Combining Texts

All the ideas for 'Mathematics without Numbers', 'De Essentia' and 'Modes of Extension: comment on Fine'

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6 ideas

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism

8698	Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
	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.
	From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Michèle Friend - Introducing the Philosophy of Mathematics 4.3
	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'?

9557	Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
	Full Idea: Hellman represents statements of pure mathematics as elliptical for modal conditionals of a certain sort.
	From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Charles Chihara - A Structural Account of Mathematics 5.3
	A reaction: It's a pity there is such difficulty in understanding conditionals (see Graham Priest on the subject). I intuit a grain of truth in this, though I take maths to reflect the structure of the actual world (with possibilities being part of that world).

10263	Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
	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.
	From: comment on Geoffrey Hellman (Mathematics without Numbers [1989]) by Stewart Shapiro - Philosophy of Mathematics 7.3
	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?

9. Objects / A. Existence of Objects / 5. Individuation / d. Individuation by haecceity

13132	A snowball's haecceity is the property of being identical with itself [Plantinga, by Westerhoff]
	Full Idea: Plantinga assumes that being identical with that snowball names a property which is that snowball's haecceity.
	From: report of Alvin Plantinga (De Essentia [1979]) by Jan Westerhoff - Ontological Categories §52
	A reaction: Only a philosopher would suggest such a bizarre way of establishing the unique individuality of a given snowball. You could hardly keep track of the snowball with just that criterion. How do you decide whether something has Plantinga's property?

18. Thought / E. Abstraction / 7. Abstracta by Equivalence

18883	Any equivalence relation among similar things allows the creation of an abstractum [Simons]
	Full Idea: Whenever we have an equivalence relation among things - such as similarity in a certain respect - we can abstract under the equivalence and consider the abstractum.
	From: Peter Simons (Modes of Extension: comment on Fine [2008], p.19)
	A reaction: This strikes me as dressing up old-fashioned psychological abstractionism in the respectable clothing of Fregean equivalences (such as 'directions'). We can actually do what Simons wants without the precision of partitioned equivalence classes.

18884	Abstraction is usually seen as producing universals and numbers, but it can do more [Simons]
	Full Idea: Abstraction as a cognitive tool has been associated predominantly with the metaphysics of universals and of mathematical objects such as numbers. But it is more widely applicable beyond this standard range. I commend its judicious use.
	From: Peter Simons (Modes of Extension: comment on Fine [2008], p.21)
	A reaction: Personally I think our view of the world is founded on three psychological principles: abstraction, idealisation and generalisation. You can try to give them rigour, as 'equivalence classes', or 'universal quantifications', if it makes you feel better.