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Ideas for 'Metaphysics', '30: Book of Amos' and 'Intro to Gdel's Theorems'

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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]
10599 For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10610 The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P]
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 / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
10619 The truths of arithmetic are just true equations and their universally quantified versions [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10618 All numbers are related to zero by the ancestral of the successor relation [Smith,P]
10608 The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / b. Baby arithmetic
10849 Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P]
10850 Baby Arithmetic is complete, but not very expressive [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / c. Robinson arithmetic
10851 Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P]
10852 Robinson Arithmetic (Q) is not negation complete [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
10068 Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
10603 The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10848 Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
10604 Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]
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 / 7. Mathematical Structuralism / a. Structuralism
9793 Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
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]