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Single Idea 18247

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy ]

Full Idea

In his early writing, Brouwer took a real number to be a Cauchy sequence determined by a rule. Later he augmented rule-governed sequences with free-choice sequences, but even then the attitude is that Cauchy sequences are potential, not actual infinities.

Gist of Idea

Brouwer saw reals as potential, not actual, and produced by a rule, or a choice

Source

report of Luitzen E.J. Brouwer (works [1930]) by Stewart Shapiro - Philosophy of Mathematics 6.6

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.200

A Reaction

This is the 'constructivist' view of numbers, as espoused by intuitionists like Brouwer.

The 5 ideas with the same theme [defining real numbers using Cauchy sequences]:

15903 A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
18251 Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
18247 Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro]
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]
18250 Cauchy gave a necessary condition for the convergence of a sequence [Lavine]