Ideas of Oliver,A/Smiley,T, by Theme

[British, fl. 2006, Both at the University of Cambridge.]

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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
 14239 The empty set is usually derived from Separation, but it also seems to need Infinity 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! 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 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 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.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
 14243 The unit set may be needed to express intersections that leave a single member 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)
5. Theory of Logic / G. Quantification / 6. Plural Quantification
 14234 If you only refer to objects one at a time, you need sets in order to refer to a plurality 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' 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.
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
 14245 Logical truths are true no matter what exists - but predicate calculus insists that something exists 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)
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
 14246 If mathematics purely concerned mathematical objects, there would be no applied mathematics 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.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
 14247 Sets might either represent the numbers, or be the numbers, or replace the numbers 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.