Nate,
You wrote that the aristotelian essence of a triangle or Spinoza's circles
seem to be as much "cultural-historical as knowledge of the object per se."
I think perhaps that the use of triangles and circles does more to reveal a
universal basis to cultural-historical phenomena than it does to show the
relativity , the absence of objective truth, that you seem to want to
demonstrate.
In particular the triangle. The triangle is the simplest figure that
defines a plane but it's properties cannot be deduced from the properties of
lines. Ilyenkov uses the triangle to illustrate the concrete universal, the
essential, and contrasts it to Aristotles notion of an "empty universal" :
" . . . the triangle is the first, the truly universal figure, which appears
also in the square, etc., as the figure which can be led back to the
simplest determination." The notion of two-dimensional figure is an empty,
an abstract universal.
Is it coincidental that Hegel singled out the triangle as the essential
figure, the figure from which all other polygons can be constructed? That
this figure is historically the first to have been fully analyzed,
specifically in the form of the right triangle, from which all other
triangles can be constructed. The discovery of the logical primacy of the
triangle was preceded by the practical application of its properties. The
Egyptians understood that the square of the hypotenuse was equal to the sum
of the squares of the sides and applied this knowledge in their
administration of property (a social relation of production). But they
didn't understand why this was true, that would await Euclid. Certainly
before the regularity expressed in Pythagoras' theorem was understood as an
analytically given property of plane geometry it must have appeared to have
some mystical power, some property inherent in the triangle as a
physical/material object (Pythagorean numerical mysticism) rather than as
the logical extrapolation of a given set of axioms. -- and this is an ideal
object an object which is purely a cultural historical product. A
transform of Egyptian cultural - historical products at the hands of Greeks
a thousand or more years later. The subsequent history of mathematics has
subsumed this universal understanding of plane geometry and there is no
culture aware of its existence that doesn't recognize its objective
existence independent of the particular systems of beliefs governing the
other areas of their culturally specific life.
The triangle is a splendid example of the logico-historical structure of
cultural artefacts that yield objective knowlege.
Paul H. Dillon
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