17 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. |
| 7306 | If the only property of a name was its reference, we couldn't explain bearerless names [Miller,A] |
| Full Idea: If having a reference were the only semantic property in terms of which we could explain the functioning of names, we would be in trouble with respect to names that simply have no bearer. | |
| From: Alexander Miller (Philosophy of Language [1998], 2.1.1) | |
| A reaction: (Miller is discussing Frege) 'Odysseus' is given as an example. Instead of switching to a bundle of descriptions, we could say that we just imagine an object which is stamped with the name. Names always try to refer. |
| 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. |
| 7322 | Constitutive scepticism is about facts, and epistemological scepticism about our ability to know them [Miller,A] |
| Full Idea: We should distinguish 'constitutive scepticism' (about the existence of certain sorts of facts) from the traditional 'epistemological scepticism' (which concedes that the sort of fact in question exists, but questions our right to claim knowledge of it). | |
| From: Alexander Miller (Philosophy of Language [1998], 4.7) | |
| A reaction: I would be inclined to call the first type 'ontological scepticism'. Miller is discussing Quine's scepticism about meaning. Atheists fall into the first group, and agnostics into the second. An important, and nicely simple, distinction. |
| 7325 | Dispositions say what we will do, not what we ought to do, so can't explain normativity [Miller,A] |
| Full Idea: Dispositional facts are facts about what we will do, not about what we ought to do, and as such cannot capture the normativity of meaning. | |
| From: Alexander Miller (Philosophy of Language [1998], 6.2) | |
| A reaction: Miller is discussing language, but this raises a nice question for all behaviourist accounts of mental events. Perhaps there is a disposition to behave in a guilty way if you do something you think you shouldn't do. (Er, isn't 'guilt' a mental event?) |
| 7324 | Explain meaning by propositional attitudes, or vice versa, or together? [Miller,A] |
| Full Idea: Grice wants to explain linguistic meaning in terms of the content of propositional attitudes, Dummett has championed the view that propositional attitudes must be explained by linguistic meaning, while Davidson says they must be explained together. | |
| From: Alexander Miller (Philosophy of Language [1998], 6.1) | |
| A reaction: A useful map. My intuition says propositional attitudes come first, for evolutionary reasons. We are animals first, and speakers second. Thought precedes language. A highly social animal flourishes if it can communicate. |
| 7323 | If truth is deflationary, sentence truth-conditions just need good declarative syntax [Miller,A] |
| Full Idea: On a deflationary concept of truth, for a sentence to possess truth-conditions it is sufficient that it be disciplined by norms of correct usage, and that it possess the syntax distinctive of declarative sentences. | |
| From: Alexander Miller (Philosophy of Language [1998], 5.3) | |
| A reaction: Idea 6337 gives the basic deflationary claim. He mentions Boghossian as source of this point. So much the worse for the deflationary concept of truth, say I. What are the truth-conditions of "Truth rotates"? |
| 7315 | 'Jones is a married bachelor' does not have the logical form of a contradiction [Miller,A] |
| Full Idea: The syntactic notion of contradiction (p and not-p) is well understood, but is no help in explaining analyticity, since "Jones is a married bachelor" is not of that syntactic form. | |
| From: Alexander Miller (Philosophy of Language [1998], 4.2) | |
| A reaction: This point is based on Quine. This means we cannot define analytic sentences as those whose denial is a contradiction, even though that seems to be true of them. Both the Kantian and the modern logical versions of analyticity are in trouble. |
| 7328 | The principle of charity is holistic, saying we must hold most of someone's system of beliefs to be true [Miller,A] |
| Full Idea: Properly construed, the principle of charity is a holistic constraint applying, not to individual beliefs, but rather to systems of belief: we must interpret a speaker so that most of the beliefs in his system are, by our lights, true. | |
| From: Alexander Miller (Philosophy of Language [1998], 8.7) | |
| A reaction: This is a lot more plausible than applying the principle to individual sentences, particularly if you are in the company of habitual ironists or constitutional liars. |
| 7329 | Maybe we should interpret speakers as intelligible, rather than speaking truth [Miller,A] |
| Full Idea: A more sophisticated version of the principle of charity holds that we interpret speakers not as necessarily having beliefs that are true by our own lights, but as having beliefs that are intelligible by our own lights. | |
| From: Alexander Miller (Philosophy of Language [1998], 8.7) | |
| A reaction: Consider Idea 4161 in the light of this. Presumably this means that we treat them as having a coherent set of beliefs, even if they seem to us to fail to correspond to reality. I prefer the stronger version that there has to be some proper truth in there. |
| 7333 | The Frege-Geach problem is that I can discuss the wrongness of murder without disapproval [Miller,A] |
| Full Idea: The main problem faced by non-cognitivism is known as the Frege-Geach problem: if I say "If murder is wrong, then getting your brother to murder people is wrong", that is an unasserted context, and I don't necessarily express disapproval of murder. | |
| From: Alexander Miller (Philosophy of Language [1998], 9.2) | |
| A reaction: The emotivist or non-cognitivist might mount a defence by saying there is some second-order or deep-buried emotion involved. Could a robot without feelings even understand what humans meant when they said "It is morally wrong"? |
| 9287 | Bruno said that ancient Egyptian magic was the true religion [Bruno, by Yates] |
| Full Idea: Giordano Bruno maintained that the magical Egyptian religion of the world was not only the most ancient but also the only true religion, which both Judaism and Christianity had obscured and corrupted. | |
| From: report of Giordano Bruno (works [1590]) by Frances A. Yates - Giordano Bruno and Hermetic Tradition Ch.1 | |
| A reaction: His beliefs were based on the Hermetic writings. No wonder he was burned at the stake. Atheists now lay flowers at his memorial in Rome. The sixteenth century was when the hunt for alternatives to established religion began. |