Letters

Examining Knowledge of Geometry

Science  02 Jun 2006:
Vol. 312, Issue 5778, pp. 1309c-1310c
DOI: 10.1126/science.312.5778.1309c

In their Report “Core knowledge of geometry in an Amazonian indigene group” (20 Jan., p. 381), S. Dehaene et al. present evidence that an isolated Amazonian group, the Mundurukú, are able to understand geometric concepts. They state that geometry constitutes “a core set of intuitions present in all humans.” I disagree with the basic concept of this investigation.

The central feature of Euclidean geometry is its demonstrative character and its logical structure, rather than graphical pictures of triangles, circles, etc. This logical system is built upon two pillars: (i) the concept of the “theoretical object,” e.g., the abstract metaphysical idea of a circle, rather than a real constructed circle; and (ii) the deductive mathematical proof, based purely on axioms and postulates.

Other civilizations dealt with geometrical figures in a more intuitive way, and their activities cannot be characterized as geometry in the Euclidean sense. Ancient civilizations other than the Greeks did not develop a demonstrative geometry. For example, the ancient Chinese never developed a theoretical geometry (13).

The topic being investigated by Dehaene et al. is simply pattern recognition. It is by no means surprising that the people tested recognized different geometric figures, since they can recognize, e.g., human faces and identify different species of tree by their silhouettes.

References

  1. 1.
  2. 2.
  3. 3.

Response

In their Report “Core knowledge of geometry in an Amazonian indigene group” (20 Jan., p. 381), S. Dehaene et al. present research documenting the Brazilian Mundurukú Indians' ability to understand concepts of geometry and to orient themselves spatially. This team, as well as the scholars mentioned in C. Holden's article “Hunter-gatherers grasp geometry” (News of the Week, 20 Jan., p. 317), might be interested to learn that over 200 years ago, the great Brazilian naturalist Alexandre Rodrigues Ferreira also observed this innate ability of indigenes. In his memoirs of Amazonian zoology and botany [published collectively as the Viagem Filosófica (1)], he posed the question “what would a European, brought up like an Indian and ignorant of geometry, geography and hydrology, do if asked about a river's direction, branching and neighboring villages?” (p. 93, my loose translation). Anticipating Dehaene et al., Ferreira conducted an experiment and asked this question of a Tapuio Indian, who by tying together several cords was able to create an approximate map of the local river and its tributaries, as well as point out the location of Indian villages. Further, Ferreira wrote that a Macuxí Indian he encountered not only drew an intelligible map of local river patterns and scaled hut outlines using a stick to trace lines in sand, but when presented with a pen and ink rendered the same idea on paper. Clearly, the naturalist understood native Brazilians not only to be reasoning individuals, but capable of understanding geometric and geographic concepts. Thus, on the basis of his own experiments, Ferreira would have agreed with Dehaene et al.'s conclusion that “geometrical knowledge arises in humans independently of instruction, experience with maps or measurement devices, or mastery of a sophisticated geometrical language.”

Amazonian Mundurukú villager reading a map to identify a hidden object.CREDIT: S. DEHAENE ET AL.

Reference

  1. 1.

Response

Wulff contrasts processes of deductive reasoning based on the axioms and postulates of Euclidean geometry with processes of visual pattern recognition, and he suggests that the latter processes underlie performance on our tests of geometrical categorization and map use. Our tasks were designed to assess geometrical concepts at a higher level of representation. Recognition of visual patterns is orientation-specific (1), yet both our tasks required that the Mundurukú abstract geometrical relations from figures that varied in orientation. The Mundurukú's globally high performance, particularly in the map task which requires a transformation from two to three dimensions, cannot plausibly be attributed to low-level processes of visual pattern recognition and implies extraction of genuine geometrical invariants.

Our tasks also do not depend on processes of deductive reasoning. Although geometry now appears as a beautiful logical construction, logic and deduction are neither necessary nor sufficient to account for core human geometrical concepts and intuitions. The central intuitions of Euclidean geometry cannot be deduced from simpler axioms, as the history of mathematics and physics attests (2): Absent the problematic and unprovable parallel postulate, Euclid's axioms and postulates support an infinite family of geometries at odds with human intuition. Geometrical intuitions nevertheless come naturally to the human mind and continue to guide commonsense reasoning about space, even in scientists who have come to believe, by deduction and experiment, that the classical three-dimensional view of space fails to capture the structure of the universe (3).

We thank Delson for drawing our attention to Alexandre Rodrigues Ferreira's report. His informal observations on map-making appear to antedate ours by two centuries. Insofar as his observations were found to be general, they would confirm that the capacity to understand maps predates the most serious intrusions of Western culture. Caution is required in evaluating such ancient reports, but both Ferreira's report and our research suggest that all human cultures share an approximate arithmetic and intuitive geometry, which are highly stable over variations in education, language, and intercultural contact. On this point, we distance ourselves from claims of a radical “incommensurability” of cognitive functions in other cultures such as the Pirahã (4, 5), which are sometimes lumped together with our own views.

Research on core knowledge of geometry is at an early stage. How do geometrical concepts emerge in children? Are these concepts unique to humans or shared by other animals? What accounts for the distinctive profile of geometrical abilities shown both by indigenous tribes and by urban Americans? Converging studies across species, ages, and cultures, using methods of psychology and neuroscience, can begin to address these questions.

References

  1. 1.
  2. 2.
  3. 3.
  4. 4.
  5. 5.

Related Content

Navigate This Article