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Ideas from 'The Principles of Mathematics' by Bertrand Russell [1903], by Theme Structure

[found in 'Principles of Mathematics' by Russell,Bertrand [Routledge 1992,978-0-415-08299-0]].

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1. Philosophy / F. Analytic Philosophy / 1. Nature of Analysis
14122 Analysis gives us nothing but the truth - but never the whole truth
1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
14109 The study of grammar is underestimated in philosophy
1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
14165 Analysis falsifies, if when the parts are broken down they are not equivalent to their sum
2. Reason / D. Definition / 13. Against Definition
14115 Definition by analysis into constituents is useless, because it neglects the whole
14159 In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives
2. Reason / F. Fallacies / 2. Infinite Regress
14148 Infinite regresses have propositions made of propositions etc, with the key term reappearing
2. Reason / F. Fallacies / 8. Category Mistake / a. Category mistakes
18002 As well as a truth value, propositions have a range of significance for their variables
3. Truth / A. Truth Problems / 5. Truth Bearers
14102 What is true or false is not mental, and is best called 'propositions'
3. Truth / H. Deflationary Truth / 1. Redundant Truth
14176 "The death of Caesar is true" is not the same proposition as "Caesar died"
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
14113 The null class is a fiction
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
15894 Russell invented the naïve set theory usually attributed to Cantor [Lavine]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
14126 Order rests on 'between' and 'separation'
14127 Order depends on transitive asymmetrical relations
4. Formal Logic / G. Formal Mereology / 1. Mereology
14121 The part-whole relation is ultimate and indefinable
5. Theory of Logic / B. Logical Consequence / 8. Material Implication
14106 Implication cannot be defined
14108 It would be circular to use 'if' and 'then' to define material implication
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
14167 The only classes are things, predicates and relations
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
14105 There seem to be eight or nine logical constants
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / c. not
18722 Negations are not just reversals of truth-value, since that can happen without negation [Wittgenstein]
5. Theory of Logic / E. Structures of Logic / 3. Constants in Logic
14104 Constants are absolutely definite and unambiguous
5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
14114 Variables don't stand alone, but exist as parts of propositional functions
5. Theory of Logic / G. Quantification / 1. Quantification
14137 'Any' is better than 'all' where infinite classes are concerned
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / a. Achilles paradox
14149 The Achilles Paradox concerns the one-one correlation of infinite classes
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
15895 Russell discovered the paradox suggested by Burali-Forti's work [Lavine]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
14151 Pure geometry is deductive, and neutral over what exists
14152 In geometry, Kant and idealists aimed at the certainty of the premisses
14154 Geometry throws no light on the nature of actual space
14153 In geometry, empiricists aimed at premisses consistent with experience
14155 Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [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 [Dummett]
14144 Ordinals result from likeness among relations, as cardinals from similarity among classes
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
14129 Ordinals presuppose two relations, where cardinals only presuppose one
14132 Properties of numbers don't rely on progressions, so cardinals may be more basic
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
14142 Ordinals are types of series of terms in a row, rather than the 'nth' instance
14141 Ordinals are defined through mathematical induction
14145 For Cantor ordinals are types of order, not numbers
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
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)
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
14123 Some quantities can't be measured, and some non-quantities are measurable
14158 Quantity is not part of mathematics, where it is replaced by order
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
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
14118 We can define one-to-one without mentioning unity
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)
14119 We do not currently know whether, of two infinite numbers, one must be greater than the other
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
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
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
14140 For every transfinite cardinal there is an infinite collection of transfinite ordinals
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
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' [Monk]
18246 Dedekind failed to distinguish the numbers from other progressions [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14125 Finite numbers, unlike infinite numbers, obey mathematical induction
14147 Denying mathematical induction gave us the transfinite
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 +
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
14117 Numbers are properties of classes
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
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
14162 Mathematics doesn't care whether its entities exist
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'
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
18003 In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Magidor]
7. Existence / A. Nature of Existence / 3. Being / a. Nature of Being
11010 Being is what belongs to every possible object of thought
7. Existence / A. Nature of Existence / 3. Being / b. Being and existence
14161 Many things have being (as topics of propositions), but may not have actual existence
7. Existence / A. Nature of Existence / 6. Criterion for Existence
14173 What exists has causal relations, but non-existent things may also have them
7. Existence / E. Categories / 3. Proposed Categories
14163 Four classes of terms: instants, points, terms at instants only, and terms at instants and points
8. Modes of Existence / A. Relations / 1. Nature of Relations
21341 Philosophers of logic and maths insisted that a vocabulary of relations was essential [Heil]
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
10586 'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness
8. Modes of Existence / A. Relations / 4. Formal Relations / b. Equivalence relation
10585 Symmetrical and transitive relations are formally like equality
9. Objects / A. Existence of Objects / 3. Objects in Thought
7781 I call an object of thought a 'term'. This is a wide concept implying unity and existence.
9. Objects / A. Existence of Objects / 5. Simples
14166 Unities are only in propositions or concepts, and nothing that exists has unity
9. Objects / B. Unity of Objects / 1. Unifying an Object / a. Intrinsic unification
14164 The only unities are simples, or wholes composed of parts
9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
14112 A set has some sort of unity, but not enough to be a 'whole'
9. Objects / D. Essence of Objects / 15. Against Essentialism
14170 Change is obscured by substance, a thing's nature, subject-predicate form, and by essences
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
14107 Terms are identical if they belong to all the same classes
11849 It at least makes sense to say two objects have all their properties in common [Wittgenstein]
10. Modality / B. Possibility / 9. Counterfactuals
22303 It makes no sense to say that a true proposition could have been false
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
10583 Abstraction principles identify a common property, which is some third term with the right relation
10582 The principle of Abstraction says a symmetrical, transitive relation analyses into an identity
10584 A certain type of property occurs if and only if there is an equivalence relation
19. Language / D. Propositions / 1. Propositions
14110 Proposition contain entities indicated by words, rather than the words themselves
19. Language / D. Propositions / 3. Concrete Propositions
19164 If propositions are facts, then false and true propositions are indistinguishable [Davidson]
19. Language / D. Propositions / 5. Unity of Propositions
19157 Russell said the proposition must explain its own unity - or else objective truth is impossible [Davidson]
14111 A proposition is a unity, and analysis destroys it
26. Natural Theory / C. Causation / 7. Eliminating causation
14175 We can drop 'cause', and just make inferences between facts
14172 Moments and points seem to imply other moments and points, but don't cause them
26. Natural Theory / D. Laws of Nature / 11. Against Laws of Nature
14174 The laws of motion and gravitation are just parts of the definition of a kind of matter
27. Natural Reality / A. Classical Physics / 1. Mechanics / a. Explaining movement
14168 Occupying a place and change are prior to motion, so motion is just occupying places at continuous times
27. Natural Reality / A. Classical Physics / 1. Mechanics / c. Forces
14171 Force is supposed to cause acceleration, but acceleration is a mathematical fiction
27. Natural Reality / C. Space / 3. Points in Space
14160 Space is the extension of 'point', and aggregates of points seem necessary for geometry
27. Natural Reality / D. Time / 3. Parts of Time / b. Instants
14156 Mathematicians don't distinguish between instants of time and points on a line
27. Natural Reality / E. Cosmology / 1. Cosmology
14169 The 'universe' can mean what exists now, what always has or will exist