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8921
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Structuralism is now common, studying relations, with no regard for what the objects might be [Hellman]
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Full Idea:
With developments in modern mathematics, structuralist ideas have become commonplace. We study 'abstract structures', having relations without regard to the objects. As Hilbert famously said, items of furniture would do.
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From:
Geoffrey Hellman (Structuralism [2007], §1)
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A reaction:
Hilbert is known as a Formalist, which suggests that modern Structuralism is a refined and more naturalist version of the rather austere formalist view. Presumably the sofa can't stand for six, so a structural definition of numbers is needed.
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8698
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Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
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Full Idea:
The modal structuralist thinks of mathematical structures as possibilities. The application of mathematics is just the realisation that a possible structure is actualised. As structures are possibilities, realist ontological problems are avoided.
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From:
report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Michèle Friend - Introducing the Philosophy of Mathematics 4.3
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A reaction:
Friend criticises this and rejects it, but it is appealing. Mathematics should aim to be applicable to any possible world, and not just the actual one. However, does the actual world 'actualise a mathematical structure'?
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10263
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Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
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Full Idea:
The usual way to show that a sentence is possible is to show that it has a model, but for Hellman presumably a sentence is possible if it might have a model (or if, possibly, it has a model). It is not clear what this move brings us.
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From:
comment on Geoffrey Hellman (Mathematics without Numbers [1989]) by Stewart Shapiro - Philosophy of Mathematics 7.3
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A reaction:
I can't assess this, but presumably the possibility of the model must be demonstrated in some way. Aren't all models merely possible, because they are based on axioms, which seem to be no more than possibilities?
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20769
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Sphaerus he was not assenting to the presence of pomegranates, but that it was 'reasonable' [Sphaerus, by Diog. Laertius]
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Full Idea:
When Sphaerus accepted pomegranates from the king, he was accused of assenting to a false presentation, to which Sphaerus replied that what he had assented to was not that they were pomegranates, but that it was reasonable that they were pomegranates.
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From:
report of Sphaerus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.177
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A reaction:
He then cited the stoic distinction between a 'graspable' presentation and a 'reasonable' one. This seems a rather helpful response to Dretske's zebra problem. I like the word 'sensible' in epistemology, because animals can be sensible.
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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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