29 ideas
| 9572 | Realists about sets say there exists a null set in the real world, with no members [Chihara] |
| Full Idea: In the Gödelian realistic view of set theory the statement that there is a null set as the assertion of the existence in the real world of a set that has no members. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It seems to me obvious that such a claim is nonsense on stilts. 'In the beginning there was the null set'? |
| 9550 | We only know relational facts about the empty set, but nothing intrinsic [Chihara] |
| Full Idea: Everything we know about the empty set is relational; we know that nothing is the membership relation to it. But what do we know about its 'intrinsic properties'? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: Set theory seems to depend on the concept of the empty set. Modern theorists seem over-influenced by the Quine-Putnam view, that if science needs it, we must commit ourselves to its existence. |
| 9562 | In simple type theory there is a hierarchy of null sets [Chihara] |
| Full Idea: In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.4) | |
| A reaction: It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong. |
| 9573 | The null set is a structural position which has no other position in membership relation [Chihara] |
| Full Idea: In the structuralist view of sets, in structures of a certain sort the null set is taken to be a position (or point) that will be such that no other position (or point) will be in the membership relation to it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It would be hard to conceive of something having a place in a structure if nothing had a relation to it, so is the null set related to singeton sets but not there members. It will be hard to avoid Platonism here. Set theory needs the null set. |
| 9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
| Full Idea: What is it about the intrinsic properties of just that one unit set in virtue of which Bill Clinton is related to just it and not to any other unit sets in the set-theoretical universe? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If we all kept pet woodlice, we had better not hold a wood louse rally, or we might go home with the wrong one. My singleton seems seems remarkably like yours. Could we, perhaps, swap, just for a change? |
| 9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
| Full Idea: The set theorist cannot tell us anything about the true relationship of membership. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it. |
| 9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
| Full Idea: ZFU set theory talks about physical objects (the urelements), and hence is in some way about the physical world. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.5) | |
| A reaction: This sounds a bit surprising, given that the whole theory would appear to be quite unaffected if God announced that idealism is true and there are no physical objects. |
| 9563 | A pack of wolves doesn't cease when one member dies [Chihara] |
| Full Idea: A pack of wolves is not thought to go out of existence just because some member of the pack is killed. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.5) | |
| A reaction: The point is that the formal extensional notion of a set doesn't correspond to our common sense notion of a group or class. Even a highly scientific theory about wolves needs a loose notion of a wolf pack. |
| 9561 | The mathematics of relations is entirely covered by ordered pairs [Chihara] |
| Full Idea: Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.2) | |
| A reaction: How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations. |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| Full Idea: In first-order logic a set of sentences is 'consistent' iff there is an interpretation (or structure) in which the set of sentences is true. ..For Frege, though, a set of sentences is consistent if it is not possible to deduce a contradiction from it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.1) | |
| A reaction: The first approach seems positive, the second negative. Frege seems to have a higher standard, which is appealing, but the first one seems intuitively right. There is a possible world where this could work. |
| 9553 | Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara] |
| Full Idea: With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.3) | |
| A reaction: The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this? |
| 10192 | We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride] |
| Full Idea: Chihara's 'constructability theory' is nominalist - mathematics is reducible to a simple theory of types. Instead of talk of sets {x:x is F}, we talk of open sentences Fx defining them. Existence claims become constructability of sentence tokens. | |
| From: report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.81 | |
| A reaction: This seems to be approaching the problem in a Fregean way, by giving an account of the semantics. Chihara is trying to evade the Quinean idea that assertion is ontological commitment. But has Chihara retreated too far? How does he assert existence? |
| 9559 | If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara] |
| Full Idea: If mathematics shares whatever confirmation accrues to the theories using it, would it not be reasonable to suppose that mathematics shares whatever disconfirmation accrues to the theories using it? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 05.8) | |
| A reaction: Presumably Quine would bite the bullet here, although maths is much closer to the centre of his web of belief, and so far less likely to require adjustment. In practice, though, mathematics is not challenged whenever an experiment fails. |
| 9566 | No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara] |
| Full Idea: Evidently, no scientific explanations of specific phenomena would collapse as a result of any hypothetical discovery that no mathematical objects exist. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.1) | |
| A reaction: It is inconceivable that anyone would challenge this claim. A good model seems to be drama; a play needs commitment from actors and audience, even when we know it is fiction. The point is that mathematics doesn't collapse either. |
| 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. |
| 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... |
| 9568 | I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara] |
| Full Idea: What I refer to as an 'equivalence class' (of line segments of a particular length) is an open sentence in my Constructibility Theory. I just use this terminology of the Platonist for didactic purposes. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.10) | |
| A reaction: This is because 'equivalence classes' is committed to the existence of classes, which is Quinean Platonism. I am with Chihara in wanting a story that avoids such things. Kit Fine is investigating similar notions of rules of construction. |
| 9547 | Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara] |
| Full Idea: Causal theories of reference seem doomed to failure for the case of reference to mathematical entities, since such entities are evidently causally inert. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.3) | |
| A reaction: Presumably you could baptise a fictional entity such as 'Polonius', and initiate a social causal chain, with a tradition of reference. You could baptise a baby in absentia. |
| 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? |
| 5049 | Intelligent pleasure is the perception of beauty, order and perfection [Leibniz] |
| Full Idea: An intelligent being's pleasure is simply the perception of beauty, order and perfection. | |
| From: Gottfried Leibniz (A Résumé of Metaphysics [1697], §18) | |
| A reaction: Leibniz seems to have inherited this from the Greeks, especially Pythagoras and Plato. Buried in Leibniz's remark I see the Christian fear of physical pleasure. He should have got out more. Must an intelligent being always be intelligent? |
| 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'. |
| 9574 | 'Gunk' is an individual possessing no parts that are atoms [Chihara] |
| Full Idea: An 'atomless gunk' is defined to be an individual possessing no parts that are atoms. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], App A) | |
| A reaction: [Lewis coined it] If you ask what are a-toms made of and what are ideas made of, the only answer we can offer is that the a-toms are made of gunk, and the ideas aren't made of anything, which is still bad news for the existence of ideas. |
| 5048 | Perfection is simply quantity of reality [Leibniz] |
| Full Idea: Perfection is simply quantity of reality. | |
| From: Gottfried Leibniz (A Résumé of Metaphysics [1697], §11) | |
| A reaction: An interesting claim, but totally beyond my personal comprehension. I presume he inherited 'quantity of reality' from Plato, e.g. as you move up the Line from shadows to Forms you increase the degree of reality. I see 'real' as all-or-nothing. |
| 5050 | Evil serves a greater good, and pain is necessary for higher pleasure [Leibniz] |
| Full Idea: Evils themselves serve a greater good, and the fact that pains are found in minds is necessary if they are to reach greater pleasures. | |
| From: Gottfried Leibniz (A Résumé of Metaphysics [1697], §23) | |
| A reaction: How much pain is needed to qualify for the 'greater pleasures'? Some people receive an awful lot. I am not sure exactly how an evil can 'serve' a greater good. Is he recommending evil? |