Parallel Lines Never Cross, Even in Remote Amazonia

By Valerie Ross | May 24, 2011 3:20 pm

What’s the News: Adults and school-age children may understand some basic principles of geometry even without formal math training at all, according to a study published online yesterday by the Proceedings of the National Academy of Sciences. Thirty members of the Mundurucú, an indigenous Amazonian group, could intuitively grasp geometric concepts about angles, lines, and points, the researchers found.

How the Heck:

  • The researchers asked eight Mundurucú children ages 7 to 13 and twenty-two Mundurucú adults to answer 21 questions about the geometry of a plane or a sphere, such as “Can a line be made to cross two other parallel-looking lines?” They also had the participants estimate angles that would complete an unfinished triangle, using their hands or a measuring tool.
  • The Mundurucú children and adults performed far better than would be expected by chance, if they were randomly answering “yes” or “no” to the questions. They answered 90% of questions about planar geometry—an imaginary, totally flat world—correctly, and 70% of questions about a spherical world correctly. When estimating angles, their answers tended to be within about 5 degrees of the correct answer.
  • In fact, the Mundurucú did about as well as French children of the same age and American adults who had had formal math education. Younger American children, however, ages 5 to 7, did poorly by comparison, answering more questions correctly than they would by chance but not as many as older children or adults.
  • These findings suggest two possible explanations: Either understanding of geometry is innate, but for some unknown reason doesn’t emerge until about age 7, or it’s acquired through “general experiences with space, such as the ways our bodies move,” says psychologist Véronique Izard, lead author of the study.

What’s the Context:

  • Nature versus nurture is a longstanding debate in psychology (and many other fields of science): what are we born with, and what do we glean from experience? Much recent research has eroded the divide between the two: it turns out that many human traits are defined by the interaction of nature and nurture.
  • As far back as the 18th century, philosopher Immanuel Kant argued that people have in-born intuitions about geometry.
  • Other research has suggested that humans have innate number sense and math aptitude.

Not So Fast:

  • This is a small study, looking mainly at one isolated group. While the comparison with French people and Americans suggested a broad human facility, it remains to be seen whether people in other groups would grasp geometry the same way, or whether a different sort of assessment would produce the same results.

Reference: Véronique Izard, Pierre Pica, Elizabeth S. Spelke, & Stanislas Dehaene. “Flexible intuitions of Euclidean geometry in an Amazonian indigene group.” Proceedings of the National Academy of Sciences online before print, May 23, 2011. DOI: 10.1073/pnas.1016686108

Image: Wikimedia Commons / HAHA VENOM

CATEGORIZED UNDER: Mind & Brain, Physics & Math
  • Kevin R. Bridges

    I always just assumed it was innate, in the way certain areas in physics seem innate to a monkey (as in, how far would I go if I swung off of this branch?) or aerodynamics to a bird. Something that isn’t understood through reasoning, but as “common sense.”

    Of course, I guess if we don ‘t have studies like this, we just assume everything is common sense, and we never learn anything.

  • A. Taylor

    I don’t agree that “the comparison with French people and Americans suggested a broad human facility,” since both the French children and American adults had had some Western education, while the Amazonian children had not.

    Also, in order to test the hypothesis that children under 7 lack this understanding of geometry, one would need to sample other children in this age range – ideally Amazonian and French. In addition to American adults, they should test Amazonian and French adults before making any suggestion about the role of age. It’s important to avoid confounding the two variables, age and nation of origin.

  • Michael Berry

    Like Kevin, I also assumed these types of things were innate, or rather learned through natural experiences of trial and error, or as the the article phrased it “acquired through general experiences with space, such as the ways our bodies move.” To me, it seems the Amazonian children might simply be extrapolating their spatial reasoning from the real world to the exercises in geometry they were given.

    Interesting study.

  • Jordan C

    Of course nature v nurture is going to rise up. After all, we are talking about people who need to understand angles in order to hunt. Hunting a moving animal is akin to understanding the trajectory and relative path of a snooker ball. If you fail at geometry in the real world where angles = kills and therefore food for your tribe/family, then you will fail at life!!

  • Juris Zars

    I am reminded of the many times, in an effort to bolster the confidence of my mathematics students at the start of the school year, I paraphrased Plato’s so called “Socretes and the Slave Boy” story. The results of the above study and the wisdom expressed by Plato point in the same direction, and the implications of this principle are of value to both the teacher and the student.

  • Tom Hanna

    If these people, in the amazon, were able to guess the angles by within 5 degrees, something tells me someone was there before to educate them on the fact that there are a total of 360 degrees in a plane!

  • joasnosmae

    Tom, i don’t think they guessed by within 5 degrees numerically. As the article states, they used measuring tools, or their hands, to estimate the angles. Duhhhhhh.

  • Ryan

    I’m sure many of these tribal people are better human specimens than much of the detritus that inhabits our “advanced” societies. In their world, the gap between success and fatal failure is much smaller.

  • Iain

    Younger American children, however, ages 5 to 7, did poorly by comparison,
    I’m assuming that the Amazonians kicked ass here as they spend most of their time in the world, not a virtual world.

  • John

    “Socratese and the Slave Boy” is in one of the late Martin Gardner’s antholigies (such as “The Mathematical Mag Pi”). A classical story.

    If we were to ask a chimp those questions (like Washo) what would the sign language answers be? Or an African Gray Parrot?

    Might we find that geometry was instinctive?

  • Holden

    Hey joasnosmae,
    Your contribution is one word too long. There’s no need for that.

  • vmaxxed

    No news, philosphers have known that since middle ages.

    The idea or concept of parallel lines is defined by Euclidean geometry as lines that never cross. So the regular answer is: No, they never cross, because then they will not be parallel according to Euclid.

    But, I’m sure the questioner is not asking about Euclidean geometry but about reality.

    In this sense some people will say: No, parallel never cross because there is no such a thing in real life. Parallel lines only exist in our mind.

    But this answer is too simplistic. There is “reality” in our “ideal” concept of parallel lines. If reality was so disconnected from our geometric and logical constructs we would not be able to interact with the real world. We can walk, see and reason reality because we find circles, lines, logical constructs and 2 as the sum of 1 + 1, in the “real” world. Or at least, a very close approximation to that, though never “perfect” circles or lines.

    Kant clearly demonstrated that these ideas are “hard coded” in our mind, not learned, because we need this idea “framework” to rationalize perception in the first place. There has never been a society where 1+1=3 and where parallel lines cross at say, 100 feet.

    Questions like these are the beginning of modern philosophy, beginning with Plato who had not other option but to conclude that these ideas have an existence of their own in some kind of idea heaven. Of course the point is not whether this idea heaven exists, but to point out the real problems: Why do we have this constructs in our mind? What is the relationship to reality? Where do ideas come from?

    So, in summary, yes, parallel lines in our mind never cross, and yes, they seem not to exist in reality. But there is a very real connection between the idea and the “real” thing, but since we can only “see” the “idea” we can not now in “reality”.


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