Combining Texts

All the ideas for 'Mahaprajnaparamitashastra', 'Mad Pain and Martian Pain' and 'What are Sets and What are they For?'

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14 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.
17. Mind and Body / C. Functionalism / 4. Causal Functionalism
Type-type psychophysical identity is combined with a functional characterisation of pain [Lewis]
     Full Idea: The materialist theory Armstrong and I proposed joins claims of type-type psychophysical identity with a behaviourist or functionalist way of characterising mental states such as pain.
     From: David Lewis (Mad Pain and Martian Pain [1980], §III)
     A reaction: Armstrong has backed off from 'type-type' identity, because the realisations of a given mental state might be too diverse to be considered of the same type. Putnam's machine functionalism allows the possibility of dualism.
17. Mind and Body / E. Mind as Physical / 1. Physical Mind
The application of 'pain' to physical states is non-rigid and contingent [Lewis]
     Full Idea: The word 'pain' is a non-rigid designator; it is a contingent matter what state the concept and the word apply to. (Note: so the sort of theory Kripke argues against is not what we propose).
     From: David Lewis (Mad Pain and Martian Pain [1980], §III)
     A reaction: I like the view that a given quale is necessarily identical to a given mental state, but that many mental states might occupy a given behavioural role. The smell of roses might occupy the behavioural role of pain. Frog pain isn't quite like ours.
17. Mind and Body / E. Mind as Physical / 7. Anti-Physicalism / b. Multiple realisability
A theory must be mixed, to cover qualia without behaviour, and behaviour without qualia [Lewis, by PG]
     Full Idea: To pass our test it seems that our theory will have to be a 'mixed' theory, to account for the Madman (whose pain has odd causes, and odd effects) and also for the Martian (who has normal causes and effects, but an odd physical state).
     From: report of David Lewis (Mad Pain and Martian Pain [1980], §II) by PG - Db (ideas)
     A reaction: A statement that 'pain' is ambiguous (qualia/causal role) would help a lot here. Martians have the causal role but no qualia, and the madman has the qualia but lacks the causal role. I say lots of different qualia might have the same causal role.
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
     Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom.
     From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88)
     A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate').