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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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7127
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If men are good you should keep promises, but they aren't, so you needn't [Machiavelli]
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Full Idea:
If all men were good, promising-breaking would not be good, but because they are bad and do not keep their promises to you, you likewise do not have to keep yours to them.
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From:
Niccolo Machiavelli (The Prince [1513], Ch.18)
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A reaction:
A rather depressing proposal to get your promise-breaking in first, based on the pessimistic view that people cannot be improved. The subsequent history of ethics in Europe showed Machiavelli to be wrong. Gentlemen began to keep their word.
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6308
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A sensible conqueror does all his harmful deeds immediately, because people soon forget [Machiavelli]
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Full Idea:
A prudent conqueror makes a list of all the harmful deeds he must do, and does them all at once, so that he need not repeat them every day, which then makes men feel secure, and gains their support by treating them well.
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From:
Niccolo Machiavelli (The Prince [1513], Ch.8)
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A reaction:
This might work for a new government in a democracy, or a new boss in a business. It sounds horribly true; dreadful deeds done a long time ago can be completely forgotten, as when reformed criminals become celebrities.
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6307
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A desire to conquer, and men who do it, are always praised, or not blamed [Machiavelli]
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Full Idea:
It is very natural and normal to wish to conquer, and when men do it who can, they always will be praised, or not blamed.
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From:
Niccolo Machiavelli (The Prince [1513], Ch.3)
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A reaction:
This view seems shocking to us, but it seems to me that this was a widely held view up until the time of Nietzsche, but came to a swift end with the invention of the machine gun in about 1885, followed by the heavy bomber and atomic bomb.
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