Principles of Mathematics Russell
Introduction
Hilbert and the Formalists theory
Hilbert refuted
Brouwer intuitionist theory
Weyl
An infinite statement
Jorgensen, ‘Treatise of Formal Logic’
‘numbers are symbols which mean nothing’ p.9
Numbers and atheism
The theory of descriptions
The abolition of classes
‘x’ wrote Waverly is equivalent to x is Scott is true for all values of ‘x’.
‘logical constructions of events’
Whatever appears to have such properties is constructed artificially in order to have them.
1) incompatibility
2) truth of all values of a propositional function
‘logic becomes much more linguistic’
True in virtue of its form
This method leaves us ‘in the lurch’ p.12
Carnap, ‘Logical Syntax of Language’
Poincare – contradicts this ‘theory of types’
Russell’s paradox
Burali- Forti paradox
The greatest ordinal of N, yet zero (0) up to N is N+1, therefore, N cannot be greatest (in magnitude) number.
i.e. ‘I am lying’ a Greek contradiction
theory of types
rules for premises – so you can have:
Socrates is a man
Socrates is a mortal
But not if
The law of contradiction is a man
The law of contradiction is a mortal
This sentence – ‘nonsense’
’Theory of types’ gives rules as to ‘permissible’ values of ‘x’.
Points, classes, instants
Ramsay: ‘virtue is triangular’
How far it is to go in the direction of ‘nominalism’ remains an unsolved question
Less Platonic, less realist.
Preface
Persuasion or police
Demonstration demon/stration
The New Logic is ‘Nominalism’
Can only be investigated (mathematical logic)
Philosophy
A ‘mental telescope’
‘Redness is a pineapple’
‘Neptune’
The notion of ‘classes’ is something ‘amiss’
Logic of Relations
Peano – specially philosophical
Frege, ‘Grundgesetze der Arithmetik’
Various original developments
No one of the component accelerations actually occurs only the resultant acceleration of which they (particles) are no part.
A view to discovering the meaning of the word ‘any;
Logical constants
The difficulty in regards to absolute motion is insoluble on a relational theory of space
G.E. Moore and philosophy
Cantor, Peano, Frege
Whitehead
E. Johnson
Chapter 1 Definition of Pure Mathematics
Logical constants
Precise, certain, exact
Idealists vs. Empiricists
appearances vs. approximation
Kant intuition or ‘a priori’
Professor Peano – Symbolic Logic refutes Kant’s intuitionist ‘a priori’
Ten principles of deduction plus ten other premises of Symbolic Logic
Symbolic Logic = Math
Leibniz
Euclid
Euclidean vs. non – Euclidean geometry a ‘relation’ between ‘p’ and ‘q’
A ‘formal implication’
‘There are variables in ‘all’ mathematical propositions’ – BR
x + y – collected into classes x + y can be substituted say with Plato & Socrates
Generalization
Symbols which stood for constants become transformed into variables
Type
pure vs applied math p.8
Euclid
Symbolic Logic as a ‘branch’ of mathematics
Chpt.2 Symbolic Logic
Boole
Peano’s, ‘Formalism’
‘implication’ is indefinable
Disjunction
Rules of inference
Definition of a proposition p.15
The axiom of parallels
Joint assertion or logical product a ‘highly artificial’ definition
The ‘principle of reduction’
Negation
The law of the excluded middle plus associative, commutative and distributive laws can be proved
The Calculus of Classes
Lewis Carrol ‘Mind’ no.#11 (1894)
ϵ – some
“Relationship between ‘E’ as the relation of whole and part between classes (due to Peano of ‘very great importance’”-BR
Frege
Class–concept
Propositional function
Peano – ‘x is an a’ is general for one variable, and that extensions of the same form are available for any number of variables
0x is a propositional function, if for every value of x, x is a proposition, determinate of when x is given.’
Thus ‘x is a man’ is a propositional function.
Verbs and adjectives
‘such that’
The roots of an equation
: of identity (class identity) (sort of) class
‘x is an a’ implies y is a u for all values of u’
‘x is an a, and x is a ‘b’ for all values of ‘x’’
x ‘such that’ u is a k implies x is a u for all values of u.
