14 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. |
| 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. |
| 19513 | A contextualist coherentist will say that how strongly a justification must cohere depends on context [DeRose] |
| Full Idea: If you are a coherentist and a contextualist, you'll probably want to hold that how strongly beliefs must cohere with one another in order to count as knowledge (if they are true), or to count as justified, is a contextually variable matter. | |
| From: Keith DeRose (The Case for Contextualism [2009], 1.09) | |
| A reaction: How exciting! He's talking about ME! Context might not only dictate the strength of the coherence, but also the range of beliefs involved. In fact all of Thagard's criteria of coherence may be subject to contextual variation. |
| 19514 | Classical invariantism combines fixed truth-conditions with variable assertability standards [DeRose] |
| Full Idea: The great rival to contextualism is classical 'invariantism' - invariantism about the truth-conditions [for knowing], combined with variable standards for warranted assertability. | |
| From: Keith DeRose (The Case for Contextualism [2009], 1.12) | |
| A reaction: That is, I take it, that we might want to assert that someone 'knows' something, when the truth is that they don't. That is, either you know or you don't, but we can bend the rules as to whether we say you know. I take this view to be false. |
| 19515 | We can make contextualism more precise, by specifying the discrimination needed each time [DeRose] |
| Full Idea: We might make the basic contextualist schema more precise ...by saying the change in content will consist in a change in the range of relevant alternatives. Higher standards would discriminate from a broader range of alternatives. | |
| From: Keith DeRose (The Case for Contextualism [2009], 1.14) | |
| A reaction: This would handle the 'fake barn' and 'disguised zebra' examples, by saying lower standards do not expect such discriminations. The zebra case has a lower standard than the barn case (because fake barns are the norm here). |
| 19510 | In some contexts there is little more to knowledge than true belief. [DeRose] |
| Full Idea: I'm inclined to accept that in certain contexts the standards for knowledge are so low that little more than true belief is required. | |
| From: Keith DeRose (The Case for Contextualism [2009], 1.6) | |
| A reaction: DeRose emphasises that 'a little more' is needed, rather than none. The example given is where 'he knew that p' means little more than 'the information that p was available to him' (in a political scandal). |
| 19516 | Contextualists worry about scepticism, but they should focus on the use of 'know' in ordinary speech [DeRose] |
| Full Idea: While skepticism has drawn much of the attention of contextualists, support for contextualism should also - and perhaps primarily - be looked for in how 'knows' is utilised in non-philosophical conversation. | |
| From: Keith DeRose (The Case for Contextualism [2009], 1016) | |
| A reaction: Contextualists say scepticism is just raising the standards absurdly high. I take it that the ordinary use of the word 'know' is obviously highly contextual, and so varied that I don't see how philosophers could 'regiment' it into invariant form. |
| 19511 | If contextualism is about knowledge attribution, rather than knowledge, then it is philosophy of language [DeRose] |
| Full Idea: Maybe contextualism isn't a theory about knowledge at all, but about knowledge attributions. As such, it is not a piece of epistemology at all, but of philosophy of language. | |
| From: Keith DeRose (The Case for Contextualism [2009], 1.7) | |
| A reaction: DeRose takes this view to be wrong. At the very least this will have to include self-attributions, by the supposed knower, because I might say 'I know that p', meaning 'but only in this rather low-standard context'. |
| 23222 | A soul with physical extension is more likely than an immaterial soul that moves bodies [Elizabeth] |
| Full Idea: I would find it easier to concede matter and extension to the soul than to concede that an immaterial thing could move and be moved by a body. | |
| From: Elizabeth, Princess of Bohemia (Letters to Descartes [1643], p.42), quoted by Matthew Cobb - The Idea of the Brain 2 | |
| A reaction: Very nicely expressed! I'm trying to imagine a ghost which finds itself stuck with a physical body which it has to drag around like a reluctant dog. She is stating the classic interaction problem which plagues all mind-body dualism. |