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Single Idea 18846
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
]
Full Idea
One of the motivations behind Cantor's and Dedekind's pioneering explorations in the field was the ambition to give real analysis a new foundation in set theory - and hence a foundation independent of geometry.
Gist of Idea
Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
Source
Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6)
Book Ref
Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.298
A Reaction
Rumfitt is inclined to think that the project has failed, although a weaker set theory than ZF might do the job (within limits).
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]
|