more from this thinker     |     more from this text

Single Idea 10236

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

Full Idea

We cannot ground mathematics in any domain or theory that is more secure than mathematics itself.

Gist of Idea

There is no grounding for mathematics that is more secure than mathematics

Source

Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.135

A Reaction

This pronouncement comes after a hundred years of hard work, notably by Gödel, so we'd better believe it. It might explain why Putnam rejects the idea that mathematics needs 'foundations'. Personally I'm prepare to found it in countable objects.

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]