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. Analysis
The study of grammar is underestimated in philosophy
Analysis gives us nothing but the truth - but never the whole truth
1. Philosophy / F. Analytic Philosophy / 5. Against Analysis
Analysis falsifies, if when the parts are broken down they are not equivalent to their sum
2. Reason / D. Definition / 12. Against Definition
Definition by analysis into constituents is useless, because it neglects the whole
In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives
2. Reason / F. Fallacies / 2. Infinite Regress
Infinite regresses have propositions made of propositions etc, with the key term reappearing
2. Reason / F. Fallacies / 8. Category Mistake / a. Category mistakes
As well as a truth value, propositions have a range of significance for their variables
3. Truth / A. Truth Problems / 5. Truth Bearers
What is true or false is not mental, and is best called 'propositions'
3. Truth / H. Deflationary Truth / 1. Redundant Truth
"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
The null class is a fiction
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Na´ve logical sets
Russell invented the na´ve set theory usually attributed to Cantor [Lavine]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
Order rests on 'between' and 'separation'
Order depends on transitive asymmetrical relations
4. Formal Logic / G. Formal Mereology / 1. Mereology
The part-whole relation is ultimate and indefinable
5. Theory of Logic / B. Logical Consequence / 8. Material Implication
Implication cannot be defined
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
The only classes are things, predicates and relations
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
There seem to be eight or nine logical constants
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / c. not
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
Constants are absolutely definite and unambiguous
5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
Variables don't stand alone, but exist as parts of propositional functions
5. Theory of Logic / G. Quantification / 1. Quantification
'Any' is better than 'all' where infinite classes are concerned
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / a. Achilles paradox
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
Russell discovered the paradox suggested by Burali-Forti's work [Lavine]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
Pure geometry is deductive, and neutral over what exists
In geometry, Kant and idealists aimed at the certainty of the premisses
In geometry, empiricists aimed at premisses consistent with experience
Geometry throws no light on the nature of actual space
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
Ordinals result from likeness among relations, as cardinals from similarity among classes
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
Some claim priority for the ordinals over cardinals, but there is no logical priority between them
Ordinals presuppose two relations, where cardinals only presuppose one
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
Ordinals are types of series of terms in a row, rather than than the 'nth' instance
For Cantor ordinals are types of order, not numbers
Ordinals are defined through mathematical induction
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
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
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
Some quantities can't be measured, and some non-quantities are measurable
Quantity is not part of mathematics, where it is replaced by order
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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
We can define one-to-one without mentioning unity
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal)
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
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
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
You can't get a new transfinite cardinal from an old one just by adding finite numbers to it
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
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
Russell tried to replace Peano's Postulates with the simple idea of 'class' [Monk]
Dedekind failed to distinguish the numbers from other progressions [Korsgaard]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Finite numbers, unlike infinite numbers, obey mathematical induction
Denying mathematical induction gave us the transfinite
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
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
Numbers are properties of classes
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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
Mathematics doesn't care whether its entities exist
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Pure mathematics is the class of propositions of the form 'p implies q'
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
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
Being is what belongs to every possible object of thought
7. Existence / A. Nature of Existence / 3. Being / b. Being and existence
Many things have being (as topics of propositions), but may not have actual existence
7. Existence / A. Nature of Existence / 7. Criterion for Existence
What exists has causal relations, but non-existent things may also have them
7. Existence / E. Categories / 3. Proposed Categories
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
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
'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
Symmetrical and transitive relations are formally like equality
9. Objects / A. Existence of Objects / 3. Objects in Thought
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
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
The only unities are simples, or wholes composed of parts
9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
A set has some sort of unity, but not enough to be a 'whole'
9. Objects / D. Essence of Objects / 15. Against Essentialism
Change is obscured by substance, a thing's nature, subject-predicate form, and by essences
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
It at least makes sense to say two objects have all their properties in common [Wittgenstein]
Terms are identical if they belong to all the same classes
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
The principle of Abstraction says a symmetrical, transitive relation analyses into an identity
Abstraction principles identify a common property, which is some third term with the right relation
A certain type of property occurs if and only if there is an equivalence relation
19. Language / D. Propositions / 1. Propositions
Proposition contain entities indicated by words, rather than the words themselves
19. Language / D. Propositions / 3. Concrete Propositions
If propositions are facts, then false and true propositions are indistinguishable [Davidson]
19. Language / D. Propositions / 5. Unity of Propositions
A proposition is a unity, and analysis destroys it
Russell said the proposition must explain its own unity - or else objective truth is impossible [Davidson]
26. Natural Theory / C. Causation / 7. Eliminating causation
Moments and points seem to imply other moments and points, but don't cause them
We can drop 'cause', and just make inferences between facts
26. Natural Theory / D. Laws of Nature / 11. Against Laws of Nature
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
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
Force is supposed to cause acceleration, but acceleration is a mathematical fiction
27. Natural Reality / C. Space-Time / 1. Space / c. Points in space
Space is the extension of 'point', and aggregates of points seem necessary for geometry
27. Natural Reality / C. Space-Time / 3. Space-Time
Mathematicians don't distinguish between instants of time and points on a line
27. Natural Reality / D. Cosmology / 1. Cosmology
The 'universe' can mean what exists now, what always has or will exist