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Single Idea 8676
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
]
Full Idea
Successful competing founding disciplines in mathematics include: the various set theories, type theory, category theory, model theory and topology.
Gist of Idea
Is mathematics based on sets, types, categories, models or topology?
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3)
Book Ref
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.29
A Reaction
Or none of the above? Set theories are very popular. Type theory is, apparently, discredited. Shapiro has a version of structuralism based on model theory (which sound promising). Topology is the one that intrigues me...
The
15 ideas
with the same theme
[existence of fundamentals as a basis for mathematics]:
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18271
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We can't prove everything, but we can spell out the unproved, so that foundations are clear
[Frege]
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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]
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17810
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The study of mathematical foundations needs new non-mathematical concepts
[Kreisel]
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9937
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I do not believe mathematics either has or needs 'foundations'
[Putnam]
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12688
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Mathematics is the formal study of the categorical dimensions of things
[Ellis]
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17776
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The ultimate principles and concepts of mathematics are presumed, or grasped directly
[Mayberry]
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17775
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If proof and definition are central, then mathematics needs and possesses foundations
[Mayberry]
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17777
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Foundations need concepts, definition rules, premises, and proof rules
[Mayberry]
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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]
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8764
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Categories are the best foundation for mathematics
[Shapiro]
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8676
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Is mathematics based on sets, types, categories, models or topology?
[Friend]
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17922
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Reducing real numbers to rationals suggested arithmetic as the foundation of maths
[Colyvan]
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18846
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Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
[Rumfitt]
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