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. |
| 16756 | Substantial forms must exist, to explain the stability of metals like silver and tin [Albertus Magnus] |
| Full Idea: There is no reason why the matter in any natural thing should be stable in its nature, if it is not completed by a substantial form. But we see that silver is stable, and tin and other metals. Therefore they will seem to be perfected by substantial forms. | |
| From: Albertus Magnus (On Minerals [1260], III.1.7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.2 | |
| A reaction: Illuminating. This may be the best reason for proposing substantial forms. Once materialism arrives, the so-called 'laws' of nature have to be imposed on the material to do the job - but what the hell is a law supposed to be? |
| 5662 | Maybe induction could never prove the existence of something unobservable [Ayer] |
| Full Idea: Some people hold that no inductive argument can give us any reason to believe in the existence of something which could not even in principle be observed. | |
| From: A.J. Ayer (The Concept of a Person [1963], §I) | |
| A reaction: I see nothing illogical in inferring the existence of a poltergeist from the recurrent flight of objects around my lounge. Only an excessive empiricism (which used to afflict Ayer) could lead to this claim. |
| 5664 | Consciousness must involve a subject, and only bodies identify subjects [Ayer] |
| Full Idea: It may not make sense to talk of states of consciousness except as the experiences of some conscious subject; and it may well be that this conscious subject can not be identified except by reference to his body. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: It strikes me that Ayer deserves more credit as a pioneer of this view. It tracks back to what may turn out to be the key difficulty for Descartes - how do you individuate a mental substance? I may identify me, but how do I identify you? |
| 5668 | People own conscious states because they are causally related to the identifying body [Ayer] |
| Full Idea: I think personal identity depends on the identity of the body, and that a person's ownership of states of consciousness consists in their standing in a special causal relation to the body by which he is identified. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: I think with this is right, with the slight reservation that Ayer talks as if there were two things which have a causal relationship, implying that the link is contingent. Better to think of the whole thing as a single causal network. |
| 5661 | We identify experiences by their owners, so we can't define owners by their experiences [Ayer] |
| Full Idea: Normally we identify experiences in terms of the persons whose experiences they are; but this will lead to a vicious circle if persons themselves are to be analysed in terms of their experiences. | |
| From: A.J. Ayer (The Concept of a Person [1963], §I) | |
| A reaction: This (from a leading empiricist) is a nice basic challenge to all empiricist accounts of personal identity. One might respond my saying that the circle is not vicious. There are two interlinked concepts (experience and persons), like day and night. |
| 5666 | Not all exerience can be remembered, as this would produce an infinite regress [Ayer] |
| Full Idea: Not every experience can be remembered; otherwise each piece of remembering, which is itself an experience, would have to be remembered, and each remembering of a remembering and so ad infinitum. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: See Idea 5667. Ayer takes for granted two sorts of consciousness - current awareness, and memory. Ayer brings out a nice difficulty for Locke's proposal, but also draws attention to what may be a very basic misunderstanding about the mind. |
| 5665 | Memory is the best proposal as what unites bundles of experiences [Ayer] |
| Full Idea: The most promising suggestion is that the bundles are tied together by means of memory. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: This is interesting for showing how Locke was essentially trying to meet (in advance) Hume's 'bundle' scepticism. Hume proposed associations as the unifying factor, instead of memories. Ayer proposes concepts as a candidate. |
| 5669 | Personal identity can't just be relations of experiences, because the body is needed to identify them [Ayer] |
| Full Idea: A Humean theory, in which a person's identity is made to depend upon relations between experiences ..is not tenable unless the experiences themselves can be identified, and that is only possible through their association with the body. | |
| From: A.J. Ayer (The Concept of a Person [1963], §IV) | |
| A reaction: This seems to me a very fruitful response to difficulties with the 'bundle' view of a person - a better response than the a priori claims of Butler and Reid, or the transcendental argument of Kant. Only a philosopher could ignore the body. |