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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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20327
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Modern attention has moved from the intrinsic properties of art to its relational properties [Lamarque/Olson]
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Full Idea:
In modern discussions, rather than look for intrinsic properties of objects, including aesthetic or formal properties, attention has turned to extrinsic or relational properties, notably of a social, historical, or 'institutional' nature.
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From:
Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 1)
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A reaction:
Lots of modern branches of philosophy have made this move, which seems to me like a defeat. We want to know why things have the relations they do. Just mapping the relations is superficial Humeanism.
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20326
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Early 20th cent attempts at defining art focused on significant form, intuition, expression, unity [Lamarque/Olson]
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Full Idea:
In the early twentieth century there were numerous attempts at defining the essence art. Significant form, intuition, the expression of emotion, organic unity, and other notions, were offered to this end.
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From:
Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 1)
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A reaction:
As far as I can see the whole of aesthetics was demolished in one blow by Marcel Duchamp's urinal. Artists announce: we will tell you what art is; you should just sit and listen. Compare the invention of an anarchic sport.
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20330
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The dualistic view says works of art are either abstract objects (types), or physical objects [Lamarque/Olson]
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Full Idea:
The dualistic view of the arts holds that works of art come in two fundamentally different kinds: those that are abstract entities, i.e. types, and those that are physical objects (tokens).
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From:
Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 2)
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A reaction:
Paintings are the main reason for retaining physical objects. Strawson 1974 argues that paintings are only physical because we cannot yet perfectly reproduce them. I agree. Works of art are types, not tokens.
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