19347
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Substance needs independence, unity, and stability (for individuation); also it is a subject, for predicates [Perkins]
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Full Idea:
For individuation, substance needs three properties: independence, to separate it from other things; unity, to call it one thing, rather than an aggregate; and permanence or stability over time. Its other role is as subject for predicates.
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From:
Franklin Perkins (Leibniz: Guide for the Perplexed [2007], 3.1)
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A reaction:
Perkins is describing the Aristotelian view, which is taken up by Leibniz. 'Substance' is not a controversial idea, if we see that it only means that the world is full of 'things'. It is an unusual philosopher wholly totally denies that.
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13165
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Geometrical proofs do not show causes, as when we prove a triangle contains two right angles [Proclus]
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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.
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From:
Proclus (Commentary on Euclid's 'Elements' [c.452], p.161-2), quoted by Paolo Mancosu - Explanation in Mathematics §5
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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.
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9569
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The origin of geometry started in sensation, then moved to calculation, and then to reason [Proclus]
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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.
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From:
Proclus (Commentary on Euclid's 'Elements' [c.452]), quoted by Charles Chihara - A Structural Account of Mathematics 9.12 n55
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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.
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