Combining Texts

All the ideas for 'Thinking About Mathematics', 'Protrepticus (frags)' and 'Interview with Philippa Foot'

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25 ideas

1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / d. Philosophy as puzzles
17949 Inquiry is the cause of philosophy [Aristotle]
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]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
18. Thought / A. Modes of Thought / 5. Rationality / b. Human rationality
23438 Full rationality must include morality [Foot]
20. Action / C. Motives for Action / 3. Acting on Reason / a. Practical reason
23437 Practical reason is goodness in choosing actions [Foot]
20. Action / C. Motives for Action / 3. Acting on Reason / c. Reasons as causes
23436 It is an odd Humean view to think a reason to act must always involve caring [Foot]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / d. Biological ethics
23431 Human defects are just like plant or animal defects [Foot]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / k. Ethics from nature
23432 Concepts such as function, welfare, flourishing and interests only apply to living things [Foot]
23433 Humans need courage like a plant needs roots [Foot]
22. Metaethics / B. Value / 1. Nature of Value / b. Fact and value
23434 There is no fact-value gap in 'owls should see in the dark' [Foot]
22. Metaethics / B. Value / 1. Nature of Value / f. Ultimate value
23439 Principles are not ultimate, but arise from the necessities of human life [Foot]
22. Metaethics / B. Value / 2. Values / a. Normativity
23435 If you demonstrate the reason to act, there is no further question of 'why should I?' [Foot]