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All the ideas for 'Thinking About Mathematics', 'Against Method' and 'Concepts without Boundaries'

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22 ideas

1. Philosophy / G. Scientific Philosophy / 1. Aims of Science
6601 Science rules the globe because of colonising power, not inherent rationality [Feyerabend]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764 Categories are the best foundation for mathematics [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
8749 Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
8752 Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8731 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
8730 'Impredicative' definitions refer to the thing being described [Shapiro]
7. Existence / D. Theories of Reality / 10. Vagueness / b. Vagueness of reality
8983 If 'red' is vague, then membership of the set of red things is vague, so there is no set of red things [Sainsbury]
7. Existence / E. Categories / 2. Categorisation
8986 We should abandon classifying by pigeon-holes, and classify around paradigms [Sainsbury]
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
8982 Vague concepts are concepts without boundaries [Sainsbury]
8984 If concepts are vague, people avoid boundaries, can't spot them, and don't want them [Sainsbury]
8985 Boundaryless concepts tend to come in pairs, such as child/adult, hot/cold [Sainsbury]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
14. Science / B. Scientific Theories / 6. Theory Holism
2561 For Feyerabend the meaning of a term depends on a whole theory [Feyerabend, by Rorty]