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Single Idea 10264

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

Full Idea

Chihara has proposal a modal primitive, a 'constructability quantifier'. Syntactically it behaves like an ordinary quantifier: Φ is a formula, and x a variable. Then (Cx)Φ is a formula, read as 'it is possible to construct an x such that Φ'.

Gist of Idea

Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ'

Source

report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.230

A Reaction

We only think natural numbers are infinite because we see no barrier to continuing to count, i.e. to construct new numbers. We accept reals when we know how to construct them. Etc. Sounds promising to me (though not to Shapiro).

The 22 ideas from Charles Chihara

8758 We could talk of open sentences, instead of sets [Chihara, by Shapiro]
10265 Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro]
8759 We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro]
10264 Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro]
18151 Could we replace sets by the open sentences that define them? [Chihara, by Bostock]
10192 We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride]
9547 Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara]
9549 The set theorist cannot tell us what 'membership' is [Chihara]
9551 What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara]
9550 We only know relational facts about the empty set, but nothing intrinsic [Chihara]
9552 Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara]
9553 Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara]
9559 If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara]
9561 The mathematics of relations is entirely covered by ordered pairs [Chihara]
9562 In simple type theory there is a hierarchy of null sets [Chihara]
9563 A pack of wolves doesn't cease when one member dies [Chihara]
9566 No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara]
9568 I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara]
9571 ZFU refers to the physical world, when it talks of 'urelements' [Chihara]
9572 Realists about sets say there exists a null set in the real world, with no members [Chihara]
9573 The null set is a structural position which has no other position in membership relation [Chihara]
9574 'Gunk' is an individual possessing no parts that are atoms [Chihara]