Combining Texts

Ideas for 'Unconscious Cerebral Initiative', 'works' and 'An Axiomatization of Set Theory'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

---

2 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy

18247	Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro]
	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.
	From: report of Luitzen E.J. Brouwer (works [1930]) by Stewart Shapiro - Philosophy of Mathematics 6.6
	A reaction: This is the 'constructivist' view of numbers, as espoused by intuitionists like Brouwer.

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

13672	All the axioms for mathematics presuppose set theory [Neumann]
	Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory.
	From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2
	A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this.