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.
|
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.
|
13127
|
Categories can't overlap; they are either disjoint, or inclusive [Sommers, by Westerhoff]
|
|
Full Idea:
Fred Sommers, in his treatment of types, says that two ontological categories cannot overlap; they are either disjoint, or one properly includes the other. This is sometimes referred to as Sommers' Law.
|
|
From:
report of Fred Sommers (Types and Ontology [1963], p.355) by Jan Westerhoff - Ontological Categories §24
|
|
A reaction:
The 'types', of course, go back to Bertrand Russell's theory of types, which is important in discussions of ontological categories. Carnap pursued it, trying to derive ontological categories from grammatical categories. 85% agree with Sommers.
|