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Single Idea 8764
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
]
Full Idea
There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics.
Gist of Idea
Categories are the best foundation for mathematics
Source
Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7)
Book Ref
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.272
A Reaction
He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties.
The
15 ideas
from 'Thinking About Mathematics'
8725
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Rationalism tries to apply mathematical methodology to all of knowledge
[Shapiro]
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8729
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Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
[Shapiro]
|
8730
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'Impredicative' definitions refer to the thing being described
[Shapiro]
|
8731
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Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
[Shapiro]
|
8744
|
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
[Shapiro]
|
8749
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Term Formalism says mathematics is just about symbols - but real numbers have no names
[Shapiro]
|
8750
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Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
[Shapiro]
|
8752
|
Deductivism says mathematics is logical consequences of uninterpreted axioms
[Shapiro]
|
8753
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Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
[Shapiro]
|
8760
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Numbers do not exist independently; the essence of a number is its relations to other numbers
[Shapiro]
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8761
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A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
[Shapiro]
|
8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
[Shapiro]
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8762
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Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
[Shapiro]
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8764
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Categories are the best foundation for mathematics
[Shapiro]
|
18249
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Cauchy gave a formal definition of a converging sequence.
[Shapiro]
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