Combining Texts

All the ideas for 'Good and Evil', 'What are Sets and What are they For?' and 'Does Consciousness Exist?'

unexpand these ideas     |    start again     |     specify just one area for these texts

---

13 ideas

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 [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.

14240	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.

14241	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)

14242	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

14243	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

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.

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 [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

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.

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 [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.

15. Nature of Minds / B. Features of Minds / 1. Consciousness / e. Cause of consciousness

9286	Consciousness is not a stuff, but is explained by the relations between experiences [James]
	Full Idea: Consciousness connotes a kind of external relation, and not a special stuff or way of being. The peculiarity of our experiences, that they not only are, but are known, is best explained by their relations to one another, the relations being experiences.
	From: William James (Does Consciousness Exist? [1904], §3)
	A reaction: This view has suddenly caught people's interest. It might be better than the higher/lower relationship, which seems to leave the basic problem untouched. Does a whole network of relations between experiences gradually 'add up' to consciousness?

17. Mind and Body / E. Mind as Physical / 3. Eliminativism

9285	'Consciousness' is a nonentity, a mere echo of the disappearing 'soul' [James]
	Full Idea: 'Consciousness' is the name of a nonentity. ..Those who cling to it are clinging to a mere echo, the faint rumour left behind by the disappearing 'soul' upon the air of philosophy. ..I deny that it stands for an entity, but it does stand for a function.
	From: William James (Does Consciousness Exist? [1904], Intro)
	A reaction: This kind of view is often treated as being preposterous, but I think it is correct. No one is denying the phenomenology, but it is the ontology which is at stake. Either you are a substance dualist, or mind must be eliminated as an 'entity'.

22. Metaethics / C. The Good / 1. Goodness / a. Form of the Good

22489	'Good' is an attributive adjective like 'large', not predicative like 'red' [Geach, by Foot]
	Full Idea: Geach puts 'good' in the class of attributive adjectives, such as 'large' and 'small', contrasting such adjectives with 'predicative' adjectives such as 'red'.
	From: report of Peter Geach (Good and Evil [1956]) by Philippa Foot - Natural Goodness Intro
	A reaction: [In Analysis 17, and 'Theories of Ethics' ed Foot] Thus any object can simply be red, but something can only be large or small 'for a rat' or 'for a car'. Hence nothing is just good, but always a good so-and-so. This is Aristotelian, and Foot loves it.