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Ideas of Philip Kitcher, by Text

[British, b.1947, Studied at Cambridge, then pupil of Thomas Kuhn. Professor at Columbia.]

1984 The Nature of Mathematical Knowledge
p.64 Kitcher says maths is an idealisation of the world, and our operations in dealing with it
Intro p.5 Mathematical knowledge arises from basic perception
Intro p.12 My constructivism is mathematics as an idealization of collecting and ordering objects
01.2 p.17 A 'warrant' is a process which ensures that a true belief is knowledge
01.3 p.22 Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge
01.4 p.24 A priori knowledge comes from available a priori warrants that produce truth
01.6 p.29 We have some self-knowledge a priori, such as knowledge of our own existence
02.2 p.45 In long mathematical proofs we can't remember the original a priori basis
02.3 p.46 Mathematical a priorism is conceptualist, constructivist or realist
03.1 p.50 Kant's intuitions struggle to judge relevance, impossibility and exactness
03.1 p.50 If mathematics comes through intuition, that is either inexplicable, or too subjective
03.2 p.53 Intuition is no basis for securing a priori knowledge, because it is fallible
03.3 p.61 Mathematical intuition is not the type platonism needs
04.1 p.65 Conceptualists say we know mathematics a priori by possessing mathematical concepts
04.6 p.86 If meaning makes mathematics true, you still need to say what the meanings refer to
04.I p.68 Analyticity avoids abstract entities, but can there be truth without reference?
05.2 p.92 We derive limited mathematics from ordinary things, and erect powerful theories on their basis
06.1 p.105 The old view is that mathematics is useful in the world because it describes the world
06.1 p.107 Abstract objects were a bad way of explaining the structure in mathematics
06.2 p.108 Arithmetic is made true by the world, but is also made true by our constructions
06.2 p.109 Arithmetic is an idealizing theory
06.2 p.111 We develop a language for correlations, and use it to perform higher level operations
06.3 p.112 A one-operation is the segregation of a single object
06.4 p.124 Real numbers stand to measurement as natural numbers stand to counting
06.5 p.142 Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability
06.5 p.142 Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori)
06.5 p.143 Intuitionists rely on assertability instead of truth, but assertability relies on truth
06.5 p.144 Idealisation trades off accuracy for simplicity, in varying degrees
07.5 p.176 Complex numbers were only accepted when a geometrical model for them was found
07.5 p.176 The defenders of complex numbers had to show that they could be expressed in physical terms
09.3 p.206 The interest or beauty of mathematics is when it uses current knowledge to advance undestanding
09.4 p.212 The 'beauty' or 'interest' of mathematics is just explanatory power
10.2 p.238 With infinitesimals, you divide by the time, then set the time to zero
2000 A Priori Knowledge Revisited
§II p.69 Knowing our own existence is a priori, but not necessary
§II p.69 Many necessities are inexpressible, and unknowable a priori
§VII p.86 Classical logic is our preconditions for assessing empirical evidence
§VII p.87 I believe classical logic because I was taught it and use it, but it could be undermined