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Single Idea 8760

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism ]

Full Idea

The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers.

Gist of Idea

Numbers do not exist independently; the essence of a number is its relations to other numbers

Source

Stewart Shapiro (Thinking About Mathematics [2000], 10.1)

Book Ref

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.258

A Reaction

This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate.

The 15 ideas from 'Thinking About Mathematics'

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]