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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
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Pure mathematics is the relations between all possible objects, and is thus formal ontology
[Husserl, by Velarde-Mayol]
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17880
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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
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Mathematics is the formal study of the categorical dimensions of things
[Ellis]
|
17776
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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
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Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
[Mayberry]
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10236
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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
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Reducing real numbers to rationals suggested arithmetic as the foundation of maths
[Colyvan]
|
18846
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Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
[Rumfitt]
|