Evolution Versus Invention: a numerical case
David and Ann Premack’s 2005 commentary on Science got it right (Evolution Versus Invention — Premack and Premack 307 (5710): 673b — Science):
…it is not language per se that is critical for understanding the transition from analog to digital numeracy, but the change from foraging that did not require exact numbers to technologies that did.
There were talking about a set of recent papers on the number systems (or the lack thereof) in huntergatherer societies.
Hunter-gatherer groups, whose languages contain only a few number words ("one, two, many" in one case; "one" to "five" in another), pose a problem. Despite having number words, they perform only approximate numerical calculations, even when problems contain only numbers for which they have words.
… hunter-gatherer number words are not comparable to our own. Even the smattering of data reported by these authors indicates that hunter- gatherer number words name approximate magnitudes, not exact sets of objects as ours do. There is then no paradox in the fact that hunter-gatherers perform only approximate numerical calculations.
The way I think about it: "one, two, many" are individual symbols in the Judy DeLoache (2004) sense: something that somebody intents to represent something else. The label "two" may mean 2 or 3 depending on my pleasure, as long as it’s used in accord with a vague sociolinguistic convention (social contract of intentionality). What is lacking:
- it’s not a symbol system, i.e., individual symbols, rather than combinations of such, are the unit of representation.
- the social-linguistic ocnventions are … well, conventions. They are not codified to be precise.
What it takes is a leap from symbols to symbol systems. As the Premacks wrote:
… they then invented systems for representing exact numeracy: dots, bars, and a shell standing for zero (Mayan) (2); ropes and knots (Incas) (3); fingers and fists (sub-Saharan tribes) (4); the abacus (Chinese) (5); Roman and Arabic numerals; and so forth. The systems are all combinatorial: The use of a base (e.g., 5, 10, 20) permits forming new numbers by combining existing numbers. They are also recursive: New numbers are formed by adding one. Both recursion and a combinatorial approach are long-standing human capacities evident in, for example, both language and (to a lesser extent) social behavior.
A cognitive revolution occurred when symbols are combined to represent the state of the affairs. The pressure for such an unobvious move (we have data to show that preschoolers don’t get this) is clearly stated in Premacks’ remarks — the surplus of goods and increase of trade.
With due respect, thought, I’d disagree with the point about recursion. A symbol system does not have to be recursive, although the most mature and efficient ones are. I think many old number naming systems are not recursive (link to wikipedia). Recursion may be a more advanced benchmark, but at the minimal, a symbol system has to be combinatorial.
Premack and Premack end the commentary with this:
Although hunter-gatherers had little need for exact numbers, one can imagine that no owner of stored goods would wish to receive three casks of oil when he had stored four, nor would the individual who had stored the goods wish to return five casks when he had stored four.
Which I think reflects the theory of mind (Premack & Woodruff, 1978) of a particular culture. Anecdotally, I have seen reports that not all peoples are equally obsesed with accounting. I am sure you have had many occasions where you gave X number of things (are dollars things or just symbols of things?) and gladly walked away.
Ahh, symbols adn happiness, that’s topic for another day.