., Reynolds, F., & Lloyd, D. (2008). Numerical thought with and without words: Evidence from indigenous Australian children. Proceedings of the National Academy of Sciences of the USA, 105, 13179-13184.

Dantzig, T. (1941) Number: the Language of Science, Nature. Edited by J. Mazur. Pi Press New York. doi: 10.1038/147009a0.

Dehaene, S. (2011) The number sense: How The Mind Creates Mathematics, Revised and Updated Edition.

Ifrah, G. (1994) The universal history of numbers.