George Gamow begins the first chapter of his delightful book "One, two three...infinity" with an apocryphal story about two Hungarian aristocrats who play a simple game. In this game, one of them merely has to say a number, and then the other one has to say out a number larger than the first number. Then the first aristocrat has to say out an even bigger number and so on. This is how the conversation proceeds:
Aristrocrat 1 (after thinking for twenty minutes): OK! 3
Aristocrat 2 (after thinking for an hour): Sorry! I give up.
No doubt this gentle slander was Gamow the prankster making fun of Hungarian scientists, many of who were among the most brilliant of the century. But further on in this fascinating chapter on the 'Mathematics of Infinity', Gamow talks about a real tribe of people in the deepest jungles of Africa called the 'Hottentots'. Apparently these people speak one of the most phonetically constrained languages on earth. The consequence-they really cannot count beyond three. So that if the Hottentot has to say how many hogs he killed for dinner, or how many enemies he slew in battle, and if the number exceeds three, he will simply say 'many'. With such a limited power of expression, how does a Hottentot father know, for example, how to share his goats equally among his two sons? In doing this, the Hottentots display what is one of the simplest but most ingenious methods of counting ever, and one which has deep reperscussions even in the most abstract and bizzare reaches of mathematics: Comparison. The father will simply line up the first goat for the first brother with the first goat for the second brother, the second goat for the first brother with the second goat for the second brother, and so on. If in the end, no goats are left over, that means that the goats have been equally distributed. This is how the Hottentots have got over their debilitating difficulty of counting beyond three. This method of establishing a one to one correspondence between things for counting them is so simple, that one would think it must have been discovered by all the arithmetically challenged people in the world.
Apparently not. In some recent amazing research published by linguist Peter Gordon from Columbia University, a tribe of people has been discovered in the jungles of the Amazon, which simply cannot count beyond three, no matter what. The reason, Gordon claims is that these people simply don't have NAMES for numbers beyond three. This conclusion roils up a long debate which connects the fields of Linguistics, Philosophy, Cognition and Mathematics: the relation between language and the real world. To do this, Gordon travelled into the Amazon jungle to visit the Piraha tribe, a remarkable group of only 200 people or so, who are some of the last artefacts of the simple life on earth. To test their counting skills, Gordon conducted the simplest tests. For example, he placed a row of ten batteries on the table and asked the Piraha to duplicate the row. Negative. Next, he drew successively, rows of one, two, three and ten lines on a sheet of paper and asked them to duplicate them. Negative. Lastly, he placed candy in a box which had pictures of a certain number of butterflies on them. He then shuffled the box with others and asked the Pirahas to pick the one which had the candy in it. In this case, even with the lucrative reward inside the box, the Pirahas could not pick the right one if the butterfly number exceeded three.
For me, this was an astounding finding, precisely because I used to think that anyone who would have a problem counting beyond three would at least not have a problem when it came to DUPLICATING objects greater than three in number, by applying the simple process of comparison demonstrated above by our friends the Hottentots. I still have to read the original article because it's not gotten published in detail yet. One interesting question springs to my mind. Can the Piraha at least distinguish between, say, twenty matches and thiry matches? If that is the case, then it would seem that our mind has a remarkable ability for 'counting without counting'. The real question is, what exactly happens in out brain when we count? Is counting just a conditioned reflex incorporated by parents and teachers in us as children, so that "one" is instantly identified with the abstract entity "1" in mathematics and so on? Or is counting an innate act wired in our brain at birth? Whatever the case is, one would expect that simple duplication does not need counting: you merely need to assure for example, that you place a battery in front of every battery in the initial row, no matter how many there are. In fact, if someone does this and correctly duplicates the row of batteries, I would NOT say that he is 'counting' in the literal sense of the term. From this point of view, I would deem Gordon's experiments as being inconclusive to whether the tribe was 'counting' or not. But ironically, it seems that the tables have been completely turned on us, because the tribe could not even do duplication. The only conlusions that eminent researchers have drawn from this fact is that languaage must be irrevocably linked with math. However, I fail to see again how a 'duplication' experiment can say the final word about 'counting'.
Why were the Pirahas unable to duplicate the row of batteries? I don't know, although I would agree that the observation is very fascinating.
Ludwig Wittgenstein once said that all our knowledge about the world is made possible through language, and that without language, even thought and logic do not exist. At face value, this seems to be a valid conclusion. What happens when I try to solve a logical puzzle? I say to myself, either silently or loudly, "Ok, this is the case...hence....therefore" and so on, until I reach the solution to the puzzle through more or less articulated expressions. But a more introspective analysis may make us think that the logic inherent in the world and in nature would be independent of language. However, any step on our part towards resolving this matter is doomed, because we can only use language to talk about language! This is a long standing and probably the most famous and intractable puzzle in logic, mathematics and philosophy. How do we talk about the nature of logic itself, but not using logic? How do we dissect the most abstract intimacies of language, without using language at all? Wittgenstein was very much aware of this self engulfing problem, like the serpent who perpetually continues to swallow his own tail, even as the tail moves away from him all the time. That is why the last paragraph of his famous work "Tractatus Logico Philosophicus" contains a most profound and provocative phrase; "Whereof we cannot talk, thereof we must remain silent". Wiggenstein would surely have loved to know about this discovery.
The problems of language and counting are as old as humanity itself. Gordon has made a path breaking discovery and one that would surely shed light on fundamental issues in a variety of fields. However, I am skeptical about his concusions drawn from the 'Duplication' experiments, and amazed by the results themselves. In fact, such events are quite profound, so that I would need to think more about this discovery and not draw any rash judgements from it. But Gordon also provides a marvelous example of that central principle common to all acts of discovery and progress; experiment and observation, something which can relegate even the most beautiful theory to non-existence, and elevate even the most mundane sounding theory to eternal glory. Science and Society, both benefit from this almost final judge of contention. As to Wittgenstein's proclamation, maybe the following can give a possible answer:
"Whereof one cannot speak, thereof one should confirm by observation"!