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Single Idea 9937

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics ]

Full Idea

I do not believe mathematics either has or needs 'foundations'.

Gist of Idea

I do not believe mathematics either has or needs 'foundations'

Source

Hilary Putnam (Mathematics without Foundations [1967])

Book Ref

'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.295

A Reaction

Agreed that mathematics can function well without foundations (given that the enterprise got started with no thought for such things), the ontology of the subject still strikes me as a major question, though maybe not for mathematicians.

The 15 ideas with the same theme [existence of fundamentals as a basis for mathematics]:

18271 We can't prove everything, but we can spell out the unproved, so that foundations are clear [Frege]
21224 Pure mathematics is the relations between all possible objects, and is thus formal ontology [Husserl, by Velarde-Mayol]
17880 Integers and induction are clear as foundations, but set-theory axioms certainly aren't [Skolem]
17810 The study of mathematical foundations needs new non-mathematical concepts [Kreisel]
9937 I do not believe mathematics either has or needs 'foundations' [Putnam]
12688 Mathematics is the formal study of the categorical dimensions of things [Ellis]
17776 The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
17775 If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
17777 Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
17804 Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
10236 There is no grounding for mathematics that is more secure than mathematics [Shapiro]
8764 Categories are the best foundation for mathematics [Shapiro]
8676 Is mathematics based on sets, types, categories, models or topology? [Friend]
17922 Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan]
18846 Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry) [Rumfitt]