Existence
x ‘such that’ if u is a k implies u is contained in c for all values of u, then for all values of c, x is a c.
Class syllogism
C.S. Pierce
G.E. Moore
Class of ‘referents’
Class of ‘relata’ R + R’
Two relations may have the same ‘extension’ without being identical xRy
The identity of two classes which results from the primitive propositions as to identity of classes to establish…p.25
Peano’s ‘Symbolic Logic’
The fixed relation
The fixed term
Denoting
Peano 1897
Class, term implication
The simultaneous affirmation of two propositions
The notion of definition and the negation of a proposition
Peano’s definition p. 28
1) if a is a class, ‘x is y are ‘a’s’ is to mean x is an a and y is an a
2) if a and b are classes, every a is a b, means x is an ‘a’ implies that x is a b.
3) the u’s such that’ x is an a, means the class of a (‘a perfectly worthless definition which has been abandoned’ – BR)
Such that f(x) – F(x) is not a proposition at all, but a ‘propositional function’.
Padoa
1) every class is contained in itself means: every proposition implies itself
2) the product of two classes, is a class
3) if a, b, be classes, their logical product ab, is contained in ‘a’ and contained in ‘b’
So:”if K be a class of classes the logical product of K consists of all terms belonging to every class that belongs to K’.
Two forms of syllogism:
1) that if a, b, c, be classes, and ‘a’ is contained in ‘b’ and ‘x’ is an ‘a’, then ‘x’ is a ‘b’
2) if a, b, c, be classes, and a is contained in b, and b is contained in c, then a is contained in c.
“This second form of the syllogism is (of course) the very ‘life’ of all forms of reasoning.” – BR
Peano, “Formulaire’ 1901
6) negation
7) disjunction or ‘the logical sum of two classes. A or b is defined as the negation of the logical product of not-a, and not-b, i.e. as the class of terms which are both not-a , and not-b
8) the ‘null’ class (1897) a null class is defined as ‘null’ when it is contained in every class
‘The Formulaire’
Chpt III Implication and Formal Implication
“…Inference and the intrusion of a psychological element” – BR.
Material implication: the relation in virtue of which it is possible for us validly to infer.
Plum pudding
Euclid: one particular proposition is deduced from another
Implication
Lewis Carroll: ‘What the Tortoise said to Achilles’ (a puzzle)
The notion of ‘therefore’
‘p’ implies, ‘q’ asserts and implication
Grave difficulties p.35
‘therefore’
Every syllogism is held to have two premises
‘formal implication’: far more difficult.
‘all’ men are mortal, ‘every’ man is mortal, and ‘any’ man is mortal
Allow our ‘x’ to take all values without exception
Assertions
“The ‘verb’ deserves far more respect than is thus paid to it.” – BR
“The ‘verb’ – the ‘distinguishing mark’ of propositions.” – BR
Propositions divided into subject (i.e. Socrates) and the assertion (is a man)
‘A’ is before ‘B’ implies ‘B’ is after ‘A’
Notion of a propositional function, and the notion of an assertion are more fundamental than the notion of class. ‘formal implication’ is involved in all the rules of inference.
