Ideas from 'The Principles of Mathematics' by Bertrand Russell [1903], by Theme Structure
[found in 'Principles of Mathematics' by Russell,Bertrand [Routledge 1992,9780415082990]].
green numbers give full details 
back to texts

expand these ideas
1. Philosophy / F. Analytic Philosophy / 1. Analysis
14109

The study of grammar is underestimated in philosophy

14122

Analysis gives us nothing but the truth  but never the whole truth

1. Philosophy / F. Analytic Philosophy / 5. Against Analysis
14165

Analysis falsifies, if when the parts are broken down they are not equivalent to their sum

2. Reason / D. Definition / 12. 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 partwhole 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 truthvalue, 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 oneone correlation of infinite classes

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. BuraliForti's paradox
15895

Russell discovered the paradox suggested by BuraliForti'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

14153

In geometry, empiricists aimed at premisses consistent with experience

14154

Geometry throws no light on the nature of actual space

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
14144

Ordinals result from likeness among relations, as cardinals from similarity among classes

18254

Russell's approach had to treat real 5/8 as different from rational 5/8 [Dummett]

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
14142

Ordinals are types of series of terms in a row, rather than than the 'nth' instance

14145

For Cantor ordinals are types of order, not numbers

14141

Ordinals are defined through mathematical induction

14139

Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic

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 nonquantities 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 onetoone 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 [Korsgaard]

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
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 / 7. Criterion for Existence
14173

What exists has causal relations, but nonexistent 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, subjectpredicate form, and by essences

9. Objects / F. Identity among Objects / 7. Indiscernible Objects
11849

It at least makes sense to say two objects have all their properties in common [Wittgenstein]

14107

Terms are identical if they belong to all the same classes

18. Thought / E. Abstraction / 7. Abstracta by Equivalence
10582

The principle of Abstraction says a symmetrical, transitive relation analyses into an identity

10583

Abstraction principles identify a common property, which is some third term with the right relation

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
14111

A proposition is a unity, and analysis destroys it

19157

Russell said the proposition must explain its own unity  or else objective truth is impossible [Davidson]

26. Natural Theory / C. Causation / 7. Eliminating causation
14172

Moments and points seem to imply other moments and points, but don't cause them

14175

We can drop 'cause', and just make inferences between facts

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. SpaceTime / 1. Space / c. Points in space
14160

Space is the extension of 'point', and aggregates of points seem necessary for geometry

27. Natural Reality / C. SpaceTime / 3. SpaceTime
14156

Mathematicians don't distinguish between instants of time and points on a line

27. Natural Reality / D. Cosmology / 1. Cosmology
14169

The 'universe' can mean what exists now, what always has or will exist
