Euclid did not divorce geometry from the empirical world. ;)
^^^^ CB: I believe he did. He was Greek ( Were any of the Greek's materialists ? Aristotle ? Democratus "discovered" atoms by thinking ,not by experiment). There are no empirical points, infinitely small in Euclid's geometry. They are infinitely small, how could we see them ? His geometry is "pure", i.e. not polluted by the empirical. The shortest distance between two points is a straight lne by definition, not measure. Euclid's geometry, like Descartes analytical geometry/Cartesian coordinates, can be drawn , but their truth does not depend on any empirical measure. A compass and protractor are heuristic, not precise empirical devices in a geometry class. Euclidean proofs are not done by measure, but by logic. Euclid's geometry is a deductive ,not inductive/empirical system. He may have used round or triangular etc. objects in nature as heuristics suggesting his ideas, but the ideas themselves are deduced logically from axioms, mathematically.
http://en.wikipedia.org/wiki/Euclid His Elements is the most successful textbook in the history of mathematics. In it, the principles of Euclidean geometry are deduced from a small set of axioms. Euclid's method of proving mathematical theorems by logical deduction from accepted principles remains the backbone of all mathematics, imbuing that field with its characteristic rigor."
^^^^
I think what distinguishes Descartes is the algebraization of geometry, not removing it from sensuous experience. That was what the ancient Platonists were all about.
^^^^^
CB: I think the lack of empiricism is clearer in analytical geometry. Cartesian coordinates and graphing are "pictures" of ideas, not of objective reality. Similarly with Euclid's forms.
Plato, famously, Euclid and Descartes are philosophical idealists all. I think Plato is an objective idealist ( like Hegel as opposed to subjective idealists like Hume), meaning , I think , that even what we view with our senses are ideas. Isn't empirical reality an imperfect reflection of the ideal forms for Plato, like shadows on a cave wall ? ( I always wonder if those in Plato's cave are cavemen ( and women) )
--- Charles Brown <charlesb at cncl.ci.detroit.mi.us> wrote:
> Well there was this guy called Euclid...
>
Mataiotes mataioteton, eipen ho Ekklasiastes, mataiotes mataioteton, ta panta mataiotes.