Bradley, ‘Logic’
Chpt. IV Proper Names, Adjectives and Verbs p. 42
Philosophical grammar
Substantives, adjectives, verbs general and proper names
Doctrine of number, notion of the variable
Bradley
Term = unit, individual, entity
G.E. Moore
Term – ‘any’ term or ‘some’ term
Things and concepts
Subject/predicate propositions i.e. ‘Socrates is human’
‘this is one’, ‘this is a number’
Is/being human/humanity
Inextricable difficulties p.46
One as adjective
One as number
‘any’ ‘cognate’ terms
A contradiction p. 46
Terms which are concepts differ from those which are not. i.e. subjective, terms of relation
Logic – conceptual diversity versus numerical diversity
Mathematics
Bradley’s ‘Logic’
‘meaning’ is a notion confusedly compounded of logical and psychological elements
Non-psychological sense
‘the confusion is largely due to the notion that words occur in propositions, which in turn is due to the notion that propositions are essentially ‘mental’ and are to be identified with cognition’ – BR
In regard to verbs 48
Felton killed Buckingham vs. (headline) ‘killing no murder’
i.e. killing – the verb becomes a ‘verbal noun’
the falsehood of Caesar’s death is never equivalent to ‘Caesar died’
psychological
“the notion of ‘truth’ belongs no more to the principles of mathematics than to the principles of everything else” – BR
‘A’ is quite different from ‘Socrates’ (is)
‘being’ of ‘A’
Verbs ‘is’ and ‘from’
A referent, a relatum
Verb as a term
A ‘difference’ – ‘complex notion’
Irrelevant
Denial or affirmation
The notion of infinity
A differs from B
Inadmissible
Tenable
‘even if differences did differ they would still have to have something in common’ – BR
The general relation of difference a ‘propositional’ concept
V Denoting
An ‘undue mixture of psychology’
The ‘shadowy limbo of the logic book’ -BR
Of the concept of ‘any’ number
Almost all propositions that contain the phrase ‘any number’
The ‘Times’:
‘Man, (in fact) does not die’
The whole theory of definition of identity of classes, of symbolism and of the variable, is wrapped up in the theory of ‘denoting’
A is A is One, A is human, A is an entity, A is a unit, A is a man (these are classes)
The class is the sum or conjunction of all terms which have the given predicate.
“’Predicates’ are the simplest type of ‘concepts’”
There is connected with every predicate a great variety of closely allied concepts i.e. – ‘human’:
We have man, men, all men, every man, any man, the human race…. Of which all except the first are twofold.
A denoting concept, an object denoted a ‘vast apparatus’ of complex terms a phrase containing one of the words (the six) : ‘all’ , ‘every’, ‘any’, ‘a’, ‘some’, and ‘the’
Always denotes ‘x’ is a ‘u’ is a propositional function when and only when ‘u’ is a ‘class-concept’
Brown and Jones
‘human’ a ‘predicate’
‘man’ a ‘class – concept’
1) Brown and Jones
A genuine combination a class no less
a) a numerical conjunction
b) propositional conjunction
2) Brown and Jones severally (separately) equivalent but not identical with:
Brown is PCTMS
Jones is PCTMS
All collectively and all distributively each of every one the ‘logical product’
3) ‘any’ is defined ‘either’ the ‘variable’ conjunction
4) ‘a’ suitor – a variable ‘disjunction’
5) the ‘constant disjunction’
Something absolutely peculiar p.58
‘Neither one nor many’, ‘all’, ‘as’, ‘every a’, ‘any a’, ‘an a’, and ‘some a’
“The possibility of giving a collection by ‘class-concept’ is ‘highly important’” – BR
‘any a’ – denotes a ‘variable a’
‘an a’ is a variable disjunction
‘some a’ a ‘constant disjunction’
The nature and the properties of the various ways of combining terms are of ‘vital’ importance to the principle of mathematics. p.55-61.
A ‘definite’ something different in each if the five cases ‘the’ king, ‘the’ prime-minister, and so-on
“a ‘puzzling paradox’ to the symbolic mind.” = BR
abstract
‘identity’: must be something.
The ‘illegitimate’ kind
Edward VII is the king
A class of having only one term (seventh Edward’s)
The ‘is’ states ‘pure’ identity
A logical discussion
Chpt VII Classes
One of the most difficult and important problems of mathematical philosophy
Extension, intension
Philosophy
Symbolic logic
‘lair’
‘theory’ of denoting p.67
Of great importance
Peano
Featherless biped 68
Frege
The notion of a ‘whole’
Purely psychological
A collection
The notion of ‘and’ – Bolzano
Diversity is implies by ‘and’
The distinction of ‘being’ and ‘existence’ p.71
“Existence is important” – BR
A and B are two
A is one and B is one
A and B are yellow
A is yellow and B is yellow
‘and’ : and addition of ‘individuals’
‘all’ – all ‘u’’s
‘all’ and ‘u’’s
Every, any, some, a, and ‘the’
Language a ‘misleading’ guide.
Infinite classes’ – the inmost’ secret of our power to deal with infinity.
