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Ideas for 'Thinking About Mathematics', 'The Logic of Decision' and 'Mathematics and the Metaphysicians'

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4 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10059 In mathematic we are ignorant of both subject-matter and truth [Russell]
     Full Idea: Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
     From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.76)
     A reaction: A famous remark, though Musgrave is rather disparaging about Russell's underlying reasoning here.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
     Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures.
     From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
     A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]
     Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε.
     From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4)
     A reaction: The sequence is 'Cauchy' if N exists.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
7556 A collection is infinite if you can remove some terms without diminishing its number [Russell]
     Full Idea: A collection of terms is infinite if it contains as parts other collections which have as many terms as it has; that is, you can take away some terms of the collection without diminishing its number; there are as many even numbers as numbers all together.
     From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.86)
     A reaction: He cites Dedekind and Cantor as source for these ideas. If it won't obey the rule that subtraction makes it smaller, then it clearly isn't a number, and really it should be banned from all mathematics.