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Ideas for 'Thinking About Mathematics', 'What is an Idea?' and 'Metaphysics'

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29 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
560 Mathematical precision is only possible in immaterial things [Aristotle]
9076 Mathematics studies the domain of perceptible entities, but its subject-matter is not perceptible [Aristotle]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
10958 Perhaps numbers are substances? [Aristotle]
13273 Pluralities divide into discontinous countables; magnitudes divide into continuous things [Aristotle]
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 / A. Nature of Mathematics / 3. Nature of Numbers / m. One
12074 The one in number just is the particular [Aristotle]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
17844 The unit is stipulated to be indivisible [Aristotle]
17845 If only rectilinear figures existed, then unity would be the triangle [Aristotle]
17859 Units came about when the unequals were equalised [Aristotle]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
17861 Two men do not make one thing, as well as themselves [Aristotle]
646 When we count, are we adding, or naming numbers? [Aristotle]
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 / b. Greek arithmetic
17843 The idea of 'one' is the foundation of number [Aristotle]
17850 Each many is just ones, and is measured by the one [Aristotle]
17851 Number is plurality measured by unity [Aristotle]
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
9793 Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
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 / 1. Mathematical Platonism / a. For mathematical platonism
13738 It is a simple truth that the objects of mathematics have being, of some sort [Aristotle]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
12339 Aristotle removes ontology from mathematics, and replaces the true with the beautiful [Aristotle, by Badiou]
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]