Combining Texts

Ideas for 'Thinking About Mathematics', 'Human Freedom and the Self' and 'Particle Physics'

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2 ideas

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.