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All the ideas for 'Thinking About Mathematics', 'Capitalism and Community' and 'The Problem of Knowledge'

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26 ideas

2. Reason / F. Fallacies / 1. Fallacy
19463 Induction assumes some uniformity in nature, or that in some respects the future is like the past [Ayer]
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]
11. Knowledge Aims / B. Certain Knowledge / 5. Cogito Critique
19461 Knowing I exist reveals nothing at all about my nature [Ayer]
19459 To say 'I am not thinking' must be false, but it might have been true, so it isn't self-contradictory [Ayer]
19460 'I know I exist' has no counterevidence, so it may be meaningless [Ayer]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
14. Science / A. Basis of Science / 6. Falsification
19464 We only discard a hypothesis after one failure if it appears likely to keep on failing [Ayer]
14. Science / C. Induction / 2. Aims of Induction
19462 Induction passes from particular facts to other particulars, or to general laws, non-deductively [Ayer]
24. Political Theory / D. Ideologies / 11. Capitalism
22252 Capitalism may actually be the best way to foster community [Conway,D]
22253 Capitalism is just the market, with optional limited government, and perhaps democracy [Conway,D]
22256 Capitalism prefers representative democracy, which avoids community decision-making [Conway,D]
22255 Capitalism breaks up extended families, and must then provide welfare for the lonely people [Conway,D]
22257 Capitalism is anti-community, by only valuing individuals, and breaking up families [Conway,D]