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