Ideas from 'The Nature of Mathematical Knowledge' by Philip Kitcher [1984], by Theme Structure
[found in 'The Nature of Mathematical Knowledge' by Kitcher,Philip [OUP 1984,0-19-503541-0]].
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4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
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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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6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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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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12392
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Mathematical a priorism is conceptualist, constructivist or realist
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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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12426
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The 'beauty' or 'interest' of mathematics is just explanatory power
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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12395
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Real numbers stand to measurement as natural numbers stand to counting
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
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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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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
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18071
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A one-operation is the segregation of a single object
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
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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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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
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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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6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
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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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18061
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Mathematical intuition is not the type platonism needs
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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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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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12387
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Mathematical knowledge arises from basic perception
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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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18065
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We derive limited mathematics from ordinary things, and erect powerful theories on their basis
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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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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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12423
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Analyticity avoids abstract entities, but can there be truth without reference?
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
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18069
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Arithmetic is an idealizing theory
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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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18070
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We develop a language for correlations, and use it to perform higher level operations
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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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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
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18063
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Conceptualists say we know mathematics a priori by possessing mathematical concepts
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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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9. Objects / A. Existence of Objects / 2. Abstract Objects / b. Need for abstracta
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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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12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori
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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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12. Knowledge Sources / A. A Priori Knowledge / 6. A Priori from Reason
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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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12. Knowledge Sources / A. A Priori Knowledge / 9. A Priori from Concepts
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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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12. Knowledge Sources / A. A Priori Knowledge / 10. A Priori as Subjective
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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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13. Knowledge Criteria / A. Justification Problems / 1. Justification / a. Justification issues
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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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13. Knowledge Criteria / A. Justification Problems / 1. Justification / c. Defeasibility
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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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15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
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18075
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Idealisation trades off accuracy for simplicity, in varying degrees
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