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Ideas for 'talk', 'Politics' and 'The Principles of Mathematics'

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6. Mathematics / A. Nature of Mathematics / 2. Geometry
14152 In geometry, Kant and idealists aimed at the certainty of the premisses [Russell]
14154 Geometry throws no light on the nature of actual space [Russell]
14151 Pure geometry is deductive, and neutral over what exists [Russell]
14153 In geometry, empiricists aimed at premisses consistent with experience [Russell]
14155 Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
18254 Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett]
14144 Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
14128 Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell]
14129 Ordinals presuppose two relations, where cardinals only presuppose one [Russell]
14132 Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
14141 Ordinals are defined through mathematical induction [Russell]
14142 Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell]
14139 Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell]
14145 For Cantor ordinals are types of order, not numbers [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
14146 We aren't sure if one cardinal number is always bigger than another [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
14135 Real numbers are a class of rational numbers (and so not really numbers at all) [Russell]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
14123 Some quantities can't be measured, and some non-quantities are measurable [Russell]
14158 Quantity is not part of mathematics, where it is replaced by order [Russell]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
14120 Counting explains none of the real problems about the foundations of arithmetic [Russell]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
14118 We can define one-to-one without mentioning unity [Russell]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
14119 We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
14133 There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
14134 Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
14143 ω names the whole series, or the generating relation of the series of ordinal numbers [Russell]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
14138 You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell]
14140 For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
14124 Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
7530 Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
18246 Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14147 Denying mathematical induction gave us the transfinite [Russell]
14125 Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
14116 Numbers were once defined on the basis of 1, but neglected infinities and + [Russell]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
14117 Numbers are properties of classes [Russell]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9977 Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
14162 Mathematics doesn't care whether its entities exist [Russell]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
14103 Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
21555 For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
18003 In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]