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Single Idea 8691

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory ]

Full Idea

The Russell/Whitehead type theory reduces mathematics to a consistent founding discipline, but is criticised for not really being logic. They could not prove the existence of infinite sets, and introduced a non-logical 'axiom of reducibility'.

Gist of Idea

The Russell/Whitehead type theory was limited, and was not really logic

Source

comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Michèle Friend - Introducing the Philosophy of Mathematics 3.6

Book Ref

Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.70

A Reaction

To have reduced most of mathematics to a founding discipline sounds like quite an achievement, and its failure to be based in pure logic doesn't sound too bad. However, it seems to reduce some maths to just other maths.

The 23 ideas with the same theme [theory that maths is a hierarchy of set types]:

10607 Frege's logic has a hierarchy of object, property, property-of-property etc. [Frege, by Smith,P]
10093 The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman]
8691 The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
21555 For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
18003 In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]
23457 Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell]
21556 Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey]
10418 Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
10047 Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
23478 Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell]
6409 The 'simple theory of types' distinguishes levels among properties [Ramsey, by Grayling]
21557 Russell confused use and mention, and reduced classes to properties, not to language [Quine, by Lackey]
18127 Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
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]
21703 Types are 'ramified' when there are further differences between the type of quantifier and its range [Linsky,B]
21714 The ramified theory subdivides each type, according to the range of the variables [Linsky,B]
21721 Higher types are needed to distinguished intensional phenomena which are coextensive [Linsky,B]
10092 In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
10094 The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
10095 Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
17901 Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
16308 Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach]