|
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.
|
|
8921
|
Structuralism is now common, studying relations, with no regard for what the objects might be [Hellman]
|
|
Full Idea:
With developments in modern mathematics, structuralist ideas have become commonplace. We study 'abstract structures', having relations without regard to the objects. As Hilbert famously said, items of furniture would do.
|
|
From:
Geoffrey Hellman (Structuralism [2007], §1)
|
|
A reaction:
Hilbert is known as a Formalist, which suggests that modern Structuralism is a refined and more naturalist version of the rather austere formalist view. Presumably the sofa can't stand for six, so a structural definition of numbers is needed.
|
|
14193
|
'Substance theorists' take modal properties as primitive, without structure, just falling under a sortal [Paul,LA]
|
|
Full Idea:
Some deep essentialists resist the need to explain the structure under de re modal properties, taking them as primitive. One version (which we can call 'substance theory') takes them to fall under a sortal concept, with no further explanation.
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], §1)
|
|
A reaction:
A very helpful identification of what Wiggins stands for, and why I disagree with him. The whole point of essences is to provide a notion that fits in with sciences, which means they must have an explanatory role, which needs structures.
|
|
14195
|
If an object's sort determines its properties, we need to ask what determines its sort [Paul,LA]
|
|
Full Idea:
If the substance essentialist holds that the sort an object belongs to determines its de re modal properties (rather than the other way round), then he needs to give an (ontological, not conceptual) explanation of what determines an object's sort.
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], §1)
|
|
A reaction:
See Idea 14193 for 'substance essentialism'. I find it quite incredible that anyone could think that a thing's sort could determine its properties, rather than the other way round. Even if sortals are conventional, they are not arbitrary.
|
|
14196
|
Substance essentialism says an object is multiple, as falling under various different sortals [Paul,LA]
|
|
Full Idea:
The explanation of material constitution given by substance essentialism is that there are multiple objects. A person is essentially human-shaped (falling under the human sort), while their hunk of tissue is accidentally human-shaped (as tissue).
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], §1)
|
|
A reaction:
At this point sortal essentialism begins to look crazy. Persons are dubious examples (with sneaky dualism involved). A bronze statue is essentially harder to dent than a clay one, because of its bronze. If you remake it of clay, it isn't the same statue.
|
|
14190
|
Deep essentialist objects have intrinsic properties that fix their nature; the shallow version makes it contextual [Paul,LA]
|
|
Full Idea:
Essentialism says that objects have their properties essentially. 'Deep' essentialists take the (nontrivial) essential properties of an object to determine its nature. 'Shallow' essentialists substitute context-dependent truths for the independent ones.
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], Intro)
|
|
A reaction:
If the deep essence determines a things nature, we should not need to say 'nontrivial'. This is my bete noire, the confusion of essential properties with necessary ones, where necessary properties (or predicates, at least) can indeed be trivial.
|
|
14189
|
'Modal realists' believe in many concrete worlds, 'actualists' in just this world, 'ersatzists' in abstract other worlds [Paul,LA]
|
|
Full Idea:
A 'modal realist' believes that there are many concrete worlds, while the 'actualist' believes in only one concrete world, the actual world. The 'ersatzist' is an actualist who takes nonactual possible worlds and their contents to be abstracta.
|
|
From:
L.A. Paul (In Defense of Essentialism [2006], Intro)
|
|
A reaction:
My view is something like that modal realism is wrong, and actualism is right, and possible worlds (if they really are that useful) are convenient abstract fictions, constructed (if we have any sense) out of the real possibilities in the actual world.
|