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Ideas for 'The Sayings of Confucius', 'Knowledge and the Philosophy of Number' and 'Replies on 'Limits of Abstraction''

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4 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 / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
23626 Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack]
     Full Idea: The transfinite ordinal numbers are important in the theory of proofs, and essential in the theory of recursive functions and computability. Mathematics would be incomplete without them.
     From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.1)
     A reaction: Hossack offers this as proof that the numbers are not human conceptual creations, but must exist beyond the range of our intellects. Hm.