An air of ‘magic’
‘nothing’ is not ‘nothing’
Plato’s ‘sophist’
‘all’ and ‘nothing’
The ‘null’ class
Denotes but does not denote anything
‘chimeras are animals’
A new difficulty p.75
Puzzling p.77
ϵ – The fundamental relation is that of subject and predicate: i.e. ‘Socrates is human’
Socrates has ‘humanity’
Socrates is a man ϵ
Peano’s E (‘is a’)
Between Socrates is a man ϵ
Peano regards ‘a’ as the fundamental relation
ϵ
A ‘battleground’
Beheaded English king 1649
Men, man, mortals
What exactly are we saying?
Predicates ‘predictability’
Socrates is human
‘human’ as predicate
Socrates has humanity’
‘humanity’ a term of relation
Although any symbolic treatment must work largely with class-concepts and intension
Classes and extension are logically more fundamental for the principles of mathematics
VIII Propositional Functions
The x’s ‘such that’ x is an ‘a’ are the class ‘a’.
Being an x ‘such that’ so and so the latter form being necessary because so and so is a propositional function containing ‘x’.
xRa = when R is a given relation and ‘a’ is a ‘given’ term.
The children of Israel ‘such that’
‘such that’ is roughly equivalent to ‘who’ or ‘which’
The ‘indefinability’ of propositional functions 80
Every relation which is ‘many-one’ defines a function
Socrates is a man
X is a man
Socrates, Plato, Aristotle
‘x’ is a man
To be a ‘man’ is to suffer
‘man’ is different (a term)
If R be a fixed relation and ‘a’ a fixed term ‘Ra’ is a perfectly definite assertion
X E b > x E c
X R a > x R c
Assertions, assertions concerning Socrates (x)
A very great difficulty
0x = x is a man implies x is a mortal
Curious and difficult
xRy where R is a constant relation, while x and y are independently variable
classes of referents and relata with respect to R i.e. parents and children, masters and servants, husbands and wives
Ry new and difficult notion
xRx i.e. class of suicides or of self-made men – the relation of a term to itself
if R is a relation implies ‘aRb’ the 0R
aRb = a and b are fixed terms
R is a variable relation
aRb > 0(R) provided R is a relation
the couple with sense
ox > ox > ox is a proposition for all values of x, and is true when and only when ox is true.
Propositional functions must be accepted as ‘ultimate data’
The theory of relations
Not unimportant & legitimate
“X’s such that 0x where 0x is a propositional function, we are introducing a notion of which, in the calculus of proposition only a very shadowy use is made – I mean the notion of ‘truth.’” – BR
Chpt IX The Variable
X’s such that 0x’
The ‘variable’ is perhaps the most distinctly mathematical of all notions.
0x, the propositional function is what is denoted by the proposition of the form 0x in which x occurs.
From the formal standpoint the variable is….
The characteristic notion of mathematics. It is the method of stating the general theorems.
Once again, ‘all men’, ‘any man’, to Russell a self-evident truth a different meaning than the concept ‘man’
Dedekind and Stolz
Proteus 91
n – the notion of ‘any’
formal (true) variable and restricted variable
‘any’ denotes a term of a class
Logic, Arithmetic, Geometry
Any ‘a’ is a ‘b’ = x is an ‘a’ implies x is a b
Any, some. All
A relation of ‘any’ term of x is an ‘a’ to ‘some’ term of x is a b
0x i.e. ‘x is a man implies x is mortal’
Propositional function 0x
x ϵ a > x ϵ b a class extension
0x – Yx
The variable a very complex logical entity
Exist or not to exist
True – therefore – no such class
Variables have a kind of ‘individuality’
Any term has some relation to any term
Successive steps
Double integration
A variable is not any term simply, but any term as entering into a propositional function
The notion of class, of denoting and of ‘any’ are fundamental, being pre-supposed in the symbolism employed.
Propositions
a) subject – predicate
b) in which a relation is asserted between the terms
c) two terms are said to be two
chpt IX Relations
a and b are two identical with
b and a are two
symbolized by aRb
different from bRa
from one to the other
bRa
referent, relatum the sense of a relation not capable of a definition
the ‘converse’ of R is R’ – oppositeness
being ‘is’ or has being
one is one or has unity
concept is conceptual, a term is a term
class concept is a class-concept
the general theory of relations
the ‘domain’ of the relation
the logical sum of the two
field of relation
predicate predicable
aRb a and Rb
relation paternity\domain: father
Converse domain: children
Field of fathers and children together
Double variability and the relative product
Non-commutative
The relative product of in-laws e.g.
