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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 29 ideas from 'works'

13736 Quinean metaphysics just lists the beings, which is a domain with no internal structure [Schaffer,J on Quine]
3302 Set theory is full of Platonist metaphysics, so Quine aimed to keep it separate from logic [Quine, by Benardete,JA]
10211 Quine wants V = L for a cleaner theory, despite the scepticism of most theorists [Quine, by Shapiro]
3336 Two things can never entail three things [Quine, by Benardete,JA]
8453 If we had to name objects to make existence claims, we couldn't discuss all the real numbers [Quine]
10311 No sense can be made of quantification into opaque contexts [Quine, by Hale]
10538 Finite quantification can be eliminated in favour of disjunction and conjunction [Quine, by Dummett]
10793 Quine thought substitutional quantification confused use and mention, but then saw its nominalist appeal [Quine, by Marcus (Barcan)]
3751 Universals are acceptable if they are needed to make an accepted theory true [Quine, by Jacquette]
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]
15782 Quine wants identity and individuation-conditions for possibilia [Quine, by Lycan]
10667 A logically perfect language could express all truths, so all truths must be logically expressible [Quine, by Hossack]
3325 For Quine everything exists theoretically, as reference, predication and quantification [Quine, by Benardete,JA]
16021 Quine says we can expand predicates easily (ideology), but not names (ontology) [Quine, by Noonan]
8534 Quine says the predicate of a true statement has no ontological implications [Quine, by Armstrong]
10295 Quine suggests that properties can be replaced with extensional entities like sets [Quine, by Shapiro]
3322 Quine says that if second-order logic is to quantify over properties, that can be done in first-order predicate logic [Quine, by Benardete,JA]
6078 Quine brought classes into semantics to get rid of properties [Quine, by McGinn]
8479 Don't analyse 'red is a colour' as involving properties. Say 'all red things are coloured things' [Quine, by Orenstein]
15783 Definite descriptions can't unambiguously pick out an object which doesn't exist [Lycan on Quine]
2796 For Quine the only way to know a necessity is empirically [Quine, by Dancy,J]
8450 Quine's empiricism is based on whole theoretical systems, not on single mental events [Quine, by Orenstein]
3868 To proclaim cultural relativism is to thereby rise above it [Quine, by Newton-Smith]
4713 For Quine, theories are instruments used to make predictions about observations [Quine, by O'Grady]
4712 Quine says there is no matter of fact about reference - it is 'inscrutable' [Quine, by O'Grady]
7330 The principle of charity only applies to the logical constants [Quine, by Miller,A]
17862 Essence gives an illusion of understanding [Quine, by Almog]
7970 Quine is committed to sets, but is more a Class Nominalist than a Platonist [Quine, by Macdonald,C]