14 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. |
7444 | Type-type psychophysical identity is combined with a functional characterisation of pain [Lewis] |
Full Idea: The materialist theory Armstrong and I proposed joins claims of type-type psychophysical identity with a behaviourist or functionalist way of characterising mental states such as pain. | |
From: David Lewis (Mad Pain and Martian Pain [1980], §III) | |
A reaction: Armstrong has backed off from 'type-type' identity, because the realisations of a given mental state might be too diverse to be considered of the same type. Putnam's machine functionalism allows the possibility of dualism. |
7445 | The application of 'pain' to physical states is non-rigid and contingent [Lewis] |
Full Idea: The word 'pain' is a non-rigid designator; it is a contingent matter what state the concept and the word apply to. (Note: so the sort of theory Kripke argues against is not what we propose). | |
From: David Lewis (Mad Pain and Martian Pain [1980], §III) | |
A reaction: I like the view that a given quale is necessarily identical to a given mental state, but that many mental states might occupy a given behavioural role. The smell of roses might occupy the behavioural role of pain. Frog pain isn't quite like ours. |
7443 | A theory must be mixed, to cover qualia without behaviour, and behaviour without qualia [Lewis, by PG] |
Full Idea: To pass our test it seems that our theory will have to be a 'mixed' theory, to account for the Madman (whose pain has odd causes, and odd effects) and also for the Martian (who has normal causes and effects, but an odd physical state). | |
From: report of David Lewis (Mad Pain and Martian Pain [1980], §II) by PG - Db (ideas) | |
A reaction: A statement that 'pain' is ambiguous (qualia/causal role) would help a lot here. Martians have the causal role but no qualia, and the madman has the qualia but lacks the causal role. I say lots of different qualia might have the same causal role. |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |