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All the ideas for 'fragments/reports', 'Intro to Gdel's Theorems' and 'The Principles of Mathematics'

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138 ideas

1. Philosophy / F. Analytic Philosophy / 1. Nature of Analysis
Analysis gives us nothing but the truth - but never the whole truth [Russell]
1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
The study of grammar is underestimated in philosophy [Russell]
1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
Analysis falsifies, if when the parts are broken down they are not equivalent to their sum [Russell]
2. Reason / D. Definition / 13. Against Definition
Definition by analysis into constituents is useless, because it neglects the whole [Russell]
In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives [Russell]
2. Reason / F. Fallacies / 2. Infinite Regress
Infinite regresses have propositions made of propositions etc, with the key term reappearing [Russell]
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 [Russell]
3. Truth / A. Truth Problems / 5. Truth Bearers
What is true or false is not mental, and is best called 'propositions' [Russell]
3. Truth / H. Deflationary Truth / 1. Redundant Truth
"The death of Caesar is true" is not the same proposition as "Caesar died" [Russell]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
The null class is a fiction [Russell]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
There cannot be a set theory which is complete [Smith,P]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
Order rests on 'between' and 'separation' [Russell]
Order depends on transitive asymmetrical relations [Russell]
4. Formal Logic / G. Formal Mereology / 1. Mereology
The part-whole relation is ultimate and indefinable [Russell]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order arithmetic can prove new sentences of first-order [Smith,P]
5. Theory of Logic / B. Logical Consequence / 8. Material Implication
It would be circular to use 'if' and 'then' to define material implication [Russell]
Implication cannot be defined [Russell]
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
The only classes are things, predicates and relations [Russell]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
There seem to be eight or nine logical constants [Russell]
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 on Russell]
5. Theory of Logic / E. Structures of Logic / 3. Constants in Logic
Constants are absolutely definite and unambiguous [Russell]
5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
Variables don't stand alone, but exist as parts of propositional functions [Russell]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'partial function' maps only some elements to another set [Smith,P]
A 'total function' maps every element to one element in another set [Smith,P]
An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P]
The 'range' of a function is the set of elements in the output set created by the function [Smith,P]
Two functions are the same if they have the same extension [Smith,P]
5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P]
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P]
5. Theory of Logic / G. Quantification / 1. Quantification
'Any' is better than 'all' where infinite classes are concerned [Russell]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
A 'natural deduction system' has no axioms but many rules [Smith,P]
5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
No nice theory can define truth for its own language [Smith,P]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P]
A 'surjective' ('onto') function creates every element of the output set [Smith,P]
A 'bijective' function has one-to-one correspondence in both directions [Smith,P]
5. Theory of Logic / K. Features of Logics / 3. Soundness
If everything that a theory proves is true, then it is 'sound' [Smith,P]
Soundness is true axioms and a truth-preserving proof system [Smith,P]
A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P]
5. Theory of Logic / K. Features of Logics / 4. Completeness
A theory is 'negation complete' if it proves all sentences or their negation [Smith,P]
'Complete' applies both to whole logics, and to theories within them [Smith,P]
A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P]
5. Theory of Logic / K. Features of Logics / 7. Decidability
'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P]
A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P]
Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P]
A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P]
A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P]
The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P]
The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P]
5. Theory of Logic / K. Features of Logics / 9. Expressibility
Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P]
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / a. Achilles paradox
The Achilles Paradox concerns the one-one correlation of infinite classes [Russell]
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 [Russell, by Lavine]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
In geometry, Kant and idealists aimed at the certainty of the premisses [Russell]
Geometry throws no light on the nature of actual space [Russell]
Pure geometry is deductive, and neutral over what exists [Russell]
In geometry, empiricists aimed at premisses consistent with experience [Russell]
Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett]
Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell]
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 [Russell]
Ordinals presuppose two relations, where cardinals only presuppose one [Russell]
Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Ordinals are defined through mathematical induction [Russell]
Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell]
Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell]
For Cantor ordinals are types of order, not numbers [Russell]
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 [Russell]
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) [Russell]
