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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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19891
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Rule of law is superior to autonomy, because citizens can see what is expected [Hooker,R]
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Full Idea:
Men saw that to live by one man's will became the cause of all men's misery. This contrained them to come unto laws wherein all men might see their duty beforehand, and know the penalties of transgressing them.
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From:
Richard Hooker (Of the Laws of Ecclesiastical Polity [1593], I s.10), quoted by John Locke - Second Treatise of Government 111 n1
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A reaction:
One British school has a single rule, that pupils 'shall always treat other people with respect'. Presumably the rulers, as well as the pupils, must decide when this is transgressed. The rule of law may be preferable.
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17005
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Natural things observe certain laws, and things cannot do otherwise if they retain their forms [Hooker,R]
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Full Idea:
Things natural …do so necessarily observe their certain laws, that as long as they keep those forms which give them their being they cannot possibly be apt or inclinable to do otherwise than they do.
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From:
Richard Hooker (Of the Laws of Ecclesiastical Polity [1593], 1.3.4), quoted by Marc Lange - Laws and Lawmakers 1.2
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A reaction:
Cited by some as the beginnings of the idea of 'laws of nature', but it is striking that Hooker says the laws are controlled by 'forms' (which are Aristotelian essences). This is an essentialist view of laws, not a regularity or divine power one.
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