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Ideas from 'Mathematics is Megethology' by David Lewis [1993], by Theme Structure

[found in 'Philosophia Mathematica' (ed/tr -) [- ,]].

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4. Formal Logic / F. Set Theory ST / 1. Set Theory
10807 Mathematics reduces to set theory, which reduces, with some mereology, to the singleton function
                        Full Idea: It is generally accepted that mathematics reduces to set theory, and I argue that set theory in turn reduces, with some aid of mereology, to the theory of the singleton function.
                        From: David Lewis (Mathematics is Megethology [1993], p.03)
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
10809 We can accept the null set, but not a null class, a class lacking members
                        Full Idea: In my usage of 'class', there is no such things as the null class. I don't mind calling some memberless thing - some individual - the null set. But that doesn't make it a memberless class. Rather, that makes it a 'set' that is not a class.
                        From: David Lewis (Mathematics is Megethology [1993], p.05)
                        A reaction: Lewis calls this usage 'idiosyncratic', but it strikes me as excellent. Set theorists can have their vital null class, and sensible people can be left to say, with Lewis, that classes of things must have members.
10812 The null set is not a little speck of sheer nothingness, a black hole in Reality
                        Full Idea: Should we accept the null set as a most extraordinary individual, a little speck of sheer nothingness, a sort of black hole in the fabric of Reality itself? Not that either, I think.
                        From: David Lewis (Mathematics is Megethology [1993], p.09)
                        A reaction: Correct!
10811 The null set plays the role of last resort, for class abstracts and for existence
                        Full Idea: The null set serves two useful purposes. It is a denotation of last resort for class abstracts that denote no nonempty class. And it is an individual of last resort: we can count on its existence, and fearlessly build the hierarchy of sets from it.
                        From: David Lewis (Mathematics is Megethology [1993], p.09)
                        A reaction: This passage assuages my major reservation about the existence of the null set, but at the expense of confirming that it must be taken as an entirely fictional entity.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
10813 What on earth is the relationship between a singleton and an element?
                        Full Idea: A new student of set theory has just one thing, the element, and he has another single thing, the singleton, and not the slightest guidance about what one thing has to do with the other.
                        From: David Lewis (Mathematics is Megethology [1993], p.12)
10814 Are all singletons exact intrinsic duplicates?
                        Full Idea: Are all singletons exact intrinsic duplicates?
                        From: David Lewis (Mathematics is Megethology [1993], p.13)
4. Formal Logic / G. Formal Mereology / 1. Mereology
10806 Megethology is the result of adding plural quantification to mereology
                        Full Idea: Megethology is the result of adding plural quantification, as advocated by George Boolos, to the language of mereology.
                        From: David Lewis (Mathematics is Megethology [1993], p.03)
5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
10816 We can use mereology to simulate quantification over relations
                        Full Idea: We can simulate quantification over relations using megethology. Roughly, a quantifier over relations is a plural quantifier over things that encode ordered pairs by mereological means.
                        From: David Lewis (Mathematics is Megethology [1993], p.18)
                        A reaction: [He credits this idea to Burgess and Haven] The point is to avoid second-order logic, which quantifies over relations as ordered n-tuple sets.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10808 Mathematics is generalisations about singleton functions
                        Full Idea: We can take the theory of singleton functions, and hence set theory, and hence mathematics, to consist of generalisations about all singleton functions.
                        From: David Lewis (Mathematics is Megethology [1993], p.03)
                        A reaction: At first glance this sounds like a fancy version of the somewhat discredited Greek idea that mathematics is built on the concept of a 'unit'.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10815 We don't need 'abstract structures' to have structural truths about successor functions
                        Full Idea: We needn't believe in 'abstract structures' to have general structural truths about all successor functions.
                        From: David Lewis (Mathematics is Megethology [1993], p.16)
9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
10810 I say that absolutely any things can have a mereological fusion
                        Full Idea: I accept the principle of Unrestricted Composition: whenever there are some things, no matter how many or how unrelated or how disparate in character they may be, they have a mereological fusion. ...The trout-turkey is part fish and part fowl.
                        From: David Lewis (Mathematics is Megethology [1993], p.07)
                        A reaction: This nicely ducks the question of when things form natural wholes and when they don't, but I would have thought that that might be one of the central issues of metaphysicals, so I think I'll give Lewis's principle a miss.