display all the ideas for this combination of texts
5 ideas
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. |
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) |
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) |
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) |
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) |