The paths we travel

Jacob Schor ND, FABNO

September 14, 2014


In New England, toll roads are still called turnpikes, a term descended from 14th century Middle English that refers to an early version of the toll booth that employed wooden pikes to limit passage until a fee was paid.  In California highways are called Freeways.  The term "highway" itself descends from the fact that Roman roads were elevated above the surrounding landscape and literally were high ways.

North of us, Calgary has its own term for large limited access roadways: they call these roads ‘Trails,’ as in Deerfoot Trail, Blackfoot Trail, Macleod Trail, Edmonton Trail, and so on.

MacLeod Trail in downtown Calgary, 2011

This use of the word trail conjures an image that these roads lay atop early trails used by fur trappers and indigenous peoples.   Though likely a false assumption on my part,  still, the notion is hard to resist, that these big roads evolved from narrow passages through the forests and across the prairie.

The contrast between natural ‘trails’ and architect and engineer-designed pathways is well illustrated at the newly innovated off-leash dog area at Cherry Creek State Park.  Serious money has been spent over the past year updating this section of the park.  The bottomland east of the creek is now encircled by a 6-foot-fence with double-gated access points to prevent the dogs from escaping.  Pathways have been cut through the sod, the lined with black plastic landscaping fabric and covered with gravel.    Walking these designer paths is quite a different experience from following the various trails and old wagon tracks that predated the park’s improvements, and actually predated the park’s existence.

The primary goal in designing these paths seems to have been to maximize the distance one can walk in the park.  While the path more or less stays near the fence perimeter the path curves back and forth like a sine wave, the opposite of a straight line.

The variations in topography do not appear to have entered into the calculations  and though all of the fenced off leash area might be considered flood plane or bottomland (depending on your viewpoint, from river or bluffs) there are subtle variations that become obvious when it rains.  Particularly when it rains hard.


After the 2013 rains, the pathway quickly turned into a riverbed


The heavy rains that came down a year ago in the fall of 2013 made these subtle variations in landscape obvious as many of the pathways quickly turned into streambeds.


In a matter of hours these new rivers eroded away not just the trails but also the roadbed and much to the surprise of the rangers, who like to write citations if dogs are unleashed outside of specified fenced area, created some rather deep erosions.



“I guess it’s deeper than it looks” is what the park ranger on the right said to me.




An old wagon track still weaves through the center of the dog park.  It follows neither a straight line nor does it weave randomly. Instead it traces an almost imperceptible contour line of high points across what most people would think of as flat grassland.  This old trail doesn’t wash away when it rains.  It’s been there at least a hundred years. One can spot the foundations of long abandoned home scattered through the park. After last year’s big rain, I found an old penny, long lost as it was minted in 1919.



The old wagon track takes a seemingly random line, yet it stays high and dry even with heavy rain.



Perhaps it’s just contrariness, or my predilection for rarely choosing to do things the easy way, but I find walking this old ‘roadway’ rather pleasant ,certainly more interesting and in a way more soothing than walking on the newly designed paths.  There is something about the meandering route of the old way that seems to make more sense.  There is a sense of purpose, of going somewhere

Here where I shift mental gears and start citing research studies.


In 2006 Elisabeth Spelke of Harvard University along with three colleagues (Stanislas Dehaene, Veronique Izard, and Pierre Pica) wrote a paper on the innate cognitive understandings of indigenous Indians living in the rainforests of the Amazon. ("Core Knowledge of Geometry in an Amazonian Indigene Group") and were able to show that “… all humans, regardless of language or schooling, possess a core set of geometrical intuitions.” These researchers have been traveling to remote areas along the Cururu River in Brazil to administer math tests  to members of the Munduruku tribe.


This wasn’t the first trip up the river for these researchers.  Dehaene, Izard and Pica had been there years before to study the mathematical cognitive ability of the Munduruku. In 2004 they reported that they had sought, “To clarify the relation between language and arithmetic, …. [and so] studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range”

[if the name Stanislas Dehaene rings a bell, it is because I  mentioned his research in a recent article on reading and writing.  He is the author of the book “Reading in the Brain: the new science of how we read.” He is also the author of “The Number Sense: how the mind creates mathematics.” For the curious, he is also the subject of a fascinating  feature article in the March 3, 2008 issue of The New Yorker .]

Back to the Amazon:

These scientists expanded on this understanding in 2011 when they showed that certain basic concepts in Euclidian mathematics are also innate. ("Flexible intuitions of Euclidean geometry in an Amazonian indigene group.")    



Writing in The New York Times in 2006, Nicholas Bakalar summed up much of this research,

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.

We apparently do not need to study Pythagoras to know that the hypotenuse is the shortest distance between two arms of a triangle.

Walk west for three days, make a right turn and walk north for four days, what’s the shortest way to get home?  These Amazon kids who have never seen a straight line or drawn a picture of a triangle, know that it’ll take about five days to trace the hypotenuse of the triangle by walking southeast back to their starting point.

If our brains have a built in concept of Euclidian mathematics, geometry and a spatial concept of the world that quickly understands maps and the geometric concepts represented by them, is it a big stretch of the imagination to wonder if the brain intuitively knows whether we are choosing an appropriate trail when we pick a route betweens points A and B?  It would certainly explain the irritation my dear wife displays when my virtual GPS girlfriend picks a route that is less efficient than what she has suggested that I take.  I had thought it was jealously.


The majority of paths we must follow in our modern world probably make little sense deep within our brains.  They are neither the shortest distance between points nor do they provide an obvious advantage in economy of effort. They are neither high and dry or short and straight.  Does some internal GPS unit squalk at our choices, recalculating over and over as we go about out daily routes?  Do we mute our unconscious mind over and over?  Or would part of us prefer to jump a few backyard fences and head straight to our desination.  Perhaps this is the urge that gets translated into people purchasing large 4 x 4 trucks.  The fellow driving that Hummer in front of you on the way to work may merely be responding to his inner mathematician, and his choice in vehicles is the result of a deep Euclidian calculus that knows the shortest distance between two points may be offroad.

Dehaene S1, Izard V, Pica P, Spelke E. Core knowledge of geometry in an Amazonian indigene group. Science. 2006 Jan 20;311(5759):381-4.


Does geometry constitute a core set of intuitions present in all humans, regardless of their language or schooling? We used two nonverbal tests to probe the conceptual primitives of geometry in the Mundurukú, an isolated Amazonian indigene group. Mundurukú children and adults spontaneously made use of basic geometric concepts such as points, lines, parallelism, or right angles to detect intruders in simple pictures, and they used distance, angle, and sense relationships in geometrical maps to locate hidden objects. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

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Science. 2004 Oct 15;306(5695):499-503.

Exact and approximate arithmetic in an Amazonian indigene group.

Pica P1Lemer CIzard VDehaene S.

Author information


Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail inexact arithmetic with numbers larger than 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exactnumber and arithmetic.

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PMID: 15486303

[PubMed - indexed for MEDLINE] 

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Proc Natl Acad Sci U S A. 2011 Jun 14;108(24):9782-7. doi: 10.1073/pnas.1016686108. Epub 2011 May 23.

Flexible intuitions of Euclidean geometry in an Amazonian indigene group.

Izard V1Pica PSpelke ESDehaene S.

Author information


Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometryalso includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ~180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclideangeometry, even in the absence of training in mathematics.

PMID: 21606377

PMC3116380 Free PMC Article