The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
Some quantities can't be measured, and some non-quantities are measurable [Russell]
Quantity is not part of mathematics, where it is replaced by order [Russell]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
Counting explains none of the real problems about the foundations of arithmetic [Russell]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
We can define one-to-one without mentioning unity [Russell]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
The truths of arithmetic are just true equations and their universally quantified versions [Smith,P]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
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 [Russell]
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 [Russell]
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 [Russell]
For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell]
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 [Russell]
All numbers are related to zero by the ancestral of the successor relation [Smith,P]
The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / b. Baby arithmetic
Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P]
Baby Arithmetic is complete, but not very expressive [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / c. Robinson arithmetic
Robinson Arithmetic (Q) is not negation complete [Smith,P]
Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P]
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' [Russell, by Monk]
Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Denying mathematical induction gave us the transfinite [Russell]
The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P]
Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]
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 + [Russell]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
Numbers are properties of classes [Russell]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Ordinals can't be defined just by progression; they have intrinsic qualities [Russell]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Mathematics doesn't care whether its entities exist [Russell]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]
7. Existence / A. Nature of Existence / 3. Being / a. Nature of Being
Being is what belongs to every possible object of thought [Russell]
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 [Russell]
7. Existence / A. Nature of Existence / 6. Criterion for Existence
What exists has causal relations, but non-existent things may also have them [Russell]
7. Existence / E. Categories / 3. Proposed Categories
Four classes of terms: instants, points, terms at instants only, and terms at instants and points [Russell]
8. Modes of Existence / A. Relations / 1. Nature of Relations
Philosophers of logic and maths insisted that a vocabulary of relations was essential [Russell, by 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 [Russell]
8. Modes of Existence / A. Relations / 4. Formal Relations / b. Equivalence relation
Symmetrical and transitive relations are formally like equality [Russell]
8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation
The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P]
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. [Russell]
9. Objects / A. Existence of Objects / 5. Simples
Unities are only in propositions or concepts, and nothing that exists has unity [Russell]
9. Objects / B. Unity of Objects / 1. Unifying an Object / a. Intrinsic unification
The only unities are simples, or wholes composed of parts [Russell]
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' [Russell]
9. Objects / D. Essence of Objects / 15. Against Essentialism
Change is obscured by substance, a thing's nature, subject-predicate form, and by essences [Russell]
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
Terms are identical if they belong to all the same classes [Russell]
It at least makes sense to say two objects have all their properties in common [Wittgenstein on Russell]
10. Modality / B. Possibility / 9. Counterfactuals
It makes no sense to say that a true proposition could have been false [Russell]
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Abstraction principles identify a common property, which is some third term with the right relation [Russell]
The principle of Abstraction says a symmetrical, transitive relation analyses into an identity [Russell]
A certain type of property occurs if and only if there is an equivalence relation [Russell]
19. Language / D. Propositions / 1. Propositions
Proposition contain entities indicated by words, rather than the words themselves [Russell]
19. Language / D. Propositions / 3. Concrete Propositions
If propositions are facts, then false and true propositions are indistinguishable [Davidson on Russell]
19. Language / D. Propositions / 5. Unity of Propositions
A proposition is a unity, and analysis destroys it [Russell]
Russell said the proposition must explain its own unity - or else objective truth is impossible [Russell, by Davidson]
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius]
26. Natural Theory / C. Causation / 7. Eliminating causation
We can drop 'cause', and just make inferences between facts [Russell]
Moments and points seem to imply other moments and points, but don't cause them [Russell]
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 [Russell]
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 [Russell]
27. Natural Reality / A. Classical Physics / 1. Mechanics / c. Forces
Force is supposed to cause acceleration, but acceleration is a mathematical fiction [Russell]
27. Natural Reality / C. Space / 3. Points in Space
Space is the extension of 'point', and aggregates of points seem necessary for geometry [Russell]
27. Natural Reality / D. Time / 3. Parts of Time / b. Instants
Mathematicians don't distinguish between instants of time and points on a line [Russell]
27. Natural Reality / E. Cosmology / 1. Cosmology
The 'universe' can mean what exists now, what always has or will exist [Russell]
27. Natural Reality / G. Biology / 3. Evolution
Archelaus said life began in a primeval slime [Archelaus, by Schofield]