33 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. |
| 18151 | Could we replace sets by the open sentences that define them? [Chihara, by Bostock] |
| Full Idea: Chihara proposes to replace all sets by reference to the open sentences that define them. | |
| From: report of Charles Chihara (Ontology and the Vicious Circle Principle [1973]) by David Bostock - Philosophy of Mathematics 9.B.4 | |
| A reaction: This depends on predicativism, because that stipulates the definitions will be available (cos if it ain't definable it ain't there). Chihara went on to define the open sentences in terms of the possibility of uttering them. Cf. propositional functions. |
| 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. |
| 8758 | We could talk of open sentences, instead of sets [Chihara, by Shapiro] |
| Full Idea: Chihara's programme is to replace talk of sets with talk of open sentences. Instead of speaking of the set of all cats, we talk about the open sentence 'x is a cat'. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
| A reaction: As Shapiro points out, this is following up Russell's view that sets should be replaced with talk of properties. Chihara is expressing it more linguistically. I'm in favour of any attempt to get rid of sets. |
| 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? |
| 10265 | Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro] |
| Full Idea: Chihara's system is a version of type theory. Translate thus: replace variables of sets of type n with level n variables over open sentences, replace membership/predication with satisfaction, and high quantifiers with constructability quantifiers. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4 |
| 8759 | We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro] |
| Full Idea: Chihara's system is similar to simple type theory; he replaces each type with variables over open sentences, replaces membership (or predication) with satisfaction, and replaces quantifiers over level 1+ variables with constructability quantifiers. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
| A reaction: This is interesting for showing that type theory may not be dead. The revival of supposedly dead theories is the bread-and-butter of modern philosophy. |
| 10264 | Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro] |
| Full Idea: Chihara has proposal a modal primitive, a 'constructability quantifier'. Syntactically it behaves like an ordinary quantifier: Φ is a formula, and x a variable. Then (Cx)Φ is a formula, read as 'it is possible to construct an x such that Φ'. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4 | |
| A reaction: We only think natural numbers are infinite because we see no barrier to continuing to count, i.e. to construct new numbers. We accept reals when we know how to construct them. Etc. Sounds promising to me (though not to Shapiro). |
| 12270 | Being is one [Melissus, by Aristotle] |
| Full Idea: Being is one. | |
| From: report of Melissus (fragments/reports [c.443 BCE]) by Aristotle - Topics 104b23 | |
| A reaction: I can only really understand this in terms of physics, as the belief that ultimately there is one simple theory which explains everything. That project doesn't look terribly promising, despite the lovely simplifications of modern physics. |
| 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. |
| 13437 | A CAR and its major PART can become identical, yet seem to have different properties [Gallois] |
| Full Idea: At t1 there is a whole CAR, and a PART of it, which is everything except the right front wheel. At t2 the wheel is removed, leaving just PART, so that CAR is now PART. But PART was a proper part of CAR, and CAR had the front wheel. Different properties! | |
| From: André Gallois (Occasions of Identity [1998], 1.II) | |
| A reaction: [compressed summary] The problem is generated by appealing to Leibniz's Law. My immediate reaction is that this is the sort of trouble you get into if you include such temporal truths about things as 'properties'. |
| 16233 | Gallois hoped to clarify identity through time, but seems to make talk of it impossible [Hawley on Gallois] |
| Full Idea: A problem for Gallois is that he leaves us no way to talk about questions of genuine identity through time, and thus undercuts one motivation for his own position. | |
| From: comment on André Gallois (Occasions of Identity [1998]) by Katherine Hawley - How Things Persist 5.8 | |
| A reaction: Gallois seems to need a second theory of identity to support his Occasional Identity theory. Two things need an identity each, before we can say that the two identities coincide. (Time to read Gallois!) |
| 16025 | If things change they become different - but then no one thing undergoes the change! [Gallois] |
| Full Idea: If things really change, there can't literally be one thing before and after the change. However, if there isn't one thing before and after the change, then no thing has really undergone any change. | |
| From: André Gallois (Identity over Time [2011], Intro) | |
| A reaction: [He cites Copi for this way of expressing the problem of identity through change] There is an obvious simple ambiguity about 'change' in ordinary English. A change of property isn't a change of object. Painting a red ball blue isn't swapping it. |
| 16026 | 4D: time is space-like; a thing is its history; past and future are real; or things extend in time [Gallois] |
| Full Idea: We have four versions of Four-Dimensionalism: the relativistic view that time is space-like; a persisting thing is identical with its history (so objects are events); past and future are equally real; or (Lewis) things extend in time, with temporal parts. | |
| From: André Gallois (Identity over Time [2011], §2.5) | |
| A reaction: Broad proposed the second one. I prefer 3-D: at any given time a thing is wholly present. At another time it is wholly present despite having changed. It is ridiculous to think that small changes destroy identity. We acquire identity by dying?? |
| 14755 | Gallois is committed to identity with respect to times, and denial of simple identity [Gallois, by Sider] |
| Full Idea: Gallois's core claim is that the identity relation holds with respect to times, ...and he must claim that there is no such thing as the relation of identity simpliciter. | |
| From: report of André Gallois (Occasions of Identity [1998]) by Theodore Sider - Four Dimensionalism 5.5 | |
| A reaction: Gallois is essentially responding to the statue and clay problem, but it seems a bit drastic to entirely change our concept of two things being identical, such as Hesperus and Phosphorus. 'Identity' seems to have several meanings; let's sort them out. |
| 16231 | Occasional Identity: two objects can be identical at one time, and different at others [Gallois, by Hawley] |
| Full Idea: Gallois' Occasional Identity Thesis is that objects can be identical at one time without being identical at all times. | |
| From: report of André Gallois (Occasions of Identity [1998]) by Katherine Hawley - How Things Persist 5.4 | |
| A reaction: The analogy is presumably with two crossing roads being identical at one place but not at others. It is a major misunderstanding to infer from Special Relativity that time is just like space. |
| 16027 | If two things are equal, each side involves a necessity, so the equality is necessary [Gallois] |
| Full Idea: The necessity of identity: a=b; □(a=a); so something necessarily = a; so something necessarily must equal b; so □(a=b). [A summary of the argument of Marcus and Kripke] | |
| From: André Gallois (Identity over Time [2011], §3) | |
| A reaction: [Lowe 1982 offered a response] The conclusion seems reasonable. If two things are mistakenly thought to be different, but turn out to be one thing, that one thing could not possibly be two things. In no world is one thing two things! |
| 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. |
| 3059 | There is no real motion, only the appearance of it [Melissus, by Diog. Laertius] |
| Full Idea: There is no such thing as real motion, but there only appears to be such. | |
| From: report of Melissus (fragments/reports [c.443 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.4.3 |
| 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. |
| 5100 | The void is not required for change, because a plenum can alter in quality [Aristotle on Melissus] |
| Full Idea: There is no need for void to be the cause of all change, because it is perfectly possible for a plenum to alter qualitatively (which is something Melissus overlooked). | |
| From: comment on Melissus (fragments/reports [c.443 BCE]) by Aristotle - Physics 214a27 | |
| A reaction: In modern physics this presumably gives us fluctuations in a force field. Motion is like a cat being digested by a python. The atomist claim that emptiness is needed if anything is to move still has intuitive appeal. |
| 456 | Nothing could come out of nothing [Melissus] |
| Full Idea: If Nothing existed, in no way could anything come into being out of nothing. | |
| From: Melissus (fragments/reports [c.443 BCE], B1), quoted by (who?) - where? |