11 ideas
| 14231 | We should always apply someone's theory of meaning to their own utterances [Liggins] |
| Full Idea: We should interpret philosophers as if their own theory of the meaning of their utterances were true, whether or not we agree with that theory. | |
| From: David Liggins (Nihilism without Self-Contradiction [2008], 8) | |
| A reaction: This seems to give legitimate grounds for some sorts of ad hominem objections. It would simply be an insult to a philosopher not to believe their theories, and then apply them to what they have said. This includes semantic theories. |
| 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. |
| 14232 | We normally formalise 'There are Fs' with singular quantification and predication, but this may be wrong [Liggins] |
| Full Idea: It is quite standard to interpret sentences of the form 'There are Fs' using a singular quantifier and a singular predicate, but this tradition may be mistaken. | |
| From: David Liggins (Nihilism without Self-Contradiction [2008], 8) | |
| A reaction: Liggins is clearly in support of the use of plural quantification, referring to 'there are some xs such that'. |
| 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. |
| 16458 | Semantic vagueness involves alternative and equal precisifications of the language [Lewis] |
| Full Idea: If vagueness is semantic indeterminacy, then wherever we have vague statements, we have several alternative precisifications of the vague language involved, all with equal claims of being 'intended'. | |
| From: David Lewis (Vague Identity: Evans misunderstood [1988], p.318) |
| 14233 | Nihilists needn't deny parts - they can just say that some of the xs are among the ys [Liggins] |
| Full Idea: We can interpret '..is a part of..' as '..are among..': the xs are a part of the ys just when the xs are among the ys (though if the ys are 'one' then they would not have parts). | |
| From: David Liggins (Nihilism without Self-Contradiction [2008], 9) | |
| A reaction: The trouble is that this still leaves us with gerrymandered 'parts', in the form of xs that are scattered randomly among the ys. That's not what we mean by 'part'. No account of identity works if it leaves out coherent structure. |