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 1975,0045100209]].
green numbers give full details 
back to texts

expand these ideas
1. Philosophy / F. Analytic Philosophy / 4. Ordinary Language
14456

'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity

2. Reason / D. Definition / 3. Types of Definition
14426

A definition by 'extension' enumerates items, and one by 'intension' gives a defining property

2. Reason / F. Fallacies / 8. Category Mistake / a. Category mistakes
8468

The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false [Orenstein]

3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
14454

An argument 'satisfies' a function φx if φa is true

4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
14453

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
14427

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
14428

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
14440

We may assume that there are infinite collections, as there is no logical reason against them

14447

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
14443

The British parliament has one representative selected from each constituency

14444

Choice is equivalent to the proposition that every class is wellordered

14445

Choice shows that if any two cardinals are not equal, one must be the greater

14446

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
14459

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
14461

Propositions about classes can be reduced to propositions about their defining functions

4. Formal Logic / F. Set Theory ST / 7. Natural Sets
8469

Russell's proposal was that only meaningful predicates have sets as their extensions [Orenstein]

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
8745

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
14452

All the propositions of logic are completely general

5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
14462

In modern times, logic has become mathematical, and mathematics has become logical

5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
10057

Logic can only assert hypothetical existence

14464

Logic can be known a priori, without study of the actual world

12444

Logic is concerned with the real world just as truly as zoology

5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
10450

Russell admitted that even names could also be used as descriptions [Bach]

14457

Names are really descriptions, except for a few words like 'this' and 'that'

14458

Asking 'Did Homer exist?' is employing an abbreviated description

5. Theory of Logic / F. Referring in Logic / 1. Naming / f. Names eliminated
7311

The only genuine proper names are 'this' and 'that'

5. Theory of Logic / F. Referring in Logic / 2. Descriptions / a. Descriptions
14455

'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not

6. Mathematics / A. Nature of Mathematics / 2. Geometry
14442

If straight lines were like ratios they might intersect at a 'gap', and have no point in common

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
14438

New numbers solve problems: negatives for subtraction, fractions for division, complex for equations

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
13510

Could a number just be something which occurs in a progression? [Hart,WD]

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

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
14421

Discovering that 1 is a number was difficult

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
14424

Numbers are needed for counting, so they need a meaning, and not just formal properties

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
14420

Infinity and continuity used to be philosophy, but are now mathematics

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
14431

The definition of order needs a transitive relation, to leap over infinite intermediate terms

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
14422

Any founded, nonrepeating series all reachable in steps will satisfy Peano's axioms

14423

'0', 'number' and 'successor' cannot be defined by Peano's axioms

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

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

6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
14465

Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men'

6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
13414

For Russell, numbers are sets of equivalent sets [Benacerraf]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
14449

There is always something psychological about inference

7. Existence / A. Nature of Existence / 1. Nature of Existence
14463

Existence can only be asserted of something described, not of something named

7. Existence / D. Theories of Reality / 6. Fictionalism
14429

Classes are logical fictions, made from defining characteristics

8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
14430

If a relation is symmetrical and transitive, it has to be reflexive

14432

'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a

9. Objects / D. Essence of Objects / 3. Individual Essences
14435

The essence of individuality is beyond description, and hence irrelevant to science

10. Modality / B. Possibility / 8. Conditionals / c. Truthfunction conditionals
12197

Inferring q from p only needs p to be true, and 'notp or q' to be true

14450

All forms of implication are expressible as truthfunctions

10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
14460

If something is true in all possible worlds then it is logically necessary

14. Science / B. Scientific Theories / 1. Scientific Theory
14433

Mathematically expressed propositions are true of the world, but how to interpret them?

19. Language / D. Propositions / 1. Propositions
14451

Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts
