The free Abstract of the article, the full text of which requires subscription or payment, provides inter alia as follows:
This study is of importance for my alleged decipherment of the Phaistos Disc as posing a "pre-Euclidean" postulate of parallel lines, i.e. a proposed "mathematical" content which has been summarily dismissed out of hand without serious consideration by mainstream scholars as something that simply could not have been in an Ancient Greek era more than a thousand years prior to Euclid."... 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 Euclidean geometry, even in the absence of training in mathematics." [emphasis added]
Based on this study of an indigenous Amazonian group, the authors of the PNAS article conclude that certain basic geometric understanding, even concerning parallel lines, exists in cultures who do not even have fixed geometric concepts in their language.
Accordingly, the early date for the pre-Euclidean parallel postulate which my decipherment of the Phaistos Disc mandates is no longer as improbable as it might have appeared prior to this Amazonian tribe study.
We should never underestimate the ancients' innate geometric understanding.
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