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. |
| 5163 | Basic propositions refer to a single experience, are incorrigible, and conclusively verifiable [Ayer] |
| Full Idea: There is a class of empirical propositions, which I call 'basic propositions', which can be verified conclusively, since they refer solely to the contents of a single experience, which are incorrigible. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.13) | |
| A reaction: A classic statement of empirical foundationalism. I sort of agree that 'single experiences' are a 'given' for philosophy, but is questionable whether there is anything which could both be a single experience AND give rise to a proposition. |
| 5167 | The argument from analogy fails, so the best account of other minds is behaviouristic [Ayer] |
| Full Idea: There are too many objections to the argument from analogy, so I am inclined to revert to a 'behaviouristic' interpretation of propositions about other people's experiences. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.26) | |
| A reaction: It seems odd to vote for behaviourism on one issue, if you aren't a general subscriber. It is one thing to say that behaviour is the best evidence for your explanation, quite another to equate the other mind with its behaviour. |
| 5164 | A statement is meaningful if observation statements can be deduced from it [Ayer] |
| Full Idea: In the improved version, a statement was verifiable, and consequently meaningful, if 'some observation-statement can be deduced from it in conjunction with certain other premises, without being deducible from those other premises alone'. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.15) | |
| A reaction: I.Berlin showed that any statement S could pass this test, because if you assert 'S' and 'If S then O', these two statements entail O, which could be some random observation. Hence a 1946 revised version had to be produced. |
| 5165 | Directly verifiable statements must entail at least one new observation statement [Ayer] |
| Full Idea: A statement is directly verifiable if it is either itself an observation-statement,or is such that in conjunction with one or more observation-statements it entails at least one observation-statement which is not deducible from these other premises alone. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.17) | |
| A reaction: This is the 1946 revised version of the Verification Principle, which was then torpedoed by an elaborate counterexample from Alonzo Church. Ayer thereafter abandoned attempts to find a precise statement of it. |
| 5166 | The principle of verification is not an empirical hypothesis, but a definition [Ayer] |
| Full Idea: I wish the principle of verification to be regarded, not as an empirical hypothesis, but as a definition. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.21) | |
| A reaction: This is Ayer's attempt to meet the well known objection of 'turning the tables' on his theory (by asking whether it is tautological or empirically verifiable). However, if it is just a definition, then presumably it is completely arbitrary… |
| 5162 | Sentences only express propositions if they are meaningful; otherwise they are 'statements' [Ayer] |
| Full Idea: I suggest that every grammatically significant indicative sentence expresses a 'statement', but the word 'proposition' will be reserved for what is expressed by sentences that are literally meaningful. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.10) | |
| A reaction: We don't have to accept Ayer's over-fussy requirements for what is meaningful to accept that this is a good distinction. Every day we hear statements from people (e.g. politicians) in which we can fish in vain for the underlying proposition. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 5168 | Moral approval and disapproval concerns classes of actions, rather than particular actions [Ayer] |
| Full Idea: The common objects of moral approval and disapproval are not particular actions so much as classes of actions. | |
| From: A.J. Ayer (Introduction to 'Language Truth and Logic' [1946], p.27) | |
| A reaction: This 1946 revision of his pure emotivism looks like a move towards Hare's prescriptivism, where classes, rules and principles are seen as the window-dressing of emotivism. It's still a bad theory. |