11 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. |
| 16025 | If things change they become different - but then no one thing undergoes the change! [Gallois] |
| Full Idea: If things really change, there can't literally be one thing before and after the change. However, if there isn't one thing before and after the change, then no thing has really undergone any change. | |
| From: André Gallois (Identity over Time [2011], Intro) | |
| A reaction: [He cites Copi for this way of expressing the problem of identity through change] There is an obvious simple ambiguity about 'change' in ordinary English. A change of property isn't a change of object. Painting a red ball blue isn't swapping it. |
| 16026 | 4D: time is space-like; a thing is its history; past and future are real; or things extend in time [Gallois] |
| Full Idea: We have four versions of Four-Dimensionalism: the relativistic view that time is space-like; a persisting thing is identical with its history (so objects are events); past and future are equally real; or (Lewis) things extend in time, with temporal parts. | |
| From: André Gallois (Identity over Time [2011], §2.5) | |
| A reaction: Broad proposed the second one. I prefer 3-D: at any given time a thing is wholly present. At another time it is wholly present despite having changed. It is ridiculous to think that small changes destroy identity. We acquire identity by dying?? |
| 16027 | If two things are equal, each side involves a necessity, so the equality is necessary [Gallois] |
| Full Idea: The necessity of identity: a=b; □(a=a); so something necessarily = a; so something necessarily must equal b; so □(a=b). [A summary of the argument of Marcus and Kripke] | |
| From: André Gallois (Identity over Time [2011], §3) | |
| A reaction: [Lowe 1982 offered a response] The conclusion seems reasonable. If two things are mistakenly thought to be different, but turn out to be one thing, that one thing could not possibly be two things. In no world is one thing two things! |
| 20422 | The experience of expression and communication are intermingled in art [Croce] |
| Full Idea: It is very difficult to perceive the frontier between expression and communication in actual fact, for the two processes usually alternate rapidly and are almost intermingled. | |
| From: Benedetto Croce (The Essence of Aesthetic [1912]), quoted by Gary Kemp - Croce and Collingwood | |
| A reaction: [text unsure] I think he is getting at seeing the painting (or whatever) as a physical object, and seeing it as the experience which results from the object. The alternation of the objective and subjective views. Reminds me of Thomas Nagel. |