R, S, RS, x, z, term y to which x has relation R s to z
Bradley, ‘Appearance and Reality’
In the calculus of relations it is classes of couples that are relevant but the symbolism deals with them by means of relations
The ‘endless regress’
X The Contradiction
Predicates not predicable of themselves
Cantor’s ‘there can be no greatest cardinal number’
‘Bicycle is a teapot’
‘Teapot is a bicycle’
Ǝu = some ‘u’
The doctrine of logical types
Quadratic forms 104
0(a) ‘men’ in two different usages
The key to the whole mystery. P.105
Smile, ‘Self-help’
Hegel
Any, every, but not all
Part II
Chpt XI Definition of Cardinal Number
The whole theory of cardinal number is a branch of logic.
Enumeration
‘all’
Property of classes
Counting
The relation of simultaneity
Between classes
Abstraction
Nominal 114
A number as a class of classes i.e. couple, or trio
Paradox
A philosophical difficulty
XII Addition and Multiplication
The theory of integers
Multiplication – A.N. Whitehead
Chpt XIII Finite and Infinite
The mathematical theory of finite and infinite as it appears in the theory of cardinal numbers
Cantor
Two classes: u’ and u’’ – add one or subtract one
Cantor
XIV Theory of Finite Numbers
Peano, ‘Formulaire’
0, finite integer, and successor of.
XV
Addition of Terms and Addition of Classes
Not of the nature of proof, but of the nature of exhortation
The notion of ‘one’
Man¹ ‘a featherless biped’
An entity Frege termed ‘werthver laufe’, Russell referred to as the ‘class as one’
Counting has a psychological aspect
A ‘collection’
‘and’
‘A’ and ‘B’
‘A’ and ‘B’ and ‘C’
Psychologically
Intension
The logical theory of infinity
Chpt XVI Whole and Part
Hegelian writers
Units – simple and complex
A complex unit is a whole
Its parts are other units, whether simple or complex, which are pre-supposed in it.
‘man’ is a concept of which ‘mortal’ is a part.
A is greater than B > B is less than A
Red
R of whole and part – an indefinable and ultimate relation
Class of many and the class of one aggregate
Definite as soon as its constituents are known
Aggregate of aggregates
A differs from B
Parts of A and B and differences
The difference between ‘kinds of wholes’ is important 140
Collections
Aggregates and unities
One collection, one manifold
There really is a single term
Theory of fractions depends partly on aggregates
3) magnitude of divisibility analysis gives us the truth and nothing but the truth, yet it can never give the whole truth
Aggregate of aggregate of aggregate
The theory of whole and part is less fundamental logically than that of predicates or class-concepts or propositional functions
Chpt XVII Infinite Wholes
Leibniz
Kant
A finite space
Only finite in a psychological sense
Cantor’s solution 144
Universe
A continuum
Merely psychological
The stretch of fractions from 0 to 1 has three simple parts 1/3, ½, 2/3,
We know of nothing suggesting of a beginning of time and space.
XVIII Ratios and Fractions
Ratios: relations between integers
Fractions: relations between aggregates or rather between their magnitudes of divisibility
A + n = b a + mn = b 0 = mn
Powers of relation being defined by relative multiplication
10(to the fifth power)
Series of types – progressions
nth power
greatest havoc and disastrous consequences p.150
the magnitude of divisibility
24 hrs. 12/12, or 23/1
Extensionally: by actual enumeration of their terms
Part III
Quantity – The Meaning of Magnitude
Euclid
Descartes
Vieta
Co-ordinate geometry
Weierstrasse
Dedekind
Cantor
Arithmetic
New arithmetic – more than non-Euclidean geometry
Fatal to the Kantian theory of a priori & intuition
Logical calculus
Projective geometry
The theory of groups
A matter of considerable importance p.160
Some ‘psychical existents’, i.e. – Pleasure and pain are quantitative
Bradley (footnote)
Euclid’s axiom ‘ The whole is greater than the part’
Equal to the ‘standard’ yard in Exchequer
Pleasures
Greater than, less than, equal to
A = A p.163
If equality be analyzable two equal terms must both be related to some third term
‘very peculiar’ p.164
‘intensity’ of pleasure
Principle of Abstraction p.166
Temporal, spatio or spatio-temporal
Herr Meinong 168
Zero
XX The Range of Quantity
Prima – facie
Poets – ‘O ruddier than the cherry’
The ideal ruddiness, others more or less ruddy.
