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Single Idea 12453
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
]
Full Idea
Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness.
Gist of Idea
Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness
Source
Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.80
A Reaction
[compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals.
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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