Combining Philosophers

Ideas for Chrysippus, Micklethwait,J/Wooldridge,A and Bertrand Russell

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80 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10059 In mathematic we are ignorant of both subject-matter and truth [Russell]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
14151 Pure geometry is deductive, and neutral over what exists [Russell]
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]
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]
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
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]
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
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]
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 / e. Ordinal numbers
14139 Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell]
14142 Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell]
14141 Ordinals are defined through mathematical induction [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 / 3. Nature of Numbers / i. Reals from cuts
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 / b. Quantity
14158 Quantity is not part of mathematics, where it is replaced by order [Russell]
14123 Some quantities can't be measured, and some non-quantities are measurable [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]
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 / e. Counting by correlation
14118 We can define one-to-one without mentioning unity [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
14133 There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
14119 We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
14420 Infinity and continuity used to be philosophy, but are now mathematics [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]
7556 A collection is infinite if you can remove some terms without diminishing its number [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 / 2. Proof in Mathematics
17627 It seems absurd to prove 2+2=4, where the conclusion is more certain than premises [Russell]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
10052 Geometry is united by the intuitive axioms of projective geometry [Russell, by Musgrave]
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]
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
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]
14422 Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
14423 '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14125 Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
14147 Denying mathematical induction gave us the transfinite [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 / 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 / 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 / 4. Mathematical Empiricism / a. Mathematical empiricism
17628 Arithmetic was probably inferred from relationships between physical objects [Russell]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
5399 Maths is not known by induction, because further instances are not needed to support it [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
14103 Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
13414 For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf]
6108 Maths can be deduced from logical axioms and the logic of relations [Russell]
6423 We tried to define all of pure maths using logical premisses and concepts [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]
10418 Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
10047 Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
23478 Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell]
23457 Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell]
21556 Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
21718 Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
21570 Numbers are just verbal conveniences, which can be analysed away [Russell]
6424 Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess [Russell]
6425 Formalism can't apply numbers to reality, so it is an evasion [Russell]
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
6104 Numbers are classes of classes, and hence fictions of fictions [Russell]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
6426 Intuitionism says propositions are only true or false if there is a method of showing it [Russell]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
21559 We need rules for deciding which norms are predicative (unless none of them are) [Russell]
21558 'Predicative' norms are those which define a class [Russell]
18126 A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock]
18128 Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock]
18124 Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock]
21568 A one-variable function is only 'predicative' if it is one order above its arguments [Russell]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
14449 There is always something psychological about inference [Russell]