Whites, blacks, redder
The relation of ‘similarity’
Pleasure – money analogy
Differential – co –efficients
Velocity and acceleration
Bentham – ‘Quantity of pleasures being equal, pushpin is as good as poetry.’
XXI
Numbers as Expressing Magnitude and Measurement
Psychophysical parallelism p.177
Kant’s ‘Anticipation of Perception’
Existence and magnitude
Twice as happy
XXII Zero
The ‘pure’ zero of magnitude
The zero Kant has in mind. In his refutation of Mendelsohn
Proof of the immortality of the soul p.184
The meaning of zero is a question of much difficulty
Zero pleasure said to be pleasure the least or limiting
Bettazzi
Magnitude of its kind
No pleasure/pain
No distance/distance
Not pleasure ~p
XXIII Infinity, the Infinitesimal and Continuity
‘almost all mathematical ideas present one great difficulty: the difficulty of infinity.’ – BR
an antinomy
Appears to be soluble by a correct philosophy of ‘any’
The ‘axiom of finitude’
The ‘axiom of Archimedes’ 187
The last term
The ‘philosophical axiom of infinity’ 189
Expand 9 points
The apparent antinomies may be considered ‘solved’ 193
Greater and lesser
Asymmetrical transitive relations
Part IV Order
XXIV The Genesis of Series
Dedekind, Cantor, Peano, and von Staudt
Progression
Complex
Considerable difficulties
The whole philosophy of space and time depend upon the view we take of ‘order’
The ‘ordinal’ definition of numbers
a,b,c – b is ‘between’ a and c
Given four terms a,b,c and d – a and c are ‘separated’ by ‘b’ and ‘d’
Let there be a collection of terms
…has to one other term a certain asymmetrical relation
Intransitive (of course)
Next after, next before
Consecutive, the beginning and the end
Bolzano 201
Professor Schroder
Fifty million people
Connected: a) a series may have two ends b) it may have no end c) it may have no end and be open d) it may have no end and be closed
Must some finite numbers (n) of steps which will take us from one end to the other, and hence ‘n + 1’ is the number of terms of the series.
Two classes
Pix and Pix’ respectively
Vivanti, Gilman
‘Mind’ #1
In a closed series, the generating relation can never be transitive
Cases of triangular relations are capable of giving rise to order
Varlati, Padoa
The few assumptions 206
Chpt…XXV The Meaning of Order
A difficult question
Purely philosophical interest
‘between’
The very meaning of ‘betweeness’ 214
There, there
Separation
Vailati five temp 214 circle diagram
Abcd = badc
Abcd = adcb
Abcd = excludes acbd
We must have abcd or acdb or adbe
Abcd and acde together imply abde
Pieri
XXVI Asymmetrical Relations
xRy, yRx symmetrical
xRy, yRz > yRx – transitive
xRy always excludes yRx – asymmetrical
xRry, yRx always excludes xRz – intransitive
deMorgan – transitiveness
diversity, implication
reflectiveness
0(u) 0(x0
Lotze, Spinoza
Chpt XXVIII Difference of Sense and Difference of Sign 207
Kant’s ‘Prolegomena’
Negation
R such that aRb > bRa
The relation of R to R is difference of sense
Theory of signs
‘Obvious and straightforward’ – BR
See diagr.right-handed, left-handed
Rays
Plain man (when sober)
Schroder’s ‘ershopft’
The usual instances of opposites
Synthetic incompatibility
pRm pRn
XXVIII On the Difference Between Open and Closed Series
Line and circle
Travel plans
Vailati
‘mean and extreme’
Three term relation see diagram
C is ‘anti-podal’ if ‘n’ is even closed or open series
Philosophical rather than mathematical importance
XXIX Progressions and Ordinal Number
Dedekind’s ordinal numbers are prior to cardinals
Helholtz, Kromiker, Dedekind
‘Logic of Relations’ RdMVII
The ‘old way’
XXX
Dedekind, ‘Theory of Number’
One – one
Cantor
Dedekind the chair of a somewhat complicated theorem 246
Definition of cardinals
The mind’s eye
The images of the system
the expression four-cornered relation
Chpt XXXI Distance
Leibniz’s controversy with Clarke
Logarithms
Non-sequitor
Archimedes
Meinong
Stretches
Non-Euclidean geometry
Whitehead
A kind of distance
1th power (converse of its relation)
Enormously complex 253
Relational addition
Postulate of Archimedes
duBois Raymond’s postulate of linearity
limits
logarithm
theory of space & time may be developed without pre-supposing distance
the arbitrary ‘first term’ of a closed series.
