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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
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The Nature of Mathematical Knowledge
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p.64
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6298
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Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Resnik]
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Intro
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p.5
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12387
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Mathematical knowledge arises from basic perception
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Intro
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p.12
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12412
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My constructivism is mathematics as an idealization of collecting and ordering objects
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01.2
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p.17
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12413
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A 'warrant' is a process which ensures that a true belief is knowledge
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01.3
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p.22
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12389
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Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge
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01.4
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p.24
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12390
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A priori knowledge comes from available a priori warrants that produce truth
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01.6
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p.29
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12416
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We have some self-knowledge a priori, such as knowledge of our own existence
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02.2
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p.45
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12418
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In long mathematical proofs we can't remember the original a priori basis
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02.3
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p.46
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12392
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Mathematical a priorism is conceptualist, constructivist or realist
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03.1
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p.50
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12420
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If mathematics comes through intuition, that is either inexplicable, or too subjective
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03.2
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p.53
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12393
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Intuition is no basis for securing a priori knowledge, because it is fallible
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03.3
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p.61
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18061
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Mathematical intuition is not the type platonism needs
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04.1
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p.65
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18063
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Conceptualists say we know mathematics a priori by possessing mathematical concepts
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04.6
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p.86
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18064
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If meaning makes mathematics true, you still need to say what the meanings refer to
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04.I
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p.68
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12423
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Analyticity avoids abstract entities, but can there be truth without reference?
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05.2
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p.92
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18065
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We derive limited mathematics from ordinary things, and erect powerful theories on their basis
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06.1
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p.105
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18066
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The old view is that mathematics is useful in the world because it describes the world
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06.1
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p.107
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18067
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Abstract objects were a bad way of explaining the structure in mathematics
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06.2
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p.108
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18068
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Arithmetic is made true by the world, but is also made true by our constructions
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06.2
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p.109
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18069
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Arithmetic is an idealizing theory
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06.2
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p.111
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18070
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We develop a language for correlations, and use it to perform higher level operations
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06.3
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p.112
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18071
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A one-operation is the segregation of a single object
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06.4
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p.124
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12395
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Real numbers stand to measurement as natural numbers stand to counting
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06.5
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p.142
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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)
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06.5
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p.143
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18074
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Intuitionists rely on assertability instead of truth, but assertability relies on truth
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06.5
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p.144
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18075
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Idealisation trades off accuracy for simplicity, in varying degrees
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07.5
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p.176
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12425
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Complex numbers were only accepted when a geometrical model for them was found
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07.5
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p.176
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18077
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The defenders of complex numbers had to show that they could be expressed in physical terms
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09.3
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p.206
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18078
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The interest or beauty of mathematics is when it uses current knowledge to advance undestanding
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09.4
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p.212
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12426
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The 'beauty' or 'interest' of mathematics is just explanatory power
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10.2
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p.238
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18083
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With infinitesimals, you divide by the time, then set the time to zero
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p.89
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p.105
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20473
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If experiential can defeat a belief, then its justification depends on the defeater's absence [Casullo]
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2000
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A Priori Knowledge Revisited
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§II
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p.69
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12428
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Many necessities are inexpressible, and unknowable a priori
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§II
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p.69
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12429
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Knowing our own existence is a priori, but not necessary
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§VII
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p.86
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12430
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Classical logic is our preconditions for assessing empirical evidence
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§VII
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p.87
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12431
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I believe classical logic because I was taught it and use it, but it could be undermined
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