Ideas from 'Thinking About Mathematics' by Stewart Shapiro [2000], by Theme Structure
[found in 'Thinking About Mathematics' by Shapiro,Stewart [OUP 2000,0-19-289306-8]].
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5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
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Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
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Cauchy gave a formal definition of a converging sequence.
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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Categories are the best foundation for mathematics
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
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Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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Numbers do not exist independently; the essence of a number is its relations to other numbers
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A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
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Term Formalism says mathematics is just about symbols - but real numbers have no names
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Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
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8752
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Deductivism says mathematics is logical consequences of uninterpreted axioms
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
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Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
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Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
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'Impredicative' definitions refer to the thing being described
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12. Knowledge Sources / C. Rationalism / 1. Rationalism
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Rationalism tries to apply mathematical methodology to all of knowledge
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