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Single Idea 8467
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
]
Full Idea
Intuitionists will not admit any numbers which are not properly constructed out of rational numbers, ...but classical mathematics appeals to the real numbers (a non-denumerable totality) in notions such as that of a limit
Gist of Idea
Intuitionists only admit numbers properly constructed, but classical maths covers all reals in a 'limit'
Source
report of Willard Quine (works [1961]) by Alex Orenstein - W.V. Quine Ch.3
Book Ref
Orenstein,Alex: 'W.V. Quine' [Princeton 2002], p.57
A Reaction
(See Idea 8454 for the categories of numbers). This is a problem for Dummett.
Related Idea
Idea 8454
The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc. [Orenstein]
The
18 ideas
with the same theme
[maths is built from intuitions and proofs]:
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9875
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Frege was completing Bolzano's work, of expelling intuition from number theory and analysis
[Frege, by Dummett]
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6426
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Intuitionism says propositions are only true or false if there is a method of showing it
[Russell]
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8728
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Intuitionist mathematics deduces by introspective construction, and rejects unknown truths
[Brouwer]
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12453
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Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness
[Brouwer]
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12454
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Intuitionists only accept denumerable sets
[Brouwer]
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1615
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Intuitionism says classes are invented, and abstract entities are constructed from specified ingredients
[Quine]
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8466
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For Quine, intuitionist ontology is inadequate for classical mathematics
[Quine, by Orenstein]
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8467
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Intuitionists only admit numbers properly constructed, but classical maths covers all reals in a 'limit'
[Quine, by Orenstein]
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10552
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Intuitionism says that totality of numbers is only potential, but is still determinate
[Dummett]
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8190
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Intuitionists rely on the proof of mathematical statements, not their truth
[Dummett]
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3908
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If maths contains unprovable truths, then maths cannot be reduced to a set of proofs
[Scruton]
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9548
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A mathematical object exists if there is no contradiction in its definition
[Waterfield]
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8753
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Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
[Shapiro]
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18788
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For intuitionists there are not numbers and sets, but processes of counting and collecting
[Mares]
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10123
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The intuitionists are the idealists of mathematics
[George/Velleman]
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10124
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Gödel's First Theorem suggests there are truths which are independent of proof
[George/Velleman]
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15928
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Intuitionism rejects set-theory to found mathematics
[Lavine]
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8707
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Intuitionists typically retain bivalence but reject the law of excluded middle
[Friend]
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