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Ideas for 'Metaphysics', 'Humean metaphysics vs metaphysics of Powers' and 'Nature and Meaning of Numbers'

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31 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]
9823 Numbers are free creations of the human mind, to understand differences [Dedekind]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
10090 Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
7524 Order, not quantity, is central to defining numbers [Dedekind, by Monk]
17452 Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
14131 Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
14437 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
18094 Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
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]
17859 Units came about when the unequals were equalised [Aristotle]
17845 If only rectilinear figures existed, then unity would be the triangle [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]
9824 In counting we see the human ability to relate, correspond and represent [Dedekind]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
9826 A system S is said to be infinite when it is similar to a proper part of itself [Dedekind]
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]
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]
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]