8 ideas
| 13655 | The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate [Putnam, by Shapiro] |
| Full Idea: Putnam claims that the Löwenheim-Skolem theorems indicate that there is no 'fact of the matter' whether all sets are constructible. | |
| From: report of Hilary Putnam (Models and Reality [1977]) by Stewart Shapiro - Foundations without Foundationalism | |
| A reaction: [He refers to the 4th and 5th pages of Putnam's article] Shapiro offers (p.109) a critique of Putnam's proposal. |
| 9915 | V = L just says all sets are constructible [Putnam] |
| Full Idea: V = L just says all sets are constructible. L is the class of all constructible sets, and V is the universe of all sets. | |
| From: Hilary Putnam (Models and Reality [1977], p.425) |
| 9913 | The Löwenheim-Skolem Theorem is close to an antinomy in philosophy of language [Putnam] |
| Full Idea: The Löwenheim-Skolem Theorem says that a satisfiable first-order theory (in a countable language) has a countable model. ..I argue that this is not a logical antinomy, but close to one in philosophy of language. | |
| From: Hilary Putnam (Models and Reality [1977], p.421) | |
| A reaction: See the rest of this paper for where he takes us on this. |
| 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? |
| 9914 | It is unfashionable, but most mathematical intuitions come from nature [Putnam] |
| Full Idea: Experience with nature is undoubtedly the source of our most basic 'mathematical intuitions', even if it is unfashionable to say so. | |
| From: Hilary Putnam (Models and Reality [1977], p.424) | |
| A reaction: Correct. I find it quite bewildering how Frege has managed to so discredit all empirical and psychological approaches to mathematics that it has become a heresy to say such things. |
| 3643 | The concept of mind excludes body, and vice versa [Descartes] |
| Full Idea: The concept of body includes nothing at all which belongs to the mind, and the concept of mind includes nothing at all which belongs to the body. | |
| From: René Descartes (Reply to Fourth Objections [1641], 225) | |
| A reaction: A headache? Hunger? The mistake, I think, is to regard the mind as entirely conscious, thus creating a sharp boundary between two aspects of our lives. As shown by blindsight, I take many of my central mental operations to be pre- or non-conscious. |