25 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) |
| 13134 | We negate predicates but do not negate names [Westerhoff] |
| Full Idea: We negate predicates but do not negate names. | |
| From: Jan Westerhoff (Ontological Categories [2005], §88) | |
| A reaction: This is a point for anyone like Ramsey who wants to collapse the distinction between particulars and universals, or singular terms and their predicates. |
| 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. |
| 13124 | Categories can be ordered by both containment and generality [Westerhoff] |
| Full Idea: Categories are usually not assumed to be ordered by containment, but also be generality. | |
| From: Jan Westerhoff (Ontological Categories [2005], §02) | |
| A reaction: I much prefer generality, which is responsive to the full picture, whereas containment seems to appeal too much to the orderly and formalised mind. Containments overlap, so we can't dream of a perfectly neat system. |
| 13117 | How far down before we are too specialised to have a category? [Westerhoff] |
| Full Idea: How far down are we allowed to go before the categories become too special to qualify as ontological categories? | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: A very nice question, because we can't deny a category to a set with only one member, otherwise the last surviving dodo would not have been a dodo. |
| 13116 | Maybe objects in the same category have the same criteria of identity [Westerhoff] |
| Full Idea: There is an idea that objects belonging to the same category have the same criteria of identity. This view was first explicitly endorsed by Frege (1884), and was later systematized by Dummett (1981). | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: This approach is based on identity between equivalence classes. Westerhoff says it means, implausibly, that the resulting categories cannot share properties. |
| 13118 | Categories are base-sets which are used to construct states of affairs [Westerhoff] |
| Full Idea: My fundamental idea is that 'form-sets' are intersubstitutable constituents of states of affairs with the same form, and 'base-sets' are special form-sets which can be used to construct other form-sets. Ontological categories are the base-sets. | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: The spirit of this is, of course, to try to achieve the kind of rigour that is expected in contemporary professional philosophy, by aiming for some sort of axiom-system that is related to a well established precise discipline like set theory. Maybe. |
| 13125 | Categories are held to explain why some substitutions give falsehood, and others meaninglessness [Westerhoff] |
| Full Idea: It is usually assumed of ontological categories that they can explain why certain substitutions make a statement false ('prime' for 'odd'), while others make it meaningless ('sweet' for 'odd', of numbers). | |
| From: Jan Westerhoff (Ontological Categories [2005], §05) | |
| A reaction: So there is a strong link between big ontological questions, and Ryle's famous identification of the 'category mistake'. The phenomenon of the category mistake is undeniable, and should make us sympathetic to the idea of categories. |
| 13126 | Categories systematize our intuitions about generality, substitutability, and identity [Westerhoff] |
| Full Idea: Systems of ontological categories are systematizations of our intuitions about generality, intersubstitutability, and identity. | |
| From: Jan Westerhoff (Ontological Categories [2005], §23) | |
| A reaction: I think we might be able to concede this without conceding the relativism about categories which Westerhoff espouses. I would claim that our 'intuitions' are pretty accurate about the joints of nature, and hence accurate about these criteria. |
| 13130 | Categories as generalities don't give a criterion for a low-level cut-off point [Westerhoff] |
| Full Idea: Categories in terms of generality, dependence and containment are unsatisfactory because of the 'cut-off point problem': they don't give an account of how far down the order we can go and be sure we are still dealing with categories. | |
| From: Jan Westerhoff (Ontological Categories [2005], §27) | |
| A reaction: I don't see why this should be a devastating objection to any theory. I have a very clear notion of a human being, but a very hazy notion of how far back towards its conception a human being extends. |
| 13131 | The aim is that everything should belong in some ontological category or other [Westerhoff] |
| Full Idea: It seems to be one of the central points of constructing systems of ontological categories that everything can be placed in some category or other. | |
| From: Jan Westerhoff (Ontological Categories [2005], §49) | |
| A reaction: After initial resistance to this, I suppose I have to give in. The phoenix (a unique mythological bird) is called a 'phoenix', though it might just be called 'John' (cf. God). If there were another phoenix, we would know how to categorise it. |
| 13123 | All systems have properties and relations, and most have individuals, abstracta, sets and events [Westerhoff] |
| Full Idea: Surveyed ontological systems show overlaps: properties and relations turn up in every system; individuals form part of five systems; abstracta, collections/sets and events are in four; facts are in two. | |
| From: Jan Westerhoff (Ontological Categories [2005], §02) | |
| A reaction: Westerhoff is a hero for doing such a useful survey. Of course, Quine challenges properties, and relations are commonly given a reductive analysis. Individuals can be challenged, and abstracta reduced. Sets are fictions. Events or facts? Etc. |
| 13115 | Ontological categories are like formal axioms, not unique and with necessary membership [Westerhoff] |
| Full Idea: I deny the absolutism of a unique system of ontological categories and the essentialist view of membership in ontological categories as necessary features. ...I regard ontological categories as similar to axioms of formalized theories. | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: The point is that modern axioms are not fundamental self-evident truths, but an economic set of basic statements from which some system can be derived. There may be no unique set of axioms for a formal system. |
| 13119 | Categories merely systematise, and are not intrinsic to objects [Westerhoff] |
| Full Idea: My conclusion is that categories are relativistic, used for systematization, and that it is not an intrinsic feature of an object to belong to a category, and that there is no fundamental distinction between individuals and properties. | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: [compressed] He calls his second conclusion 'anti-essentialist', but I think we can still get an account of (explanatory) essence while agreeing with his relativised view of categories. Wiggins might be his main opponent. |
| 13135 | A thing's ontological category depends on what else exists, so it is contingent [Westerhoff] |
| Full Idea: What ontological category a thing belongs to is not dependent on its inner nature, but dependent on what other things there are in the world, and this is a contingent matter. | |
| From: Jan Westerhoff (Ontological Categories [2005], §89) | |
| A reaction: This is aimed at those, like Wiggins, who claim that category is essential to a thing, and there is no possible world in which that things could belong to another category. Sounds good, till you try to come up with examples. |
| 13129 | Essential kinds may be too specific to provide ontological categories [Westerhoff] |
| Full Idea: Essential kinds can be very specific, and arguably too specific for the purposes of ontological categories. | |
| From: Jan Westerhoff (Ontological Categories [2005], §27) | |
| A reaction: Interesting. There doesn't seem to be any precise guideline as to how specific an essential kind might be. In scientific essentialism, each of the isotopes of tin has a distinct essence, but why should they not be categories |
| 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) |