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All the ideas for 'Letters to Remond de Montmort', 'Abstraction Reconsidered' and 'Thinking About Mathematics'

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24 ideas

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 / 4. Mathematical Empiricism / c. Against mathematical empiricism
10735 Abstraction from objects won't reveal an operation's being performed 'so many times' [Geach]
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]
10. Modality / C. Sources of Modality / 2. Necessity as Primitive
12732 Some necessary truths are brute, and others derive from final causes [Leibniz]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
15. Nature of Minds / B. Features of Minds / 1. Consciousness / c. Parts of consciousness
19438 Our large perceptions and appetites are made up tiny unconscious fragments [Leibniz]
15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind
10732 If concepts are just recognitional, then general judgements would be impossible [Geach]
18. Thought / A. Modes of Thought / 3. Emotions / c. Role of emotions
19415 Passions reside in confused perceptions [Leibniz]
18. Thought / D. Concepts / 3. Ontology of Concepts / b. Concepts as abilities
10731 For abstractionists, concepts are capacities to recognise recurrent features of the world [Geach]
18. Thought / E. Abstraction / 8. Abstractionism Critique
10733 The abstractionist cannot explain 'some' and 'not' [Geach]
10734 Only a judgement can distinguish 'striking' from 'being struck' [Geach]
28. God / A. Divine Nature / 2. Divine Nature
19439 God produces possibilities, and thus ideas [Leibniz]