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Single Idea 8753

[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism ]

Full Idea

Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle.

Gist of Idea

Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions

Source

Stewart Shapiro (Thinking About Mathematics [2000], 7.1)

Book Ref

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.174

A Reaction

The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them.

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]
8466 For Quine, intuitionist ontology is inadequate for classical mathematics [Quine, by Orenstein]
8467 Intuitionists only admit numbers properly constructed, but classical maths covers all reals in a 'limit' [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]