Ideas from 'Sets and Numbers' by Penelope Maddy [1981], by Theme Structure

[found in 'Philosophy of Mathematics: anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,0-631-21870-x]].

green numbers give full details    |     back to texts     |     unexpand these ideas

4. Formal Logic / F. Set Theory ST / 7. Natural Sets
 17824 The master science is physical objects divided into sets
 Full Idea: The master science can be thought of as the theory of sets with the entire range of physical objects as ur-elements. From: Penelope Maddy (Sets and Numbers [1981], II) A reaction: This sounds like Quine's view, since we have to add sets to our naturalistic ontology of objects. It seems to involve unrestricted mereology to create normal objects.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
 17825 Set theory (unlike the Peano postulates) can explain why multiplication is commutative
 Full Idea: If you wonder why multiplication is commutative, you could prove it from the Peano postulates, but the proof offers little towards an answer. In set theory Cartesian products match 1-1, and n.m dots when turned on its side has m.n dots, which explains it. From: Penelope Maddy (Sets and Numbers [1981], II) A reaction: 'Turning on its side' sounds more fundamental than formal set theory. I'm a fan of explanation as taking you to the heart of the problem. I suspect the world, rather than set theory, explains the commutativity.
 17826 Standardly, numbers are said to be sets, which is neat ontology and epistemology
 Full Idea: The standard account of the relationship between numbers and sets is that numbers simply are certain sets. This has the advantage of ontological economy, and allows numbers to be brought within the epistemology of sets. From: Penelope Maddy (Sets and Numbers [1981], III) A reaction: Maddy votes for numbers being properties of sets, rather than the sets themselves. See Yourgrau's critique.
 17828 Numbers are properties of sets, just as lengths are properties of physical objects
 Full Idea: I propose that ...numbers are properties of sets, analogous, for example, to lengths, which are properties of physical objects. From: Penelope Maddy (Sets and Numbers [1981], III) A reaction: Are lengths properties of physical objects? A hole in the ground can have a length. A gap can have a length. Pure space seems to contain lengths. A set seems much more abstract than its members.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
 17827 Sets exist where their elements are, but numbers are more like universals
 Full Idea: A set of things is located where the aggregate of those things is located, ...but a number is simultaneously located at many different places (10 in my hand, and a baseball team) ...so numbers seem more like universals than particulars. From: Penelope Maddy (Sets and Numbers [1981], III) A reaction: My gut feeling is that Maddy's master idea (of naturalising sets by building them from ur-elements of natural objects) won't work. Sets can work fine in total abstraction from nature.
 17830 Number theory doesn't 'reduce' to set theory, because sets have number properties
 Full Idea: I am not suggesting a reduction of number theory to set theory ...There are only sets with number properties; number theory is part of the theory of finite sets. From: Penelope Maddy (Sets and Numbers [1981], V)
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
 17823 If mathematical objects exist, how can we know them, and which objects are they?
 Full Idea: The popular challenges to platonism in philosophy of mathematics are epistemological (how are we able to interact with these objects in appropriate ways) and ontological (if numbers are sets, which sets are they). From: Penelope Maddy (Sets and Numbers [1981], I) A reaction: These objections refer to Benacerraf's two famous papers - 1965 for the ontology, and 1973 for the epistemology. Though he relied too much on causal accounts of knowledge in 1973, I'm with him all the way.
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
 17829 Number words are unusual as adjectives; we don't say 'is five', and numbers always come first
 Full Idea: Number words are not like normal adjectives. For example, number words don't occur in 'is (are)...' contexts except artificially, and they must appear before all other adjectives, and so on. From: Penelope Maddy (Sets and Numbers [1981], IV) A reaction: [She is citing Benacerraf's arguments]