Combining Texts

Ideas for 'Parmenides', 'The Reasons of Love' and 'Replies on 'Limits of Abstraction''

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6 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
10573 Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K]
     Full Idea: Because of Dedekind's definition of reals by cuts, there is a bizarre modern doctrine that there are many 1's - the natural number 1, the rational number 1, the real number 1, and even the complex number 1.
     From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
     A reaction: See Idea 10572.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
10575 Why should a Dedekind cut correspond to a number? [Fine,K]
     Full Idea: By what right can Dedekind suppose that there is a number corresponding to any pair of irrationals that constitute an irrational cut?
     From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
10574 Unless we know whether 0 is identical with the null set, we create confusions [Fine,K]
     Full Idea: What is the union of the singleton {0}, of zero, and the singleton {φ}, of the null set? Is it the one-element set {0}, or the two-element set {0, φ}? Unless the question of identity between 0 and φ is resolved, we cannot say.
     From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
10560 Set-theoretic imperialists think sets can represent every mathematical object [Fine,K]
     Full Idea: Set-theoretic imperialists think that it must be possible to represent every mathematical object as a set.
     From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
16150 One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
     Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
     From: Plato (Parmenides [c.364 BCE], 144a)
     A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
10568 Logicists say mathematics can be derived from definitions, and can be known that way [Fine,K]
     Full Idea: Logicists traditionally claim that the theorems of mathematics can be derived by logical means from the relevant definitions of the terms, and that these theorems are epistemically innocent (knowable without Kantian intuition or empirical confirmation).
     From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)