17 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. |
| 22687 | Maybe literary assessment is evaluating the artist as a suitable friend [Gaut] |
| Full Idea: An approach in Hume (elaborated by Wayne Booth) holds that literary assessment is akin to an act of befriending, for one assesses the author of a work as a suitable friend. | |
| From: Berys Gaut (The Ethical Criticism of Art [1998], 'Some') | |
| A reaction: I like the idea that art exploits our normal range of social emotions and attitudes, so I think this has some truth, but some of the best artists are so out of my league as to not even be candidates for friendship. Dostoevsky? Webster? Caravaggio? |
| 22686 | Formalists say aesthetics concerns types of beauty, or unity, complexity and intensity [Gaut] |
| Full Idea: The formal objects which individuate the aesthetic attitude may be narrowly aesthetic, as beauty, and its subspecies, such as grace and elegance, or more broadly by other formalist criteria, such as Beardley's unity, complexity and intensity. | |
| From: Berys Gaut (The Ethical Criticism of Art [1998], 'Objections 1') | |
| A reaction: I'm not sure about unity or complexity, but intensity was endorsed by Henry James. Intensity doesn't sound very 'formal'. 'Beauty' doesn't seem the right word for the wonderful 'King Lear', or even for Jane Austen novels. |
| 22690 | 'Moralism' says all aesthetic merits are moral merits [Gaut] |
| Full Idea: The view that the only aesthetic merit of works are ethical ones is known as 'moralism'. | |
| From: Berys Gaut (The Ethical Criticism of Art [1998], n 1) | |
| A reaction: [He says this view was demolished by R.W.Beardsmore in 1971] Gaut contrasts this with his own carefully modulated 'ethicism'. Moralism predominated in the eighteenth century, but now looks clearly wrong (or naïve). |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 22684 | Good ethics counts towards aesthetic merit, and bad ethics counts against it [Gaut] |
| Full Idea: I defend 'ethicism', which says that ethically admirable attitudes count toward the the aesthetic merit of a work, and ethically reprehensible attitudes count against its aesthetic merit. | |
| From: Berys Gaut (The Ethical Criticism of Art [1998], 'Ethicism') | |
| A reaction: He recognises that morally admirable works can explore unethical behaviour, and also that identifying the 'attitude' of a work is not simple. The ethics are not necessary. 'Triumph of the Will' is a classic test case. I disagree with Gaut. |
| 22689 | If we don't respond ethically in the way a work prescribes, that is an aesthetic failure [Gaut] |
| Full Idea: Our having reason not to respond in the way prescribed (because it is unethical) is a failure of the work …so that is an aesthetic failure, which is an aesthetic defect. | |
| From: Berys Gaut (The Ethical Criticism of Art [1998], 'Merited') | |
| A reaction: A key argument for Gaut's theory of 'ethicism' about literature. If 'Triumph of the Will' gets the right response from Nazi sympathisers, that is probably all aesthetic success. Jane Austen hasn't failed if she is rejected as bourgeois. |
| 22685 | Good art does not necessarily improve people (any more than good advice does) [Gaut] |
| Full Idea: Ethicism does not entail the causal thesis that good art ethically improves people, …any more than it follows that earnest ethical advice improves people. | |
| From: Berys Gaut (The Ethical Criticism of Art [1998], 'Ethicism') | |
| A reaction: How successful were sermons, in the great days of Christianity? It seems hard to disagree with Gaut's point. |