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4 ideas
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. |
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. |
18086 | Weierstrass eliminated talk of infinitesimals [Weierstrass, by Kitcher] |
Full Idea: Weierstrass effectively eliminated the infinitesimalist language of his predecessors. | |
From: report of Karl Weierstrass (works [1855]) by Philip Kitcher - The Nature of Mathematical Knowledge 10.6 |
18092 | Weierstrass made limits central, but the existence of limits still needed to be proved [Weierstrass, by Bostock] |
Full Idea: After Weierstrass had stressed the importance of limits, one now needed to be able to prove the existence of such limits. | |
From: report of Karl Weierstrass (works [1855]) by David Bostock - Philosophy of Mathematics 4.4 | |
A reaction: The solution to this is found in work on series (going back to Cauchy), and on Dedekind's cuts. |