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14234
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If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley]
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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').
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From:
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro)
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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.
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14237
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We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley]
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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.
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From:
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro)
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A reaction:
[Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it.
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14246
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If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley]
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Full Idea:
If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics.
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From:
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1)
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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.
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14247
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Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley]
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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.
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From:
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2)
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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.
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14534
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Shoemaker moved from properties as powers to properties bestowing powers [Shoemaker, by Mumford/Anjum]
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Full Idea:
Shoemaker ventured the theory in 1980 that properties just are clusters of powers, but he has subsequently abandoned this, and now thinks properties bestow their bearers with causal powers.
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From:
report of Sydney Shoemaker (Self, Body and Coincidence [1999], p.297) by S.Mumford/R.Lill Anjum - Getting Causes from Powers 1.1
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A reaction:
Like Mumford and Anjum, I prefer the earlier theory. I think taking powers as basic is the only story that really makes sense. A power is intrinsic and primitive, whereas properties are complex, messy, partly subjective, and higher level.
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