Geometry and Genetics

An indigenous group called the Mundurukú, who live in isolated villages in several Brazilian states in the Amazon jungles, have no words in their language for square, rectangle, triangle or any other geometric shape except circles.

The members use no measuring instruments or compasses, they have no maps, and their words for directions are limited to sunrise, sunset, upstream and downstream. The Mundurukú language has few words for numbers beyond five except “few” and “many,” and even those words are not used consistently.

Yet, researchers have discovered, they appear to understand many principles of geometry as well as American children do, and in some cases almost as well as American adults. An article describing the findings appears in the Jan. 20 issue of Science.

“Across cultures that live extremely different lives, we see common foundational sets of abilities,” said Elizabeth Spelke, a co-author of the paper and a professor of psychology at Harvard, “and they are not just low-level kinds of abilities that humans share with other animals, but abilities that are at the center of human thinking at its highest reaches.”

To test their understanding of geometry, the researchers presented 44 members of a Mundurukú group and 54 Americans with a series of slides illustrating various geometric concepts. Each slide had six images. Five of them were examples of the concept; one was not.

The Mundurukú subjects, tested by a native speaker of Mundurukú working with a linguist, were asked to identify the image that was “weird” or “ugly.” For example, to test the concept of right angles, a slide shows five right triangles and one isosceles triangle. The isosceles triangle is the correct answer.

In data that do not appear in the article but were presented by e-mail from the authors, Mundurukú children scored the same as American children – 64 percent right – while Mundurukú adults scored 83 percent compared with 86 percent for the American adults.

The researchers also tested the Mundurukú with maps, demonstrating that people who had never seen a map before could use one correctly to orient themselves in space and to locate objects previously hidden in containers laid out on the ground.

The indigenous people were able to use the maps to find the objects, even when they were presented with the maps at varying angles so that they had to turn them mentally to match the pattern on the ground in front of them. Dr. Spelke found this particularly significant.

“The Mundurukú, who aren’t themselves in a culture that relies on symbols of any kind, when they were presented with maps were able to spontaneously extract the geometric information in them,” she said.

The idea that an understanding of geometry may be a universal quality of the human mind dates back at least as far as Plato. In the Meno dialogue by Plato, written about 380 B.C., he describes Socrates as he elicited correct answers to geometric puzzles from a young slave who had never studied the subject.

Do these findings among the Mundurukú confirm Socrates’ contention that concepts of geometry are innate? Stanislas Dehaene, another co-author and a professor of psychology at the College of France, is not willing to go quite that far. People learn things, after all, just by living in the world.

“In our article we do not use the word ‘innate,’ ” he said in an e-mail message. “We do not know whether this core knowledge is present very early on – the youngest subjects we tested were 5 years old – or to what extent it is learned. The Mundurukú, like all of us, do interact with 3-D objects, navigate in a complex spatial environment, and so on.”

Instead, Dr. Dehaene described an innate ability, rather than an innate knowledge. “Our current thinking is that the human brain has been predisposed by millions of years of evolution to ‘internalize,’ either very early on or through very fast learning, various mental representations of the external world, including representations of space, time and number,” he explained.

“I have proposed that such representations provide a universal foundation for the cultural constructions of mathematics,” he added.

Dr. Spelke sees in these results evidence of the universality of human thought processes. “Geometry is central to the development of science and the arts,” she said. “The profile of abilities that the Mundurukú show is qualitatively very similar to what we see in our own culture. This suggests that we are finding some of the common ground at the center of human knowledge.”