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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 6298 Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Resnik]
Intro p.5 12387 Mathematical knowledge arises from basic perception
Intro p.12 12412 My constructivism is mathematics as an idealization of collecting and ordering objects
01.2 p.17 12413 A 'warrant' is a process which ensures that a true belief is knowledge
01.3 p.22 12389 Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge
01.4 p.24 12390 A priori knowledge comes from available a priori warrants that produce truth
01.6 p.29 12416 We have some self-knowledge a priori, such as knowledge of our own existence
02.2 p.45 12418 In long mathematical proofs we can't remember the original a priori basis
02.3 p.46 12392 Mathematical a priorism is conceptualist, constructivist or realist
03.1 p.50 12420 If mathematics comes through intuition, that is either inexplicable, or too subjective
03.2 p.53 12393 Intuition is no basis for securing a priori knowledge, because it is fallible
03.3 p.61 18061 Mathematical intuition is not the type platonism needs
04.1 p.65 18063 Conceptualists say we know mathematics a priori by possessing mathematical concepts
04.6 p.86 18064 If meaning makes mathematics true, you still need to say what the meanings refer to
04.I p.68 12423 Analyticity avoids abstract entities, but can there be truth without reference?
05.2 p.92 18065 We derive limited mathematics from ordinary things, and erect powerful theories on their basis
06.1 p.105 18066 The old view is that mathematics is useful in the world because it describes the world
06.1 p.107 18067 Abstract objects were a bad way of explaining the structure in mathematics
06.2 p.108 18068 Arithmetic is made true by the world, but is also made true by our constructions
06.2 p.109 18069 Arithmetic is an idealizing theory
06.2 p.111 18070 We develop a language for correlations, and use it to perform higher level operations
06.3 p.112 18071 A one-operation is the segregation of a single object
06.4 p.124 12395 Real numbers stand to measurement as natural numbers stand to counting
06.5 p.142 18072 Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori)
06.5 p.143 18074 Intuitionists rely on assertability instead of truth, but assertability relies on truth
06.5 p.144 18075 Idealisation trades off accuracy for simplicity, in varying degrees
07.5 p.176 12425 Complex numbers were only accepted when a geometrical model for them was found
07.5 p.176 18077 The defenders of complex numbers had to show that they could be expressed in physical terms
09.3 p.206 18078 The interest or beauty of mathematics is when it uses current knowledge to advance undestanding
09.4 p.212 12426 The 'beauty' or 'interest' of mathematics is just explanatory power
10.2 p.238 18083 With infinitesimals, you divide by the time, then set the time to zero
p.89 p.105 20473 If experiential can defeat a belief, then its justification depends on the defeater's absence [Casullo]
2000 A Priori Knowledge Revisited
§II p.69 12428 Many necessities are inexpressible, and unknowable a priori
§II p.69 12429 Knowing our own existence is a priori, but not necessary
§VII p.86 12430 Classical logic is our preconditions for assessing empirical evidence
§VII p.87 12431 I believe classical logic because I was taught it and use it, but it could be undermined