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Ideas for 'Commentary on 'De Anima'', 'Introducing the Philosophy of Mathematics' and 'Science without Numbers'

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33 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
8958 In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
8667 The 'integers' are the positive and negative natural numbers, plus zero [Friend]
8668 The 'rational' numbers are those representable as fractions [Friend]
8670 A number is 'irrational' if it cannot be represented as a fraction [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
8661 The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
8664 Cardinal numbers answer 'how many?', with the order being irrelevant [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
8671 The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
8663 Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
8662 The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
8669 Between any two rational numbers there is an infinite number of rational numbers [Friend]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8676 Is mathematics based on sets, types, categories, models or topology? [Friend]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
18221 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
8678 Most mathematical theories can be translated into the language of set theory [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8701 The number 8 in isolation from the other numbers is of no interest [Friend]
8702 In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
8699 Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
8696 Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]
8695 Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
8700 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
8757 The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
8681 The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
18212 Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
10261 The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro]
18218 Hilbert explains geometry, by non-numerical facts about space [Field,H]
9623 Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
8712 Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
18215 It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
8716 Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
18216 Abstractions can form useful counterparts to concrete statements [Field,H]
18214 Mathematics is only empirical as regards which theory is useful [Field,H]
18210 Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
8706 Constructivism rejects too much mathematics [Friend]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8707 Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]