11 ideas
| 21699 | Russell offered a paraphrase of definite description, to avoid the commitment to objects [Quine] |
| Full Idea: Russell's theory involved defining a term not by presenting a direct equivalent of it, but by 'paraphrasis', providing equivalents of the sentences. In this way, reference to fictitious objects can be simulated without our being committed to the objects. | |
| From: Willard Quine (Russell's Ontological Development [1966], p.75) | |
| A reaction: I hadn't quite grasped that the modern strategy of paraphrase tracks back to Russell - though it now looks obvious, thanks to Quine. Paraphrase is a beautiful way of sidestepping ontological problems. See Frege on the moons of Jupiter. |
| 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. |
| 21700 | Taking sentences as the unit of meaning makes useful paraphrasing possible [Quine] |
| Full Idea: The new freedom that Russell confers by paraphrasis (of definite descriptions) is our reward for recognising that the unit of communication is the sentence and not the word. | |
| From: Willard Quine (Russell's Ontological Development [1966], p.75) | |
| A reaction: Since many people hardly ever speak a properly formed sentence, I take propositions to be better candidates for this. However, I don't see how we can reject the compositional view (the meanings are assembled). |
| 21701 | Knowing a word is knowing the meanings of sentences which contain it [Quine] |
| Full Idea: We can say that knowing words is knowing how to work out the meanings of sentences containing them. Dictionary definitions are mere clauses in a recursive definition of the meanings of sentences. | |
| From: Willard Quine (Russell's Ontological Development [1966], p.76) | |
| A reaction: Do you have to recursively define all the sentences that might contain the word, before you can fully know the meaning of the word? He seems to credit Russell with the holistic view of sentences (though I think that starts with Frege). |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |