### 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
 18074 Intuitionists rely on assertability instead of truth, but assertability relies on truth
###### 6. Mathematics / A. Nature of Mathematics / 1. Mathematics
 6298 Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Resnik]
 12392 Mathematical a priorism is conceptualist, constructivist or realist
 18078 The interest or beauty of mathematics is when it uses current knowledge to advance undestanding
 12426 The 'beauty' or 'interest' of mathematics is just explanatory power
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
 12395 Real numbers stand to measurement as natural numbers stand to counting
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
 12425 Complex numbers were only accepted when a geometrical model for them was found
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
 18071 A one-operation is the segregation of a single object
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
 18066 The old view is that mathematics is useful in the world because it describes the world
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
 18083 With infinitesimals, you divide by the time, then set the time to zero
###### 6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
 18061 Mathematical intuition is not the type platonism needs
 12420 If mathematics comes through intuition, that is either inexplicable, or too subjective
 12393 Intuition is no basis for securing a priori knowledge, because it is fallible
###### 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
 12387 Mathematical knowledge arises from basic perception
 12412 My constructivism is mathematics as an idealization of collecting and ordering objects
 18065 We derive limited mathematics from ordinary things, and erect powerful theories on their basis
 18077 The defenders of complex numbers had to show that they could be expressed in physical terms
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
 12423 Analyticity avoids abstract entities, but can there be truth without reference?
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
 18069 Arithmetic is an idealizing theory
 18068 Arithmetic is made true by the world, but is also made true by our constructions
 18070 We develop a language for correlations, and use it to perform higher level operations
 18072 Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori)
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
 18063 Conceptualists say we know mathematics a priori by possessing mathematical concepts
 18064 If meaning makes mathematics true, you still need to say what the meanings refer to
###### 9. Objects / A. Existence of Objects / 2. Abstract Objects / b. Need for abstracta
 18067 Abstract objects were a bad way of explaining the structure in mathematics
###### 12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori
 12390 A priori knowledge comes from available a priori warrants that produce truth
###### 12. Knowledge Sources / A. A Priori Knowledge / 6. A Priori from Reason
 12418 In long mathematical proofs we can't remember the original a priori basis
###### 12. Knowledge Sources / A. A Priori Knowledge / 9. A Priori from Concepts
 12389 Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge
###### 12. Knowledge Sources / A. A Priori Knowledge / 10. A Priori as Subjective
 12416 We have some self-knowledge a priori, such as knowledge of our own existence
###### 13. Knowledge Criteria / A. Justification Problems / 1. Justification / a. Justification issues
 12413 A 'warrant' is a process which ensures that a true belief is knowledge
###### 13. Knowledge Criteria / A. Justification Problems / 1. Justification / c. Defeasibility
 20473 If experiential can defeat a belief, then its justification depends on the defeater's absence [Casullo]
###### 15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
 18075 Idealisation trades off accuracy for simplicity, in varying degrees