17 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) |
| 17809 | Gödel showed that the syntactic approach to the infinite is of limited value [Kreisel] |
| Full Idea: Usually Gödel's incompleteness theorems are taken as showing a limitation on the syntactic approach to an understanding of the concept of infinity. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 05) |
| 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) |
| 17810 | The study of mathematical foundations needs new non-mathematical concepts [Kreisel] |
| Full Idea: It is necessary to use non-mathematical concepts, i.e. concepts lacking the precision which permit mathematical manipulation, for a significant approach to foundations. We currently have no concepts of this kind which we can take seriously. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 06) | |
| A reaction: Music to the ears of any philosopher of mathematics, because it means they are not yet out of a job. |
| 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. |
| 11976 | Aristotelian essentialism says essences are not relative to specification [Lewis] |
| Full Idea: So-called 'Aristotelian essentialism' is the doctrine of essences not relative to specifications. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], III) | |
| A reaction: In other words, they are so-called 'real essences', understood as de re. Quine says essences are all de dicto, and relative to some specification. I vote for Aristotle. |
| 11978 | Causal necessities hold in all worlds compatible with the laws of nature [Lewis] |
| Full Idea: Just as a sentence is necessary if it holds in all worlds, so it is causally necessary if it holds in all worlds compatible with the laws of nature. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], V) | |
| A reaction: I don't believe in the so-called 'laws of nature', so I'm not buying that. Is there no distinction in Lewis's view between those sentences which must hold, and those which happen to hold universally? |
| 11979 | It doesn't take the whole of a possible Humphrey to win the election [Lewis] |
| Full Idea: Even if Humphrey is a modal continuant, it doesn't take the whole of him to do such things as winning. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], Post B) | |
| A reaction: This responds to Kripke's famous example, that people only care about what happens to themselves, and not to some 'counterpart' of themselves. |
| 16994 | Counterpart theory is bizarre, as no one cares what happens to a mere counterpart [Kripke on Lewis] |
| Full Idea: Probably Humphrey could not care less whether someone else, no matter how much resembling him, would have been victorious in another possible world. Thus Lewis's view seems even more bizarre that the usual transworld identification it replaces. | |
| From: comment on David Lewis (Counterpart theory and Quant. Modal Logic [1968]) by Saul A. Kripke - Naming and Necessity notes and addenda note 13 | |
| A reaction: I begin to see this as a devastating reply to a theory I previously found quite congenial. |
| 11974 | Counterparts are not the original thing, but resemble it more than other things do [Lewis] |
| Full Idea: Your counterparts resemble you closely in content and context in important respects. They resemble you more closely than do the other things in their worlds. But they are not really you. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], I) | |
| A reaction: It is a dilemma. If my counterpart were exactly me, I couldn't contemplate possibly losing a leg, or my sanity. But if my counterpart isn't exactly me, then I don't have much interest in its fate. Only essences can save us here. Cf. me tomorrow. |
| 11975 | If the closest resembler to you is in fact quite unlike you, then you have no counterpart [Lewis] |
| Full Idea: If whatever thing in world w6 it is that resembles you more closely than anything else in w6 is nevertheless quite unlike you; nothing in w6 resembles you at all closely. If so, you have no counterpart in w6. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], I) | |
| A reaction: This is the nub, because the whole theory rests on deciding whether two things resemble sufficiently 'closely'. But then we need a criterion of closeness, so we must start talking about which properties matter. Essences loom. |
| 11977 | Essential attributes are those shared with all the counterparts [Lewis] |
| Full Idea: An essential attribute of something is an attribute it shares with all its counterparts. | |
| From: David Lewis (Counterpart theory and Quant. Modal Logic [1968], III) | |
| A reaction: I don't like this. It ties essence entirely to identity, but I think essence precedes identity. Essence is a nexus of causal and explanatory powers which bestows an identity on each thing. But essence might be unstable, and identity with it. |
| 17811 | The natural conception of points ducks the problem of naming or constructing each point [Kreisel] |
| Full Idea: In analysis, the most natural conception of a point ignores the matter of naming the point, i.e. how the real number is represented or by what constructions the point is reached from given points. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 13) | |
| A reaction: This problem has bothered me. There are formal ways of constructing real numbers, but they don't seem to result in a name for each one. |