12 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. |
| 21710 | We know a universal in 'yellow differs from blue' or 'yellow resembles blue less than green does' [Russell] |
| Full Idea: We are aware of the universal 'yellow'; this universal is the subject in such judgements as 'yellow differs from blue' or 'yellow resembles blue less than green does'. | |
| From: Bertrand Russell (Knowledge by Acquaintance and Description-1 [1911], 154), quoted by Bernard Linsky - Russell's Metaphysical Logic 2.3 | |
| A reaction: This still seems one of the strongest examples in support of universals. You could hardly be talking about yellow tropes in such instances (even if the world does contain yellow tropes). |
| 8854 | My 'acquaintance' with sense-data is nothing like my knowing New York [Williams,M on Russell] |
| Full Idea: My 'acquaintance' with sense-data is nothing like my knowing New York. | |
| From: comment on Bertrand Russell (Knowledge by Acquaintance and Description-1 [1911]) by Michael Williams - Without Immediate Justification §4 | |
| A reaction: This pinpoints a nice difficulty for Russell. Williams may misrepresent Russell's account of acquaintance, but that is probably because Russell is unclear, or uncertain. The problem is when Russell claims that his acquaintance gives knowledge. |
| 1470 | Belief in an afterlife may be unverifiable in this life, but it will be verifiable after death [Hick, by PG] |
| Full Idea: Religion is capable of 'eschatological verification', by reaching evidence at the end of life, even though falsification of its claims is never found in this life; a prediction of coming to a Celestial City must await the end of the journey. | |
| From: report of John Hick (Theology and Verification [1960], III) by PG - Db (ideas) |
| 1471 | It may be hard to verify that we have become immortal, but we could still then verify religious claims [Hick, by PG] |
| Full Idea: Verification of religious claims after death is only possible if the concept of surviving death is intelligible, and we can understand the concept of immortality, despite difficulties in being certain that we had reached it. | |
| From: report of John Hick (Theology and Verification [1960], IV) by PG - Db (ideas) |
| 1469 | Some things (e.g. a section of the expansion of PI) can be verified but not falsified [Hick, by PG] |
| Full Idea: Falsification and verification are not logically equivalent. For example, you might verify the claim that there will be three consecutive sevens in the infinite expansion of PI, but you could never falsify such a claim. | |
| From: report of John Hick (Theology and Verification [1960], §II) by PG - Db (ideas) |