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'fragments/reports', 'Russell's Metaphysical Logic' and 'The Nature of Mathematical Knowledge'
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20 ideas
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
18061
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Mathematical intuition is not the type platonism needs [Kitcher]
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12393
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Intuition is no basis for securing a priori knowledge, because it is fallible [Kitcher]
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12420
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If mathematics comes through intuition, that is either inexplicable, or too subjective [Kitcher]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
12387
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Mathematical knowledge arises from basic perception [Kitcher]
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12412
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My constructivism is mathematics as an idealization of collecting and ordering objects [Kitcher]
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18065
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We derive limited mathematics from ordinary things, and erect powerful theories on their basis [Kitcher]
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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 [Kitcher]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
21723
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The task of logicism was to define by logic the concepts 'number', 'successor' and '0' [Linsky,B]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
21721
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Higher types are needed to distinguished intensional phenomena which are coextensive [Linsky,B]
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21703
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Types are 'ramified' when there are further differences between the type of quantifier and its range [Linsky,B]
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21714
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The ramified theory subdivides each type, according to the range of the variables [Linsky,B]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
12423
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Analyticity avoids abstract entities, but can there be truth without reference? [Kitcher]
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21713
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Did logicism fail, when Russell added three nonlogical axioms, to save mathematics? [Linsky,B]
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21715
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For those who abandon logicism, standard set theory is a rival option [Linsky,B]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
18069
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Arithmetic is an idealizing theory [Kitcher]
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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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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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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
18063
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Conceptualists say we know mathematics a priori by possessing mathematical concepts [Kitcher]
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18064
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If meaning makes mathematics true, you still need to say what the meanings refer to [Kitcher]
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