Ideas from 'Commentary on Euclid's 'Elements'' by Proclus [452], by Theme Structure

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14. Science / D. Explanation / 2. Types of Explanation / g. Causal explanations
Geometrical proofs do not show causes, as when we prove a triangle contains two right angles
                        Full Idea: Geometry does not ask 'why?' ..When from the exterior angle equalling two opposite interior angles it is shown that the interior angles make two right angles, this is not a causal demonstration. With no exterior angle they still equal two right angles.
                        From: Proclus (Commentary on Euclid's 'Elements' [c.452], p.161-2), quoted by Paolo Mancosu - Explanation in Mathematics 5
                        A reaction: A very nice example. It is hard to imagine how one might demonstrate the cause of the angles making two right angles. If you walk, turn left x, then turn left y, then turn left z, and x+y+z=180, you end up going in the original direction.
18. Thought / E. Abstraction / 1. Abstract Thought
The origin of geometry started in sensation, then moved to calculation, and then to reason
                        Full Idea: It is unsurprising that geometry was discovered in the necessity of Nile land measurement, since everything in the world of generation goes from imperfection to perfection. They would naturally pass from sense-perception to calculation, and so to reason.
                        From: Proclus (Commentary on Euclid's 'Elements' [c.452]), quoted by Charles Chihara - A Structural Account of Mathematics 9.12 n55
                        A reaction: The last sentence is the core of my view on abstraction, that it proceeds by moving through levels of abstraction, approaching more and more general truths.