Combining Texts

All the ideas for 'fragments/reports', 'What are Sets and What are they For?' and 'Inessential Aristotle: Powers without Essences'

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17 ideas

4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
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.
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.
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)
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.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
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)
5. Theory of Logic / G. Quantification / 6. Plural Quantification
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.
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.
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
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)
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
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.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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.
8. Modes of Existence / C. Powers and Dispositions / 1. Powers
Powers give explanations, without being necessary for some class membership [Chakravartty]
     Full Idea: Powers explain behaviours regardless of whether they are necessary for membership in a particular class of things.
     From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 3)
     A reaction: This seems right, and is important for driving a wedge between powers and essences. If there are essences, they are not simply some bunch of powers.
9. Objects / D. Essence of Objects / 5. Essence as Kind
A kind essence is the necessary and sufficient properties for membership of a class [Chakravartty]
     Full Idea: The modern concept of a kind essence is a set of intrinsic properties that are individually necessary and jointly sufficient for the membership of something in a class of things, or 'kind'.
     From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2)
     A reaction: I am always struck by the problem that the kind itself is constructed from the individuals, so circularity always seems to loom.
9. Objects / D. Essence of Objects / 15. Against Essentialism
Cluster kinds are explained simply by sharing some properties, not by an 'essence' [Chakravartty]
     Full Idea: The fact that members of some cluster kinds are subjects of causal generalizations reflects the degree to which they share causally efficacious properties, not the fact that they may be composed of essence kinds per se.
     From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2)
     A reaction: I think this is right. I am a fan of individual essences, but not of kind essences. I take kinds, and kind explanations, to be straightforward inductive generalisations from individuals. Extreme stabilities give the illusion of a kind essence.
14. Science / D. Explanation / 2. Types of Explanation / g. Causal explanations
Explanation of causal phenomena concerns essential kinds - but also lack of them [Chakravartty]
     Full Idea: Scientific practices such as prediction and explanation regarding causal phenomena are concerned not merely with kinds having essences, but also with kinds lacking them.
     From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 1)
     A reaction: Not quite clear what he has in mind, but explanation should certainly involve a coherent picture, and not just the citation of some underlying causal mechanism.
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
Musical performance can reveal a range of virtues [Damon of Ath.]
     Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice.
     From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where?
26. Natural Theory / B. Natural Kinds / 4. Source of Kinds
Some kinds, such as electrons, have essences, but 'cluster kinds' do not [Chakravartty]
     Full Idea: Many of the kinds we theorize about and experiment on today simply do not have essences. We can distinguish 'essence kinds', such as electrons, and 'cluster kinds', such as biological species.
     From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2)
     A reaction: This is an important point for essentialists. He offers a strict criterion, in Idea 15145, for mind membership, but we might allow species to have essences by just relaxing the criteria a bit, and acknowledging some vagueness, especially over time.
26. Natural Theory / D. Laws of Nature / 1. Laws of Nature
Many causal laws do not refer to kinds, but only to properties [Chakravartty]
     Full Idea: Causal laws often do not make reference to kinds of objects at all, but rather summarize relations between quantitative, causally efficacious properties of objects.
     From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 3)
     A reaction: This would only be a serious challenge if it was not possible to translate talk of properties into talk of kinds, and vice versa.