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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.
Clarification
See under 'Types of Number' for explanations of these terms
Gist of Idea
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
Source
Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
Book Ref
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.267
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.
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |