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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7517
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I could take a healthy infant and train it up to be any type of specialist I choose [Watson,JB]
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Full Idea:
Give me a dozen healthy infants, and my own specified world to bring them up in, and I'll guarantee to take any one at random and train him to become any type of specialist I might select - doctor, artist, beggar, thief - regardless of his ancestry.
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From:
J.B. Watson (Behaviorism [1924], Ch.2), quoted by Steven Pinker - The Blank Slate
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A reaction:
This was a famous pronouncement rejecting the concept of human nature as in any way fixed - a total assertion of nurture over nature. Modern research seems to be suggesting that Watson is (alas?) wrong.
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