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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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16657
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Substance, Quantity and Quality are real; other categories depend on those three [Henry of Ghent]
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Full Idea:
Among creatures there are only three 'res' belong to the three first categories: Substance, Quantity and Quality. All other are aspects [rationes] and intellectual concepts with respect to them, with reality only as grounded on the res of those three.
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From:
Henry of Ghent (Quodlibeta [1284], VII:1-2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.3
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A reaction:
Pasnau connects with the 'arrangement of being', giving an 'ontologically innocent' structure to reality. That seems to be what we all want, if only we could work out the ontologically guilty bit.
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16645
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Accidents are diminished beings, because they are dispositions of substance (unqualified being) [Henry of Ghent]
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Full Idea:
Accidents are beings only in a qualified and diminished sense, because they are not called beings, nor are they beings, except because they are dispositions of an unqualified being, a substance.
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From:
Henry of Ghent (Quodlibeta [1284], XV.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 10.4
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A reaction:
This is aimed to 'half' detach the accidents (as the Eucharist requires). Later scholastics detached them completely. Late scholastics seem to have drifted back to Henry's view. The equivocal use of 'being' here was challenged later.
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22012
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Kant says things-in-themselves cause sensations, but then makes causation transcendental! [Henry of Ghent, by Pinkard]
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Full Idea:
Kant claimed that things-in-themselves caused our sensations; but causality was a transcendental condition of experience, not a property of things-in-themselves, so the great Kant had contradicted himself.
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From:
report of Henry of Ghent (Quodlibeta [1284], Supplement) by Terry Pinkard - German Philosophy 1760-1860 04
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A reaction:
This early objection by the conservative Jacobi (who disliked Enlightenment rational religion) is the key to the dispute over whether Kant is an idealist. Kant denied being an idealist, but how can he be, if this idea is correct?
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