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