Combining Philosophers

Ideas for Reiss,J/Spreger,J, Richard Dedekind and Aristotle

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11 ideas

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
13508 Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
18096 Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
18841 Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14130 Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
17843 The idea of 'one' is the foundation of number [Aristotle]
11041 Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle]
17851 Number is plurality measured by unity [Aristotle]
17850 Each many is just ones, and is measured by the one [Aristotle]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
9793 Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
8924 Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
9153 Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K]