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Ideas for 'On the Question of Absolute Undecidability', 'Principia Mathematica' and 'Introduction to Mathematical Philosophy'

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30 ideas

6. Mathematics / A. Nature of Mathematics / 2. Geometry
14442 If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
14438 New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
13510 Could a number just be something which occurs in a progression? [Russell, by Hart,WD]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18248 A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro]
14436 A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
14439 A complex number is simply an ordered couple of real numbers [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
14421 Discovering that 1 is a number was difficult [Russell]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
14424 Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
14441 The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
14420 Infinity and continuity used to be philosophy, but are now mathematics [Russell]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
17890 There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
14431 The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
14423 '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
14422 Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
17887 PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17891 Arithmetical undecidability is always settled at the next stage up [Koellner]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / a. Defining numbers
18152 Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Russell/Whitehead, by Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
14425 A number is something which characterises collections of the same size [Russell]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
14434 What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
14465 Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
10025 Russell and Whitehead took arithmetic to be higher-order logic [Russell/Whitehead, by Hodes]
8683 Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend]
13414 For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf]
10037 'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
10093 The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman]
8691 The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
10305 In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
8684 Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Russell/Whitehead, by Friend]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
8746 To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
14449 There is always something psychological about inference [Russell]