35 ideas
| 12442 | 'Mickey Mouse is a fictional mouse' is true without a truthmaker [Azzouni] |
| Full Idea: 'Mickey Mouse is a fictional mouse' can be taken as true without have any truthmaker. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3) | |
| A reaction: There might be an equivocation over 'true' here. 'What, really really true that he IS a fictional mouse?' |
| 12439 | Truth is dispensable, by replacing truth claims with the sentence itself [Azzouni] |
| Full Idea: No truth predicate is ever indispensable, because Tarski biconditionals, the equivalences between sentences and explicit truth ascriptions to those sentences, allow us to replace explicit truth ascriptions with the sentences themselves. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.1) | |
| A reaction: Holding a sentence to be true isn't the same as saying that it is true, and it isn't the same as saying the sentence, because one might say it in an ironic tone of voice. |
| 12437 | Truth lets us assent to sentences we can't explicitly exhibit [Azzouni] |
| Full Idea: My take on truth is a fairly deflationary one: The role of the truth predicate is to enable us to assent to sentences we can't explicitly exhibit. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Intro) | |
| A reaction: Clearly this is a role for truth, as in 'I forget what he said, but I know it was true', but it isn't remotely what most people understand by true. We use 'true' about totally explicit sentences all the time. |
| 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. |
| 12446 | Names function the same way, even if there is no object [Azzouni] |
| Full Idea: Names function the same way (semantically and grammatically) regardless of whether or not there's an object that they refer to. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3 n55) | |
| A reaction: I take this to be a fairly clear rebuttal of the 'Fido'-Fido view of names (that the meaning of the name IS the dog), which never seems to quite go away. A name is a peg on which description may be hung, seems a good slogan to me. |
| 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). |
| 12447 | That all existents have causal powers is unknowable; the claim is simply an epistemic one [Azzouni] |
| Full Idea: If the argument isn't that, metaphysically speaking, anything that exists must have causal powers - how on earth would we show that? - rather, the claim is an epistemic one. Any thing we're in a position to know about we must causally interact with. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.4) | |
| A reaction: A very good point. I am attracted to causal power as a criterion for existence, but Azzouni's distinction is vital. Maybe there is just no point in even talking about things which exist but have no causal powers. |
| 12445 | If fictional objects really don't exist, then they aren't abstract objects [Azzouni] |
| Full Idea: It's robustly part of common sense that fictional objects don't exist in any sense at all, and this means they aren't abstracta either. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3) | |
| A reaction: Nice. It is so easy to have some philosopher dilute and equivocate over the word 'object' until you find yourself committed to all sorts of daft things as somehow having objectual existence. We can discuss things which don't exist in any way at all. |
| 12449 | Modern metaphysics often derives ontology from the logical forms of sentences [Azzouni] |
| Full Idea: It is widespread in contemporary metaphysics to extract commitments to various types of object on the basis of the logical form of certain sentences. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.7) | |
| A reaction: I'm with Azzouni in thinking that this procedure is a very bad idea. I'm increasingly inclined towards the wild view that people are only ontologically committed to things if they explicitly say that they are so committed. |
| 12440 | If objectual quantifiers ontologically commit, so does the metalanguage for its semantics [Azzouni] |
| Full Idea: The argument that objectual quantifiers are ontologically committing has the crucial and unnoticed presupposition that the language in which the semantics for the objectual quantifiers is couched (the 'metalanguage') also has quantifiers with commitment. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3) | |
| A reaction: That is, presumably we find ourselves ontologically committed to the existence of quantifiers, and are also looking at an infinite regress. See Idea 12439. |
| 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. |
| 12438 | In the vernacular there is no unequivocal ontological commitment [Azzouni] |
| Full Idea: There are no linguistic devices, no idioms (not 'there is', not 'exists') that unequivocally indicate ontological commitment in the vernacular. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Intro) | |
| A reaction: This seems right, since people talk in such ways about soap opera, while understanding the ontological situation perfectly well. Presumably Quine is seeking higher standards than the vernacular, if we are doing science. |
| 12441 | We only get ontology from semantics if we have already smuggled it in [Azzouni] |
| Full Idea: A slogan: One can't read ontological commitments from semantic conditions unless one has already smuggled into those semantic conditions the ontology one would like to read off. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3) | |
| A reaction: The arguments supporting this are subtle, but it's good enough for me, as I never thought anyone was ontologically committed just because they used the vagueries of language to try to say what's going on around here. |
| 12448 | Things that don't exist don't have any properties [Azzouni] |
| Full Idea: Things that don't exist don't have any properties. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.4) | |
| A reaction: Sounds reasonable! I totally agree, but that is because my notion of properties is sparse and naturalistic. If you identify properties with predicates (which some weird people seem to), then non-existents can have properties like 'absence' or 'nullity'. |
| 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. |
| 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. |
| 12450 | The periodic table not only defines the elements, but also excludes other possible elements [Azzouni] |
| Full Idea: The periodic table not only governs what elements there can be, with their properties, but also explicitly excludes others sorts of elements, because the elements are individuated by the number of discrete protons in their nuclei. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.7) | |
| A reaction: It has to be central to the thesis of scientific essentialism that the possibilities in nature are far more restricted than is normally thought, and this observation illustrates the view nicely. He makes a similar point about subatomic particles. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |