20 ideas
| 14239 | The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley] |
| Full Idea: The empty set is usually derived via Zermelo's axiom of separation. But the axiom of separation is conditional: it requires the existence of a set in order to generate others as subsets of it. The original set has to come from the axiom of infinity. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: They charge that this leads to circularity, as Infinity depends on the empty set. |
| 14240 | The empty set is something, not nothing! [Oliver/Smiley] |
| Full Idea: Some authors need to be told loud and clear: if there is an empty set, it is something, not nothing. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: I'm inclined to think of a null set as a pair of brackets, so maybe that puts it into a metalanguage. |
| 14241 | We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley] |
| Full Idea: The empty set is said to be useful to express non-existence, but saying 'there are no Us', or ¬∃xUx are no less concise, and certainly less roundabout. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) |
| 14242 | Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley] |
| Full Idea: Suppose we introduce Ω not as a term standing for a supposed empty set, but as a paradigm of an empty term, not standing for anything. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: This proposal, which they go on to explore, seems to mean that Ω (i.e. the traditional empty set symbol) is no longer part of set theory but is part of semantics. |
| 14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |
| Full Idea: Thomason says with no unit sets we couldn't call {1,2}∩{2,3} a set - but so what? Why shouldn't the intersection be the number 2? However, we then have to distinguish three different cases of intersection (common subset or member, or disjoint). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 2.2) |
| 14234 | If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley] |
| Full Idea: A 'singularist', who refers to objects one at a time, must resort to the language of sets in order to replace plural reference to members ('Henry VIII's wives') by singular reference to a set ('the set of Henry VIII's wives'). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: A simple and illuminating point about the motivation for plural reference. Null sets and singletons give me the creeps, so I would personally prefer to avoid set theory when dealing with ontology. |
| 14237 | We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley] |
| Full Idea: Plurals earn their keep in set theory, to answer Skolem's remark that 'in order to treat of 'sets', we must begin with 'domains' that are constituted in a certain way'. We can speak in the plural of 'the objects', not a 'domain' of objects. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: [Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it. |
| 14245 | Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley] |
| Full Idea: Logical truths should be true no matter what exists, so true even if nothing exists. The classical predicate calculus, however, makes it logically true that something exists. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) |
| 14246 | If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley] |
| Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) | |
| A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application. |
| 14247 | Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley] |
| Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2) | |
| A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated. |
| 20298 | The traditional a priori is justified without experience; post-Quine it became unrevisable by experience [Rey] |
| Full Idea: Where Kant and others had traditionally assumed that the a priori concerned beliefs 'justifiable independently of experience', Quine and others of the time came to regard it as beliefs 'unrevisable in the light of experience'. | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 3.7) | |
| A reaction: That throws a rather striking light on Quine's project. Of course, if the a priori is also necessary, then it has to be unrevisable. But is a bachelor necessarily an unmarried man? It is not necessary that 'bachelor' has a fixed meaning. |
| 20300 | Externalist synonymy is there being a correct link to the same external phenomena [Rey] |
| Full Idea: Externalists are typically committed to counting expressions as 'synonymous' if they happen to be linked in the right way to the same external phenomena, even if a thinker couldn't realise that they are by reflection alone. | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 4.2) | |
| A reaction: [He cites Fodor] Externalists always try to link to concrete things in the world, but most of our talk is full of generalities, abstractions and fiction which don't link directly to anything. |
| 20294 | 'Married' does not 'contain' its symmetry, nor 'bigger than' its transitivity [Rey] |
