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) |
| 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. |
| 18890 | Putnam smuggles essentialism about liquids into his proof that water must be H2O [Salmon,N on Putnam] |
| Full Idea: In the full exposition of Putnam's mechanism for generating the necessary truth that water is H2O, we find that the mechanism employs a certain nontrivial general principle of essentialism concerning liquid substances as a crucial premise. | |
| From: comment on Hilary Putnam (The Meaning of 'Meaning' [1975]) by Nathan Salmon - Reference and Essence (1st edn) 6.23.1 | |
| A reaction: This charge, that Kripke and Putnam smuggle the essentialism into their semantics, rather than deriving it, is the nub of Salmon's criticism of them. It seems to me that a new world view emerged while those two where revising the semantics. |
| 7705 | The Twin Earth theory suggests that intentionality is independent of qualia [Jacquette on Putnam] |
| Full Idea: Putnam's Twin Earth thought experiment suggests that two thinkers can have identical qualia, despite intending different objects on Earth and Twin Earth, and hence that qualia and intentionality must be logically independent of one another. | |
| From: comment on Hilary Putnam (The Meaning of 'Meaning' [1975]) by Dale Jacquette - Ontology Ch.10 | |
| A reaction: [See Idea 4099, Idea 3208, Idea 7612 for Twin Earth]. Presumably my thought of 'the smallest prime number above 10000' would be a bit thin on qualia too. Does that make them 'logically' independent? Depends what we reduce qualia or intentionality to. |
| 6029 | Whoever knows future causes knows everything that will be [Cicero] |
| Full Idea: Whoever grasps the causes of future things must necessarily grasp all that will be. | |
| From: M. Tullius Cicero (On Divination ('De divinatione') [c.46 BCE], 1.127) | |
| A reaction: Laplace stated this idea in terms of Newtonian physics (Idea 3441), but the key idea is stated more simply and clearly here. God can know the future in this way, without actually seeing it happen now. I can't think why it should not be true. |
| 4099 | If Twins talking about 'water' and 'XYZ' have different thoughts but identical heads, then thoughts aren't in the head [Putnam, by Crane] |
| Full Idea: Putnam claims that the Twins have different thoughts even though their heads are the same, so their thoughts (about 'water' or 'XYZ') cannot be in their heads. | |
| From: report of Hilary Putnam (The Meaning of 'Meaning' [1975]) by Tim Crane - Elements of Mind 4.37 | |
| A reaction: Is Putnam guilty of a simple confusion of de re and de dicto reference? |
| 12026 | We say ice and steam are different forms of water, but not that they are different forms of H2O [Forbes,G on Putnam] |
| Full Idea: Putnam presumes it is correct to say that ice and steam are forms of water, rather than that ice, water and steam are three forms of H2O. If we allow the latter, then 'water is H2O' is not an identity, but elliptical for 'water is H2O in liquid state'. | |
| From: comment on Hilary Putnam (The Meaning of 'Meaning' [1975]) by Graeme Forbes - The Metaphysics of Modality 8.2 | |
| A reaction: This nice observation seems to reveal that the word 'water' is ambiguous. I presume the ambiguity preceded the discovery of its chemical construction. Shakespeare would have hesitated over whether to say 'water is ice'. Context would matter. |
| 3208 | Does 'water' mean a particular substance that was 'dubbed'? [Putnam, by Rey] |
| Full Idea: Putnam argued that "water" refers to H2O by virtue of causal chains extending from present use back to early dubbing uses of it that were in fact dubbings of the substance H2O (although, of course, the original users of the word didn't know this). | |
| From: report of Hilary Putnam (The Meaning of 'Meaning' [1975]) by Georges Rey - Contemporary Philosophy of Mind 9.2.1 | |
| A reaction: This is the basic idea of the Causal Theory of Reference. Nice conclusion: most of us don't know what we are talking about. Maybe the experts on H2O are also wrong... |
| 3893 | Often reference determines sense, and not (as Frege thought) vice versa [Putnam, by Scruton] |
| Full Idea: Putnam argues that, Frege notwithstanding, it is often the case that reference determines sense, and not vice versa. | |
| From: report of Hilary Putnam (The Meaning of 'Meaning' [1975]) by Roger Scruton - Modern Philosophy:introduction and survey 19.6 | |
| A reaction: Does this say anything more than that once you have established a reference, you can begin to collect information about the referent? |
| 11191 | The hidden structure of a natural kind determines membership in all possible worlds [Putnam] |
| Full Idea: If there is a hidden structure, then generally it determines what it is to be a member of the natural kind, ...in all possible worlds. Put another way, it determines what we can and cannot counterfactually suppose about the natural kind. | |
| From: Hilary Putnam (The Meaning of 'Meaning' [1975], p.241) | |
| A reaction: This is the arrival of the bold new view of natural kinds (which is actually the original view - see Idea 8153). One must be careful of the necessity here. There is causal context, vagueness etc. |
| 11192 | If causes are the essence of diseases, then disease is an example of a relational essence [Putnam, by Williams,NE] |
| Full Idea: Putnam takes causes to be the essence of disease kinds, and they are distinct from the diseases they cause, both in identity and in proper parthood. These are relational properties, so Putnam gives examples of natural kinds with relational essences. | |
| From: report of Hilary Putnam (The Meaning of 'Meaning' [1975]) by Neil E. Williams - Putnam's Traditional Neo-Essentialism §4 | |
| A reaction: This seems to be a nice point, since scientific essentialism invariable takes itself to be pursuing instrinsic properties when it unravels the essences of natural kinds. Probably the best response is the Putnam has got muddled. |
| 11190 | Archimedes meant by 'gold' the hidden structure or essence of the stuff [Putnam] |
| Full Idea: When Archimedes asserted that something was gold, he was not just saying that it had the superficial characteristics of gold; he was saying that it had the same general hidden structure (the same 'essence', so to speak) as any normal piece of local gold. | |
| From: Hilary Putnam (The Meaning of 'Meaning' [1975], p.235) | |
| A reaction: This is one of the key announcements of the new scientific essentialism, and seems to me to be totally correct. Obviously Archimedes could say 'this is really gold, even if it no way appears to be gold'. |