15 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. |
| 21687 | It seems obvious to prefer the simpler of two theories, on grounds of beauty and convenience [Quine] |
| Full Idea: It is not to be wondered that theory makers seek simplicity. When two theories are equally defensible on other counts, certainly the simpler of the two is to be preferred on the score of both beauty and convenience. | |
| From: Willard Quine (On Simple Theories of a Complex World [1960], p.255) | |
| A reaction: A simple application of Ockham's Razor. Quine goes on to nicely deconstruct what is involved in simplicity, and identify a certain amount of dubious prejudice in the concept. |
| 21688 | There are four suspicious reasons why we prefer simpler theories [Quine] |
| Full Idea: We prefer simpler theories through wishful thinking, or a bias which slants the data, or a bias where the simpler hypothesis is more open to confirmation, or simpler hypotheses tolerating wider deviations in score-keeping. | |
| From: Willard Quine (On Simple Theories of a Complex World [1960], p.258) | |
| A reaction: [a compression of his summary of the paper] Quine is not dismissing our preference for simpler theories, but just very nicely inviting us to focus of aspects about which we should be cautious. |
| 17064 | 1: Coherence is a symmetrical relation between two propositions [Thagard, by Smart] |
| Full Idea: 1: Coherence and incoherence are symmetrical between pairs of propositions. | |
| From: report of Paul Thagard (Explanatory Coherence [1989], 1) by J.J.C. Smart - Explanation - Opening Address p.04 |
| 17065 | 2: An explanation must wholly cohere internally, and with the new fact [Thagard, by Smart] |
| Full Idea: 2: If a set of propositions explains a further proposition, then each proposition in the set coheres with that proposition, and propositions in the set cohere pairwise with one another. | |
| From: report of Paul Thagard (Explanatory Coherence [1989], 2) by J.J.C. Smart - Explanation - Opening Address p.04 |
| 17066 | 3: If an analogous pair explain another analogous pair, then they all cohere [Thagard, by Smart] |
| Full Idea: 3: If two analogous propositions separately explain different ones of a further pair of analogous propositions, then the first pair cohere with one another, and so do the second (explananda) pair. | |
| From: report of Paul Thagard (Explanatory Coherence [1989], 3) by J.J.C. Smart - Explanation - Opening Address p.04 |
| 17067 | 4: For coherence, observation reports have a degree of intrinsic acceptability [Thagard, by Smart] |
| Full Idea: 4: Observation reports (for coherence) have a degree of acceptability on their own. | |
| From: report of Paul Thagard (Explanatory Coherence [1989], 4) by J.J.C. Smart - Explanation - Opening Address p.04 | |
| A reaction: Thagard makes this an axiom, but Smart rejects that and says there is no reason why observation reports should not also be accepted because of their coherence (with our views about our senses etc.). I agree with Smart. |
| 17068 | 5: Contradictory propositions incohere [Thagard, by Smart] |
| Full Idea: 5: Contradictory propositions incohere. | |
| From: report of Paul Thagard (Explanatory Coherence [1989], 5) by J.J.C. Smart - Explanation - Opening Address p.04 | |
| A reaction: This has to be a minimal axiom for coherence, but coherence is always taken to be more than mere logical consistency. Mutual relevance is the first step. At least there must be no category mistakes. |
| 17069 | 6: A proposition's acceptability depends on its coherence with a system [Thagard, by Smart] |
| Full Idea: 6: Acceptability of a proposition in a system depends on its coherence with the propositions in that system. | |
| From: report of Paul Thagard (Explanatory Coherence [1989], 6) by J.J.C. Smart - Explanation - Opening Address p.04 | |
| A reaction: Thagard tried to build an AI system for coherent explanations, but I would say he has no chance with these six axioms, because they never grasp the nettle of what 'coherence' means. You first need rules for how things relate. What things are comparable? |