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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

[maths is built from intuitions and proofs]

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