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Single Idea 8707
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
]
Full Idea
An intuitionist typically retains bivalence, but rejects the law of excluded middle.
Gist of Idea
Intuitionists typically retain bivalence but reject the law of excluded middle
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2)
Book Ref
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.106
A Reaction
The idea would be to say that only T and F are available as truth-values, but failing to be T does not ensure being F, but merely not-T. 'Unproven' is not-T, but may not be F.
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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