19 ideas
| 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. |
| 16405 | To understand a name (unlike a description) picking the thing out is sufficient? [Stalnaker] |
| Full Idea: If we ask 'what must you know to understand a name?', the naïve answer is that one must know who or what it names - nothing more. (But no one would give this answer about what is needed to understand a definite description). | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 4) | |
| A reaction: Presumably this is naive because names can be full of meaning ('the Empress'), or description and reference together ('there's the man who robbed me') and so on. It's a nice starting point though. A number can serve as a name. |
| 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. |
| 8732 | It is spooky the way mathematics anticipates physics [Weinberg] |
| Full Idea: It is positively spooky how the physicist finds the mathematician has been there before him or her. | |
| From: Steven Weinberg (Lecture on Applicability of Mathematics [1986], p.725), quoted by Stewart Shapiro - Thinking About Mathematics 2.3 | |
| A reaction: This suggests that mathematics might be the study of possibilities or hypotheticals, like mental rehearsals for physics. See Hellman's modal structuralism. Maybe mathematicians are reading the mind of God, but I doubt that. |
| 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. |
| 16407 | Possible worlds allow separating all the properties, without hitting a bare particular [Stalnaker] |
| Full Idea: The possible worlds framework suggests a way to express the idea that a particular is conceptually separable from its properties without relying on the rejected picture of a bare particular. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 5) | |
| A reaction: As I read him, Stalnaker's proposal just comes down to replacing each property in turn with a different one. 'Strip away' red by making it green. It being green in w1 doesn't throw extra light. Can it be a bare particular in w37? |
| 16397 | If it might be true, it might be true in particular ways, and possible worlds describe such ways [Stalnaker] |
| Full Idea: A clarifying assumption is that if something might be true, then it might be true in some particular way. …Possible worlds begin from this, and the assumption that what might be true can be described as how a possibility might be realised. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 2) | |
| A reaction: This is a leading practitioner giving his best shot at explaining the rationale of the possible worlds approach, addressed to many sceptics. Most sceptics, I think, don't understand the qualifications the practitioners apply to their game. |
| 16399 | Possible worlds are ontologically neutral, but a commitment to possibilities remains [Stalnaker] |
| Full Idea: I argue for the metaphysical neutrality of the possible worlds framework, but I do not suggest that its use is free of ontological commitment to possibilities (ways things might be, counterfactual situations, possible states of worlds). | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 2) | |
| A reaction: Glad to hear this, as I have always been puzzled at possible aspirations to eliminate modality (such as possibility) by introducing 'possible' worlds. Commitment to possibilities I take to be basic and unavoidable. |
| 16398 | Possible worlds allow discussion of modality without controversial modal auxiliaries [Stalnaker] |
| Full Idea: The main benefit of the possible worlds move is to permit one to paraphrase modal claims in an extensional language that has quantifiers, but no modal auxiliaries, so the semantic stucture of modal discourse can be discussed without the controversies. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 2) | |
| A reaction: The strategy introduces the controversy of possible worlds instead, but since they just boil down to collections of objects with properties, classical logic can reign. Possible worlds are one strategy alongside many others. |
| 16396 | Kripke's possible worlds are methodological, not metaphysical [Stalnaker] |
| Full Idea: The possible worlds framework that Kripke introduces should be understood not as a metaphysical theory, but as a methodological framework. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], Intro) | |
| A reaction: That's certainly how I see possible worlds. I lose no sleep over whether they exist. I just take a set of possible worlds to be like cells in a spreadsheet, or records in a database. |
| 16408 | Rigid designation seems to presuppose that differing worlds contain the same individuals [Stalnaker] |
| Full Idea: A rigid designator is a designator that denotes the same individual in all possible worlds; doesn't this presuppose that the same individuals can be found in differing possible worlds? | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 5) | |
| A reaction: This is part of Stalnaker's claim that Kripke already has a metaphysics in place when he starts on his semantics and his theory of reference. Kripke needs a global domain, not a variable domain. Possibilities suggest variable domains to me. |
| 16406 | If you don't know what you say you can't mean it; what people say usually fits what they mean [Stalnaker] |
| Full Idea: If you don't know what you are saying then you don't mean what you say, and also speakers generally mean what they say (in that what they say coincides with what they mean). | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 4) | |
| A reaction: Both these thoughts seem completely acceptable and correct, but rely on something called 'meaning' that is distinct from saying. I would express this in terms of propositions, which I take to be mental events. |
| 16404 | In the use of a name, many individuals are causally involved, but they aren't all the referent [Stalnaker] |
| Full Idea: The causal theory of reference is criticised for vagueness. Causal connections are ubiquitous, and there are obviously many individuals that are causally implicated in the speaker's use of a name, but they aren't all plausible candidates for the referent. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 4) | |
| A reaction: This seems to be a very good objection. Among all the causal links back to some baptised object, we have to pick out the referential link, which needs a criterion. |
| 16403 | 'Descriptive' semantics gives a system for a language; 'foundational' semantics give underlying facts [Stalnaker] |
| Full Idea: 'Descriptive' semantics gives a semantics for the language without saying how practice explains why the semantics is right; …'foundational' semantics concerns the facts that give expressions their semantic values. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], §1) | |
| A reaction: [compressed] Sounds parallel to the syntax/semantics distinction, or proof-theoretical and semantic validity. Or the sense/reference distinction! Or object language/metalanguage. Shall I go on? |
| 16401 | To understand an utterance, you must understand what the world would be like if it is true [Stalnaker] |
| Full Idea: To understand what is said in an utterance of 'The first dog born at sea was a basset hound', one needs to know what the world would have been like in order for what was said in that utterance to be true. | |
| From: Robert C. Stalnaker (Reference and Necessity [1997], 3) | |
| A reaction: Put like that, the idea is undeniable. Understanding involves truth conditions. Does mean involve the understanding of the meaning. What do you understand when you understand a sentence? Just facts about dogs? Or something in the sentence? |