display all the ideas for this combination of texts
4 ideas
10206 | Modal operators are usually treated as quantifiers [Shapiro] |
Full Idea: It is common now, and throughout the history of philosophy, to interpret modal operators as quantifiers. This is an analysis of modality in terms of ontology. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) |
10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
Full Idea: One version of the Axiom of Choice says that for every set A of nonempty sets, there is a function whose domain is A and whose value, for every a ∈ A, is a member of a. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 1) |
10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
Full Idea: If the Axiom of Choice says we can choose one member from each of a set of non-empty sets and put the chosen elements together in a set, this licenses the constructor to do an infinite amount of choosing. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3) | |
A reaction: This is one reason why the Axiom was originally controversial, and still is for many philosophers. |
10207 | Anti-realists reject set theory [Shapiro] |
Full Idea: Anti-realists reject set theory. | |
From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro) | |
A reaction: That is, anti-realists about mathematical objects. I would have thought that one could accept an account of sets as (say) fictions, which provided interesting models of mathematics etc. |