Combining Texts

All the ideas for 'Mathematics without Numbers', 'Epistemic Justification' and 'The Poverty of Philosophy'

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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?

13. Knowledge Criteria / A. Justification Problems / 3. Internal or External / a. Pro-internalism

19679	'Access' internalism says responsibility needs access; weaker 'mentalism' needs mental justification [Kvanvig]
	Full Idea: Strong 'access' internalism says the justification must be accessible to the person holding the belief (for cognitive duty, or blame), and weaker 'mentalist' internalism just says the justification must supervene on mental features of the individual.
	From: Jonathan Kvanvig (Epistemic Justification [2011], III)
	A reaction: [compressed] I think I'm a strong access internalist. I doubt whether there is a correct answer to any of this, but my conception of someone knowing something involves being able to invoke their reasons for it. Even if they forget the source.

13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / a. Foundationalism

19678	Strong foundationalism needs strict inferences; weak version has induction, explanation, probability [Kvanvig]
	Full Idea: Strong foundationalists require truth-preserving inferential links between the foundations and what the foundations support, while weaker versions allow weaker connections, such as inductive support, or best explanation, or probabilistic support.
	From: Jonathan Kvanvig (Epistemic Justification [2011], II)
	A reaction: [He cites Alston 1989] Personally I'm a coherentist about justification, but I'm a fan of best explanation, so I'd vote for that. It's just that best explanation is not a very foundationalist sort of concept. Actually, the strong version is absurd.

24. Political Theory / D. Ideologies / 11. Capitalism

21994	The handmill gives feudalism, the steam mill capitalism [Marx]
	Full Idea: The handmill gives you society with the feudal lord; the steam mill society with the industrial capitalist.
	From: Karl Marx (The Poverty of Philosophy [1847], p.202), quoted by Peter Singer - Marx 7
	A reaction: If technology dictates social structure, then feudalism is still with us, in low-tech industries. What if the steam mill had been invented in 1300?