14596
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Call 'nominalism' the denial of numbers, properties, relations and sets [Dorr]
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Full Idea:
Just as there are no numbers or properties, there are no relations (like 'being heavier than' or 'betweenness'), or sets. I will provisionally use 'nominalism' for the conjunction of these four claims.
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From:
Cian Dorr (There Are No Abstract Objects [2008], 1)
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A reaction:
If you are going to be a nominalist, do it properly! My starting point in metaphysics is strong sympathy with this view. Right now [Tues 22nd Nov 2011, 10:57 am GMT] I think it is correct.
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14598
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Abstracta imply non-logical brute necessities, so only nominalists can deny such things [Dorr]
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Full Idea:
If there are abstract objects, there are necessary truths about these things that cannot be reduced to truths of logic. So only the nominalist, who denies that there are any such things, can adequately respect the idea that there are no brute necessities.
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From:
Cian Dorr (There Are No Abstract Objects [2008], 4)
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A reaction:
This is where two plates of my personal philosophy grind horribly against one another. I love nominalism, and I love natural necessities. They meet like a ring-species in evolution. I'll just call it a 'paradox', and move on (swiftly).
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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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11214
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We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt]
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Full Idea:
The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A.
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From:
Ian Rumfitt ("Yes" and "No" [2000], IV)
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A reaction:
[compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role).
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