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Single Idea 9608
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
]
Full Idea
Constuctivists link truth with constructive proof, but necessarily lack constructions for many highly desirable results of classical mathematics, making their account of mathematical truth rather implausible.
Gist of Idea
There are no constructions for many highly desirable results in mathematics
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.12
A Reaction
The tricky word here is 'desirable', which is an odd criterion for mathematical truth. Nevertheless this sounds like a good objection. How flexible might the concept of a 'construction' be?
The
16 ideas
with the same theme
[maths is entirely created by the human mind]:
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9916
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Convention, yes! Arbitrary, no!
[Poincaré, by Putnam]
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8684
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Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality
[Russell/Whitehead, by Friend]
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14248
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We could accept the integers as primitive, then use sets to construct the rest
[Cohen]
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15939
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For intuitionists it is constructed proofs (which take time) which make statements true
[Dummett]
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18068
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Arithmetic is made true by the world, but is also made true by our constructions
[Kitcher]
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18069
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Arithmetic is an idealizing theory
[Kitcher]
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18070
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We develop a language for correlations, and use it to perform higher level operations
[Kitcher]
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18072
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Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori)
[Kitcher]
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9222
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The objects and truths of mathematics are imperative procedures for their construction
[Fine,K]
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9223
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My Proceduralism has one simple rule, and four complex rules
[Fine,K]
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10255
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Presumably nothing can block a possible dynamic operation?
[Shapiro]
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10254
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Can the ideal constructor also destroy objects?
[Shapiro]
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10264
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Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ'
[Chihara, by Shapiro]
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9608
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There are no constructions for many highly desirable results in mathematics
[Brown,JR]
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9645
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Constructivists say p has no value, if the value depends on Goldbach's Conjecture
[Brown,JR]
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8706
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Constructivism rejects too much mathematics
[Friend]
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