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
Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability
Intuitionists rely on assertability instead of truth, but assertability relies on truth
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Kitcher says maths is an idealisation of the world, and our operations in dealing with it
Mathematical a priorism is conceptualist, constructivist or realist
The interest or beauty of mathematics is when it uses current knowledge to advance undestanding
The 'beauty' or 'interest' of mathematics is just explanatory power
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Real numbers stand to measurement as natural numbers stand to counting
6. Mathematics / A. Nature of Mathematics / 3. Numbers / j. Complex numbers
Complex numbers were only accepted when a geometrical model for them was found
6. Mathematics / A. Nature of Mathematics / 3. Numbers / o. Units
A one-operation is the segregation of a single object
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / l. Infinitesimals
With infinitesimals, you divide by the time, then set the time to zero
6. Mathematics / A. Nature of Mathematics / 7. Application of Mathematics
The old view is that mathematics is useful in the world because it describes the world
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
Kant's intuitions struggle to judge relevance, impossibility and exactness
If mathematics comes through intuition, that is either inexplicable, or too subjective
Intuition is no basis for securing a priori knowledge, because it is fallible
Mathematical intuition is not the type platonism needs
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Mathematical knowledge arises from basic perception
My constructivism is mathematics as an idealization of collecting and ordering objects
We derive limited mathematics from ordinary things, and erect powerful theories on their basis
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
Analyticity avoids abstract entities, but can there be truth without reference?
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Arithmetic is made true by the world, but is also made true by our constructions
Arithmetic is an idealizing theory
We develop a language for correlations, and use it to perform higher level operations
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
If meaning makes mathematics true, you still need to say what the meanings refer to
Conceptualists say we know mathematics a priori by possessing mathematical concepts
9. Objects / A. Existence of Objects / 2. Abstract Objects / b. Need for abstracta
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
A priori knowledge comes from available a priori warrants that produce truth
12. Knowledge Sources / A. A Priori Knowledge / 6. A Priori from Reason
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
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
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
A 'warrant' is a process which ensures that a true belief is knowledge
15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
Idealisation trades off accuracy for simplicity, in varying degrees