14 ideas
| 18889 | Ostensive definitions needn't involve pointing, but must refer to something specific [Salmon,N] |
| Full Idea: So-called ostensive definitions need not literally involve ostension, e.g. pointing, but they must involve genuine reference of some sort (in this case reference to a sample of water). | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 4.11.2) |
| 14627 | S4, and therefore S5, are invalid for metaphysical modality [Salmon,N, by Williamson] |
| Full Idea: Salmon argues that S4 and therefore S5 are invalid for metaphysical modality. | |
| From: report of Nathan Salmon (Reference and Essence (1st edn) [1981], 238-40) by Timothy Williamson - Modal Logic within Counterfactual Logic 4 | |
| A reaction: [He gives references for Salmon, and for his own reply] Salmon's view seems to be opposed my most modern logicians (such as Ian Rumfitt). |
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 18888 | Essentialism says some properties must be possessed, if a thing is to exist [Salmon,N] |
| Full Idea: The metaphysical doctrine of essentialism says that certain properties of things are properties that those things could not fail to have, except by not existing. | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 3.8.2) | |
| A reaction: A bad account of essentialism, and a long way from Aristotle. It arises from the logicians' tendency to fix objects entirely in terms of a 'flat' list of predicates (called 'properties'!), which ignore structure, constitution, history etc. |
| 18886 | Frege's 'sense' solves four tricky puzzles [Salmon,N] |
| Full Idea: Reference via sense solves Frege's four puzzles, of the informativeness of identity statements, the failure of substitutivity in attitude contexts, of negative existentials, and the truth-value of statements using nondenoting singular terms. | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 1.1.1) | |
| A reaction: These must then be compared with Kripke's three puzzles about referring via sense, and the whole debate is then spread before us. |
| 18887 | The perfect case of direct reference is a variable which has been assigned a value [Salmon,N] |
| Full Idea: The paradigm of a nondescriptional, directly referential, singular term is an individual variable. …The denotation of a variable… is semantically determined directly by the assignment of values. | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 1.1.2) | |
| A reaction: This cuts both ways. Maybe we are muddling ordinary reference with the simplicities of logical assignments, or maybe we make logical assignments because that is the natural way our linguistic thinking works. |
| 22086 | The most important aspect of a human being is not reason, but passion [Kierkegaard, by Carlisle] |
| Full Idea: Kierkegaard insisted that the most important aspect of a human being is not reason, but passion. | |
| From: report of Søren Kierkegaard (works [1845]) by Clare Carlisle - Kierkegaard: a guide for the perplexed Intro | |
| A reaction: Hume comes to mind for a similar view, but in character Hume was far more rational than Kierkegaard. |
| 18891 | Nothing in the direct theory of reference blocks anti-essentialism; water structure might have been different [Salmon,N] |
| Full Idea: There seems to be nothing in the theory of direct reference to block the anti-essentialist assertion that the substance water might have been the very same entity and yet have had a different chemical structure. | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 6.23.1) | |
| A reaction: Indeed, water could be continuously changing its inner structure, while retaining the surface appearance that gets baptised as 'water'. We make the reasonable empirical assumption, though, that structure-change implies surface-change. |