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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]:
9875
|
Frege was completing Bolzano's work, of expelling intuition from number theory and analysis
[Frege, by Dummett]
|
6426
|
Intuitionism says propositions are only true or false if there is a method of showing it
[Russell]
|
8728
|
Intuitionist mathematics deduces by introspective construction, and rejects unknown truths
[Brouwer]
|
12453
|
Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness
[Brouwer]
|
12454
|
Intuitionists only accept denumerable sets
[Brouwer]
|
1615
|
Intuitionism says classes are invented, and abstract entities are constructed from specified ingredients
[Quine]
|
8467
|
Intuitionists only admit numbers properly constructed, but classical maths covers all reals in a 'limit'
[Quine, by Orenstein]
|
8466
|
For Quine, intuitionist ontology is inadequate for classical mathematics
[Quine, by Orenstein]
|
10552
|
Intuitionism says that totality of numbers is only potential, but is still determinate
[Dummett]
|
8190
|
Intuitionists rely on the proof of mathematical statements, not their truth
[Dummett]
|
3908
|
If maths contains unprovable truths, then maths cannot be reduced to a set of proofs
[Scruton]
|
9548
|
A mathematical object exists if there is no contradiction in its definition
[Waterfield]
|
8753
|
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
[Shapiro]
|
18788
|
For intuitionists there are not numbers and sets, but processes of counting and collecting
[Mares]
|
10123
|
The intuitionists are the idealists of mathematics
[George/Velleman]
|
10124
|
Gödel's First Theorem suggests there are truths which are independent of proof
[George/Velleman]
|
15928
|
Intuitionism rejects set-theory to found mathematics
[Lavine]
|
8707
|
Intuitionists typically retain bivalence but reject the law of excluded middle
[Friend]
|