31 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? |
| 18680 | To avoid misunderstandings supervenience is often expressed negatively: no A-change without B-change [Orsi] |
| Full Idea: It is no part of supervenience that 'if p then q' entails 'if not p then not q'. To avoid such misunderstandings, it is common (though not more accurate) to describe supervenience in negative terms: no difference in A without a difference in B. | |
| From: Francesco Orsi (Value Theory [2015], 5.2) | |
| A reaction: [compressed] In other words it is important to avoid the presupposition that the given supervenience is a two-way relation. The paradigm case of supervenience is stalking. |
| 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. |
| 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. |
| 18684 | Rather than requiring an action, a reason may 'entice' us, or be 'eligible', or 'justify' it [Orsi] |
| Full Idea: Many have suggested alternative roles or sorts of reasons, which are not mandatory. Dancy says some reasons are 'enticing' rather than peremptory; Raz makes options 'eligible' rather than required; Gert says they justify rather than require action. | |
| From: Francesco Orsi (Value Theory [2015], 6.4) | |
| A reaction: The third option is immediately attractive - but then it would only justify the action because it was a good reason, which would need explaining. 'Enticing' captures the psychology in a nice vague way. |
| 18666 | Value-maker concepts (such as courageous or elegant) simultaneously describe and evaluate [Orsi] |
| Full Idea: Examples of value-maker concepts are courageous, honest, cowardly, corrupt, elegant, tacky, melodious, insightful. Employing these concepts normally means both evaluating and describing the thing or person one way or another. | |
| From: Francesco Orsi (Value Theory [2015], 1.2) | |
| A reaction: The point being that they tell you two things - that this thing has a particular value, and also why it has that value. Since I am flirting with the theory that all values must have 'value-makers' this is very interesting. |
| 18667 | The '-able' concepts (like enviable) say this thing deserves a particular response [Orsi] |
| Full Idea: The '-able' concepts, such as valuable, enviable, contemptible, wear on their sleeve the idea that the thing so evaluated merits or is worth a certain attitude or response (of valuing, envying, despising). | |
| From: Francesco Orsi (Value Theory [2015], 1.2) | |
| A reaction: Compare Idea 18666. Hence some concepts point to the source of value in the thing, and others point to the source of the value in the normative attitude of the speaker. Interesting. |
| 18685 | Final value is favoured for its own sake, and personal value for someone's sake [Orsi] |
| Full Idea: Final value is to be favoured for its own sake; personal value is to be favoured for someone's sake. | |
| From: Francesco Orsi (Value Theory [2015], 7.2) | |
| A reaction: This gives another important dimension for discussions of value. I like the question 'what gives rise to this value?', but we can also ask (given the value) why we should then promote it. Health isn't a final value, and truth isn't a personal value? |
| 18679 | Things are only valuable if something makes it valuable, and we can ask for the reason [Orsi] |
| Full Idea: If a certain object is valuable, then something other than its being valuable must make it so. ...One is always in principle entitled to an answer as to why it is good or bad. | |
| From: Francesco Orsi (Value Theory [2015], 5.2) | |
| A reaction: What Orsi calls the 'chemistry' of value. I am inclined to think that this is the key to a philosophical study of value. Without this assumption the values float free, and we drift into idealised waffle. Note that here he only refers to 'objects'. |
| 18682 | A complex value is not just the sum of the values of the parts [Orsi] |
| Full Idea: The whole 'being pleased by cats being tortured' is definitely not better, and is likely worse, than cats being tortured. So its value cannot result from a sum of the intrinsic values of the parts. | |
| From: Francesco Orsi (Value Theory [2015], 5.3) | |
| A reaction: This example is simplistic. It isn't a matter of just adding 'pleased' and 'tortured'. 'Pleased' doesn't have a standalone value. Only a rather gormless utilitarian would think it was always good if someone was pleased. I suspect values don't sum at all. |
| 18683 | Trichotomy Thesis: comparable values must be better, worse or the same [Orsi] |
| Full Idea: It is natural to assume that if we can compare two objects or states of affairs, X and Y, then X is either better than, or worse than, or as good as Y. This has been called the Trichotomy Thesis. | |
| From: Francesco Orsi (Value Theory [2015], 6.2) | |
| A reaction: This is the obvious starting point for a discussion of the difficult question of the extent to which values can be compared. Orsi says even if there was only one value, like pleasure, it might have incommensurable aspects like duration and intensity. |
| 18686 | The Fitting Attitude view says values are fitting or reasonable, and values are just byproducts [Orsi] |
| Full Idea: The main claims of the Fitting Attitude view of value are Reduction: values such are goodness are reduced to fitting attitudes, having reasons, and Normative Redundancy: goodness provides no reasons for attitudes beyond the thing's features. | |
| From: Francesco Orsi (Value Theory [2015], 8.2) | |
| A reaction: Orsi's book is a sustained defence of this claim. I like the Normative Redundancy idea, but I am less persuaded by the Reduction. |
| 18672 | Values from reasons has the 'wrong kind of reason' problem - admiration arising from fear [Orsi] |
| Full Idea: A support for the fittingness account (against the buck-passing reasons account) is the 'wrong kind of reasons' problem. There are many reasons for positive attitudes towards things which are not good. We might admire a demon because he threatens torture. | |
| From: Francesco Orsi (Value Theory [2015], 1.4) | |
| A reaction: [compressed] I like the Buck-Passing view, but was never going to claim that all reasons for positive attitudes bestow value. I only think that there is no value without a reason |
| 18677 | A thing may have final value, which is still derived from other values, or from relations [Orsi] |
| Full Idea: Many believe that final values can be extrinsic: objects which are valuable for their own sake partly thanks to their relations to other objects. ...This might depend on the value of other things...or an object's relational properties. | |
| From: Francesco Orsi (Value Theory [2015], 2.3) | |
| A reaction: It strikes me that virtually nothing (or even absolutely nothing) has final value in total isolation from other things (Moore's 'isolation test'). Values arise within a tangled network of relations. Your final value is my instrumental value. |
| 18668 | Truths about value entail normative truths about actions or attitudes [Orsi] |
| Full Idea: My guiding assumption is that truths about value, at least, regularly entail normative truths of some sort about actions or attitudes. | |
| From: Francesco Orsi (Value Theory [2015], 1.4) | |
| A reaction: Not quite as clear as it sounds. If I say 'the leaf is green' I presume a belief that it is green, which is an attitude. If I say 'shut the door' that implies an action with no value. One view says that values are entirely normative in this way. |
| 18670 | The Buck-Passing view of normative values says other properties are reasons for the value [Orsi] |
| Full Idea: Version two of the normative view of values is the Buck-Passing account, which says that 'x is good' means 'x has the property of having other properties that provide reasons to favour x'. | |
| From: Francesco Orsi (Value Theory [2015], 1.4) | |
| A reaction: [He cites Scanlon 1998:95-8] I think this is the one to explore. We want values in the world, bridging the supposed 'is-ought gap', and not values that just derive from the way human beings are constituted (and certainly not supernatural values!). |
| 18669 | Values can be normative in the Fitting Attitude account, where 'good' means fitting favouring [Orsi] |
| Full Idea: Version one of the normative view of values is the Fitting Attitude account, which says that 'x is good' means 'it is fitting to respond favourably to (or 'favour') x'. | |
| From: Francesco Orsi (Value Theory [2015], 1.4) | |
| A reaction: Brentano is mentioned. Orsi favours this view. The rival normative view is Scanlon's [1998:95-8] Buck-Passing account, in Idea 18670. I am interested in building a defence of the Buck-Passing account, which seems to suit a naturalistic realist like me. |
| 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. |