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Ideas for 'Thinking About Mathematics', 'Three Dialogues of Hylas and Philonous' and 'The Principles of Mathematics'

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12 ideas

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 / 4. Axioms for Number / a. Axioms for numbers
14124 Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
7530 Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
18246 Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14147 Denying mathematical induction gave us the transfinite [Russell]
14125 Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
14116 Numbers were once defined on the basis of 1, but neglected infinities and + [Russell]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
14117 Numbers are properties of classes [Russell]
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 / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9977 Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]