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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9316
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How is self-representation possible, does it produce a regress, and is experience like that? [Kriegel/Williford]
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Full Idea:
The difficulties with a self-representational view of consciousness are how self-representation of mental states could be possible, whether it leads to an infinite regress, and whether it can capture the actual phenomenology of experience.
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From:
U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], §3)
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A reaction:
[compressed] All of these objections strike me as persuasive, especially the first one. I'm not sure I know what self-representation is. Mirrors externally represent, and they can't represent themselves. Two mirrors together achieve something..
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9314
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Unfortunately, higher-order representations could involve error [Kriegel/Williford]
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Full Idea:
A problem for explaining consciousness by higher-order representations is that, like their first-order counterparts, they can misrepresent; there could be a subjective impression of being in a conscious state without actually being in any conscious state.
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From:
U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], §1)
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A reaction:
It sounds plausible that this is a logical possibility, but how do you assess whether it is an actual or natural possibility? Are we saying that higher-order representations are judgments, which could be true or false? Hm.
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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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