Mathematical ideas, like artistic ideas or religious ideas, are a universal part of human culture. This forthright claim isn’t made by Ascher, but her book compels me to that conclusion. Mathematics as we know it was invented by the Greeks. But mathematical ideas involving number and space, probability and logic, even graph theory and group theory—these are present in preliterate societies in North and South America, Africa, the South Pacific, and doubtless many other places if anyone bothers to look.
This is not to say that everybody can do mathematics, any more than everybody can play an instrument or succeed in politics. Many people do not have mathematical or musical or political ability. But every society has its music and its politics; so too, it seems, every society has its mathematics.
Some people count by tens, others by twenties. “There is an often-repeated idea that numerals involving cycles based on ten are somehow more logical because of human fingers. The Yuki of California are said to believe that their cycles based on eight are most appropriate for exactly the same reason. The Yuki, however, are referring to the interfinger spaces.” And how about Toba, a language of western South America, in which “the word with value five implies (two plus three), six implies (two times three), and seven implies (two times three) plus one. Then eight implies (two times four), nine implies (two times four) plus one, and ten is (two times four) plus two.”
Professor Ascher knows of three cultures that trace patterns in sand—the Bushoong in Zaire, the Tshokwe in Zaire and Angola, and the Malekula in Vanuatu (islands between Fiji and Australia formerly called the New Hebrides). Sand drawings play a different role in each culture. “Among the Malekula, passage to the Land of the Dead is dependent on figures traced in the sand. Generally the entrance is guarded by a ghost or spider-related ogre who is seated on a rock and challenges those trying to enter. There is a figure in the sand in front of the guardian and, as the ghost of the newly dead person approaches, the guardian erases half the figure. The challenge is to complete the figure which should have been learned during life, and failure results in being eaten. . . . The tales emphasize the need to know one’s figures properly and demonstrate their cultural importance by involving them in the most fundamental of questions—mortality and (survival) beyond death. The figures vary in complexity from simple closed curves to having more than one hundred vertices, some with degrees of l0 or l2.”
In all three cultures, there’s special concern for Eulerian paths—paths that can be traced through every vertex without tracing any edge more than once. (The seven bridges of Königsberg!) All three seem to know that an Eulerian path is possible if and only if there are zero or two vertices of odd degree. The Maori of New Zealand play a game of skill called mu torere. The game is played by two players; the “board” is an eight-pointed star. Each player has four markers—pebbles, or bits of broken china. Prof. Ascher shows that with any number of points except eight, the game would be uninteresting. “Mu torere, with four markers per player on an eight-pointed star, is the most enjoyable version of the game.”
The Caroline Islanders north of New Guinea cross hundreds of miles of empty ocean to Guam or Saipan. “The Caroline navigators do not use any navigational equipment such as our rulers, compasses, and charts; they travel only with what they carry in their minds.” Professor Ascher reminds us that leading anthropologists once taught that preliterate peoples were at an early stage of evolution. (Western society was advanced.) Later it was said that preliterate peoples (“savages”) had an utterly different way of thinking from us. They were prelogical. We were logical.
Nowadays anthropologists say there’s no objective way to rank societies as more or less advanced, higher or lower. Each is uniquely itself.
Professor Ascher’s research is related to ethnomathematics as an educational program. This movement asks schools to respect and use the mathematical skills pupils bring with them—even if they differ from what’s taught in school. By increasing understanding and respect for ethnomathematics, this work may benefit education.
There’s a lesson for the philosophy of mathematics. Mathematics as an abstract deductive system is associated with our culture. But people created mathematical ideas long before there were abstract deductive systems. Perhaps mathematical ideas will be here after abstract deductive systems have had their day and passed on.
