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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics

[existence of fundamentals as a basis for mathematics]

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