| Full Idea: If Bob is married to Sue, then Sue is married to Bob. If x bigger than y, and y bigger than z, x is bigger than z. The symmetry of 'marriage' or transitivity of 'bigger than' are not obviously 'contained in' the corresponding thoughts. | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 1.2) | |
| A reaction: [Also 'if something is red, then it is coloured'] This is a Fregean criticism of Kant. It is not so much that Kant was wrong, as that the concept of analyticity is seen to have a much wider application than Kant realised. Especially in mathematics. |
| 20293 | Analytic judgements can't be explained by contradiction, since that is what is assumed [Rey] |
| Full Idea: Rejecting 'a married bachelor' as contradictory would seem to have no justification other than the claim that 'All bachelors are unmarried is analytic, and so cannot serve to justify or explain that claim. | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 1.2) | |
| A reaction: Rey is discussing Frege's objection to Kant (who tried to prove the necessity of analytic judgements, on the basis of the denial being a contradiction). |
| 20297 | Analytic statements are undeniable (because of meaning), rather than unrevisable [Rey] |
| Full Idea: What's peculiar about the analytic is that denying it seem unintelligible. Far from unrevisability explaining analyticity, it seems to be analyticitiy that explains unrevisability; we only balk at denying unmarried bachelors because that's what it means! | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 3.7) | |
| A reaction: This is a criticism of Quine, who attacked analyticity when it is understood as unrevisability. Obviously we could revise the concept of 'bachelor', if our marriage customs changed a lot. Rey seems right here. |
| 20301 | The meaning properties of a term are those which explain how the term is typically used [Rey] |
| Full Idea: It may be that the meaning properties of a term are the ones that play a basic explanatory role with regard to the use of the term generally, the ones in virtue ultimately of which a term is used with that meaning. | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 4.3) | |
| A reaction: [He cites Devitt 1996, 2002, and Horwich 1998, 2005) I spring to philosophical life whenever I see the word 'explanatory', because that is the point of the whole game. They are pointing to the essence of the concept (which is explanatory, say I). |
| 20302 | An intrinsic language faculty may fix what is meaningful (as well as grammatical) [Rey] |
| Full Idea: The existence of a separate language faculty may be an odd but psychologically real fact about us, and it may thereby supply a real basis for commitments about not only what is or is not grammatical, but about what is a matter of natural language meaning. | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 4.4) | |
| A reaction: This is the Chomskyan view of analytic sentences. An example from Chomsky (1977:142) is the semantic relationships of persuade, intend and believe. It's hard to see how the secret faculty on its own could do the job. Consensus is needed. |
| 20303 | Research throws doubts on the claimed intuitions which support analyticity [Rey] |
| Full Idea: The movement of 'experimental philosophy' has pointed to evidence of considerable malleability of subject's 'intuitions' with regard to the standard kinds of thought experiments on which defenses of analytic claims typically rely. | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 4.4) | |
| A reaction: See Cappelen's interesting attack on the idea that philosophy relies on intuitions, and hence his attack on experimental philosophy. Our consensus on ordinary English usage hardly qualifies as somewhat vague 'intuitions'. |
| 20299 | If we claim direct insight to what is analytic, how do we know it is not sub-consciously empirical? [Rey] |
| Full Idea: How in the end are we going to distinguish claims or the analytic as 'rational insight', 'primitive compulsion', inferential practice or folk belief from merely some deeply held empirical conviction, indeed, from mere dogma. | |
| From: Georges Rey (The Analytic/Synthetic Distinction [2013], 4.1) | |
| A reaction: This is Rey's summary of the persisting Quinean challenge to analytic truths, in the face of a set of replies, summarised by the various phrases here. So do we reject a dogma of empiricism, by asserting dogmatic empiricism? |
| 22489 | 'Good' is an attributive adjective like 'large', not predicative like 'red' [Geach, by Foot] |
| Full Idea: Geach puts 'good' in the class of attributive adjectives, such as 'large' and 'small', contrasting such adjectives with 'predicative' adjectives such as 'red'. | |
| From: report of Peter Geach (Good and Evil [1956]) by Philippa Foot - Natural Goodness Intro | |
| A reaction: [In Analysis 17, and 'Theories of Ethics' ed Foot] Thus any object can simply be red, but something can only be large or small 'for a rat' or 'for a car'. Hence nothing is just good, but always a good so-and-so. This is Aristotelian, and Foot loves it. |