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Single Idea 18069

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

Full Idea

I construe arithmetic as an idealizing theory.

Gist of Idea

Arithmetic is an idealizing theory

Source

Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.2)

Book Ref

Kitcher,Philip: 'The Nature of Mathematical Knowledge' [OUP 1984], p.109

A Reaction

I find 'generalising' the most helpful word, because everyone seems to understand and accept the idea. 'Idealisation' invokes 'ideals', which lots of people dislike, and lots of philosophers seem to have trouble with 'abstraction'.

The 16 ideas with the same theme [maths is entirely created by the human mind]:

9916 Convention, yes! Arbitrary, no! [Poincaré, by Putnam]
8684 Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Russell/Whitehead, by Friend]
14248 We could accept the integers as primitive, then use sets to construct the rest [Cohen]
15939 For intuitionists it is constructed proofs (which take time) which make statements true [Dummett]
18068 Arithmetic is made true by the world, but is also made true by our constructions [Kitcher]
18070 We develop a language for correlations, and use it to perform higher level operations [Kitcher]
18069 Arithmetic is an idealizing theory [Kitcher]
18072 Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori) [Kitcher]
9222 The objects and truths of mathematics are imperative procedures for their construction [Fine,K]
9223 My Proceduralism has one simple rule, and four complex rules [Fine,K]
10255 Presumably nothing can block a possible dynamic operation? [Shapiro]
10254 Can the ideal constructor also destroy objects? [Shapiro]
10264 Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro]
9608 There are no constructions for many highly desirable results in mathematics [Brown,JR]
9645 Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR]
8706 Constructivism rejects too much mathematics [Friend]