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Ideas for 'Conditionals', 'works' and 'The Scope and Language of Science'

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4 ideas

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
3331 If '5' is the set of all sets with five members, that may be circular, and you can know a priori if the set has content [Benardete,JA on Frege]
     Full Idea: There is a suspicion that Frege's definition of 5 (as the set of all sets with 5 members) may be infected with circularity, …and how can we be sure on a priori grounds that 4 and 5 are not both empty sets, and hence identical?
     From: comment on Gottlob Frege (works [1890]) by José A. Benardete - Metaphysics: the logical approach Ch.14
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
8463 Maths can be reduced to logic and set theory [Quine]
     Full Idea: Researches in the foundations of mathematics have made it clear that all of (interpreted) mathematics can be got down to logic and set theory, and the objects needed for mathematics can be got down to the category of classes (and classes of classes..).
     From: Willard Quine (The Scope and Language of Science [1954], §VI)
     A reaction: This I take to be a retreat from pure logicism, presumably influenced by Gödel. So can set theory be reduced to logic? Crispin Wright is the one the study.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
16880 Frege aimed to discover the logical foundations which justify arithmetical judgements [Frege, by Burge]
     Full Idea: Frege saw arithmetical judgements as resting on a foundation of logical principles, and the discovery of this foundation as a discovery of the nature and structure of the justification of arithmetical truths and judgments.
     From: report of Gottlob Frege (works [1890]) by Tyler Burge - Frege on Knowing the Foundations Intro
     A reaction: Burge's point is that the logic justifies the arithmetic, as well as underpinning it.
8689 Eventually Frege tried to found arithmetic in geometry instead of in logic [Frege, by Friend]
     Full Idea: After the problem with Russell's paradox, Frege did not publish for fourteen years, and he then tried to re-found arithmetic in Euclidean geometry, rather than in logic.
     From: report of Gottlob Frege (works [1890], 3.4) by Michčle Friend - Introducing the Philosophy of Mathematics 3.4
     A reaction: I take it that his new road would have led him to modern Structuralism, so I think he was probably on the right lines. Unfortunately Frege had already done enough for one good lifetime.