13 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. |
| 14288 | 'If A,B' affirms that A⊃B, and also that this wouldn't change if A were certain [Jackson, by Edgington] |
| Full Idea: According to Jackson, in asserting 'If A,B' the speaker expresses his belief that A⊃B, and also indicates that this belief is 'robust' with respect to the antecedent A - the speaker would not abandon A⊃B if he were to learn that A. | |
| From: report of Frank Jackson (On Assertion and Indicative Conditionals [1979]) by Dorothy Edgington - Conditionals (Stanf) 4.2 | |
| A reaction: The point is that you must not believe A⊃B solely on the dubious grounds of ¬A. This is 'to ensure an assertable conditional is fit for modus ponens' - that is, that you really will affirm B when you learn that A is true. Nice idea. |
| 13769 | Conditionals are truth-functional, but should only be asserted when they are confident [Jackson, by Edgington] |
| Full Idea: Jackson holds that conditionals are truth-functional, but are governed by rules of assertability, rather like 'but' compared to 'and'. The belief must be 'robust' - the speaker would not abandon his belief that A⊃B if he were to learn that A. | |
| From: report of Frank Jackson (On Assertion and Indicative Conditionals [1979]) by Dorothy Edgington - Conditionals 17.3.2 | |
| A reaction: This seems to spell out more precisely the pragmatic approach to conditionals pioneered by Grice, in Idea 13767. The idea is make conditionals 'fit for modus ponens'. They mustn't just be based on a belief that ¬A. |
| 5504 | Maybe we should see persons in four dimensions, with stages or time-slices at an instant [Martin/Barresi] |
| Full Idea: Some recent philosophers have argued that we should replace the three-dimensional view of persons with a four-dimensional view according to which only time-slices, or 'stages', of persons exist at short intervals of time. | |
| From: R Martin / J Barresi (Introduction to 'Personal Identity' [2003], p.3) | |
| A reaction: At first glance this seems to neatly eliminate lots of traditional worries. But why would I want to retain my identity, if someone threatened to brainwash me. I also want to disown my inadequate earlier selves. Interesting, though. Lewis. |
| 5503 | Maybe personal identity is not vital in survival, and other continuations would suffice [Martin/Barresi] |
| Full Idea: A modern question is whether personal identity is primarily what matters in survival; that is, people might cease and be continued by others whose continuation the original people would value as much. | |
| From: R Martin / J Barresi (Introduction to 'Personal Identity' [2003], p.3) | |
| A reaction: When put like this, the proposal seems hard to grasp. It only makes sense if you don't really believe in a thing called 'personal identity'. I don't see how you can believe in it without also believing that for you it has central importance. |
| 5502 | Locke's intrinsic view of personal identity has been replaced by an externalist view [Martin/Barresi] |
| Full Idea: In modern times the Lockean intrinsic relations view of personal identity has been superseded by an extrinsic relations view (also called the 'closest-continuer' or 'externalist' view). | |
| From: R Martin / J Barresi (Introduction to 'Personal Identity' [2003], p.1) | |
| A reaction: Sounds sweeping. My suspicion is that there is a raging fashion for externalist views of everything (justification, content etc.), but this will pass. I take Parfit to be the source of the modern views. |
| 5505 | For Aristotle the psyche perishes with the body (except possibly 'nous') [Martin/Barresi] |
| Full Idea: In Aristotle's view, with the possible exception of 'nous' the psyche and all its parts come into being at the same time as its associated body; it is inseparable from the body, and perishes along with it. | |
| From: R Martin / J Barresi (Introduction to 'Personal Identity' [2003], p.8) | |
| A reaction: It is suggested that he thought there was only one 'nous', which all humans share (p.9). If he wants to claim that one part is immortal, he doesn't have much evidence. If psyche is the form of the body, it is bound to perish. |