Combining Philosophers

Ideas for Lynch,MP/Glasgow,JM, Peter Smith and Plato

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23 ideas

6. Mathematics / A. Nature of Mathematics / 2. Geometry
8726 Geometry can lead the mind upwards to truth and philosophy [Plato]
9867 It is absurd to define a circle, but not be able to recognise a real one [Plato]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
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 / 4. Using Numbers / f. Arithmetic
10619 The truths of arithmetic are just true equations and their universally quantified versions [Smith,P]
13155 If you add one to one, which one becomes two, or do they both become two? [Plato]
9865 Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10608 The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
10618 All numbers are related to zero by the ancestral of the successor relation [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / b. Baby arithmetic
10850 Baby Arithmetic is complete, but not very expressive [Smith,P]
10849 Baby arithmetic covers addition and multiplication, but no general facts about numbers [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
10604 Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]
10848 Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
9863 We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato]
9864 In pure numbers, all ones are equal, with no internal parts [Plato]
8727 Geometry is not an activity, but the study of unchanging knowledge [Plato]
10216 We master arithmetic by knowing all the numbers in our soul [Plato]
16150 One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
9861 The same thing is both one and an unlimited number at the same time [Plato]