Ideas from 'Introduction to Mathematical Philosophy' by Bertrand Russell [1919], by Theme Structure
[found in 'Introduction to Mathematical Philosophy' by Russell,Bertrand [George Allen and Unwin 1975,0-04-510020-9]].
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1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
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14456
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'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity
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2. Reason / D. Definition / 3. Types of Definition
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14426
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A definition by 'extension' enumerates items, and one by 'intension' gives a defining property
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2. Reason / F. Fallacies / 9. Category Mistake / a. Category mistakes
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8468
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The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false [Orenstein]
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3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
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14454
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An argument 'satisfies' a function φx if φa is true
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4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
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14453
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The Darapti syllogism is fallacious: All M is S, all M is P, so some S is P' - but if there is no M?
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
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14427
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We can enumerate finite classes, but an intensional definition is needed for infinite classes
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
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14428
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Members define a unique class, whereas defining characteristics are numerous
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
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14440
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We may assume that there are infinite collections, as there is no logical reason against them
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14447
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Infinity says 'for any inductive cardinal, there is a class having that many terms'
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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14443
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The British parliament has one representative selected from each constituency
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14444
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Choice is equivalent to the proposition that every class is well-ordered
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14445
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Choice shows that if any two cardinals are not equal, one must be the greater
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14446
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We can pick all the right or left boots, but socks need Choice to insure the representative class
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
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14459
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Reducibility: a family of functions is equivalent to a single type of function
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
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14461
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Propositions about classes can be reduced to propositions about their defining functions
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4. Formal Logic / F. Set Theory ST / 7. Natural Sets
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8469
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Russell's proposal was that only meaningful predicates have sets as their extensions [Orenstein]
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4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
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8745
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Classes are logical fictions, and are not part of the ultimate furniture of the world
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5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
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14452
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All the propositions of logic are completely general
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5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
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14462
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In modern times, logic has become mathematical, and mathematics has become logical
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5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
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10057
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Logic can only assert hypothetical existence
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12444
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Logic is concerned with the real world just as truly as zoology
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14464
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Logic can be known a priori, without study of the actual world
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5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
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14458
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Asking 'Did Homer exist?' is employing an abbreviated description
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10450
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Russell admitted that even names could also be used as descriptions [Bach]
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14457
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Names are really descriptions, except for a few words like 'this' and 'that'
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5. Theory of Logic / F. Referring in Logic / 1. Naming / f. Names eliminated
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7311
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The only genuine proper names are 'this' and 'that'
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5. Theory of Logic / F. Referring in Logic / 2. Descriptions / a. Descriptions
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14455
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'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not
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6. Mathematics / A. Nature of Mathematics / 2. Geometry
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14442
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If straight lines were like ratios they might intersect at a 'gap', and have no point in common
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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14438
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New numbers solve problems: negatives for subtraction, fractions for division, complex for equations
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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13510
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Could a number just be something which occurs in a progression? [Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
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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
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
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14439
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A complex number is simply an ordered couple of real numbers
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
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14421
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Discovering that 1 is a number was difficult
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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14424
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Numbers are needed for counting, so they need a meaning, and not just formal properties
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
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14441
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The formal laws of arithmetic are the Commutative, the Associative and the Distributive
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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14420
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Infinity and continuity used to be philosophy, but are now mathematics
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
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14431
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The definition of order needs a transitive relation, to leap over infinite intermediate terms
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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14423
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'0', 'number' and 'successor' cannot be defined by Peano's axioms
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14422
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Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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14425
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A number is something which characterises collections of the same size
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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14434
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What matters is the logical interrelation of mathematical terms, not their intrinsic nature
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
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14465
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Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men'
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
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13414
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For Russell, numbers are sets of equivalent sets [Benacerraf]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
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14449
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There is always something psychological about inference
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7. Existence / A. Nature of Existence / 1. Nature of Existence
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14463
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Existence can only be asserted of something described, not of something named
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7. Existence / D. Theories of Reality / 7. Fictionalism
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14429
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Classes are logical fictions, made from defining characteristics
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8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
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14430
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If a relation is symmetrical and transitive, it has to be reflexive
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14432
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'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a
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9. Objects / D. Essence of Objects / 3. Individual Essences
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14435
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The essence of individuality is beyond description, and hence irrelevant to science
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10. Modality / B. Possibility / 8. Conditionals / c. Truth-function conditionals
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12197
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Inferring q from p only needs p to be true, and 'not-p or q' to be true
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14450
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All forms of implication are expressible as truth-functions
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10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
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14460
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If something is true in all possible worlds then it is logically necessary
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14. Science / B. Scientific Theories / 1. Scientific Theory
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14433
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Mathematically expressed propositions are true of the world, but how to interpret them?
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19. Language / D. Propositions / 1. Propositions
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14451
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Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts
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