Part V Infinity and Continuity
XXXII The Correlation of Series
Weierstrasse and Cantor
A ‘complete’ transformation
‘Anzhal’ becomes irrelevant
Able to dispense entirely with the infinitesimal
When it is admitted that mathematical induction may be denied without contradiction, the supposed antinomies of infinity and continuity one and all disappear – BR
Homography – an example of correlation
Theory of time p; 265
Relative/absolute
Space/time
Mutatis mutatatids
Couturat, “de L’Infinite Mathematique’ part I book II
Law
Logic of Relations
Complete/incomplete series
A generating relation and vice-versa
XXXIII Real Numbers
Compact
Segments
Segments – : upper and lower limits
XXIV Limits and Irrational Numbers
X = div. 2, x2 -2x – 1 = 0
Thus x = 2 +1/x = 2 + ½ +x , 1/, and x – 1 = 1 + ½ +1/2 + 1/x = etc.
The successive convergents to the contrived fraction 1 + ½ +21/2 +1/1/+….
Zusammengehrig <> coherent
The fraction 14159….
1416
S/1 an
Ascending or descending
+ Î- ϵ
LimAv = b
V = ∞
Wholly new
Conclusion 286
XXXV Cantor’s First Definition of Continuity
BR Humor 287
‘compactness’
Incommensurables tenth book of Euclid
Zuzzamenhangend
Bien enchainee
Number Î or anything Îasymmetrical, transitive relation
Axiom of Archimedes
Meinong
Log x/y
XXXVI Ordinal Continuity
Couterat
Continuity arbitrary
A v is a v an upper segment
U and V endless compact series
XXXVII Transitive Cardinals
A +b = B + a and a + (b + a) = (a + B) + c – ab = ba. A(bc) = (ab)c, a(b +c) = ab +ac
Aᵇac= a, acbc, = (ab)c, (ab)c = abc
A) Every transfinite collection contains others as parts whose number is a
B) every transfinite collection which is part of one whose number is a0
C) no finite collection is similar to any proper part of itself
D) every transfinite collection is similar to some proper part of itself
Denumerable
1, ½, 21/3, ¼, 2/3, 3/2, 4, 15, a, § 2a⁰ II – a
Chpt XXXVIII Transfinite Ordinals
Do not obey commutative law
Even more interesting than transitive cardinals
The first principle of formation 313
Nothing objectionable to imagining a new number call it ‘w’
1,2,3…..v w is 1st number after v
Thus: w + 1, w+2….w + v,…..:
Then a new one call it 2w
First after all previous v and w + v
Thus the second principle of formation of real integers
The next number greater than all of those (limit)
A number having no immediate predecessor since the series has no last term
Thus w is simply the name of the progression, or of the ‘generating’ relations of series of this class.
Therefore, the existence of ‘w’ is not open to question when the segment theory is adopted…
A principle of limitation: (Hemmungsprincip)
A + x x b and x+ a = b
Y(a + b) = ya + yb
A + b, a, b are multipliers
w + 1, 1 + w and 1 + w ¹w +1
1 + w = w
W, w,.2 w . 3, w², w³, wʷ…
ξ + w = w + 1
ξ + a = b endless type if a represents and endless type, while b represents a terminating type, and: can never produce the type β
consider ξ+ w = w . 2
thus ξ= w + n where n is zero or any infinite number thus w + n + w = w . z
when ξ has infinite number of values
let M an N be two series of the type a and β, in N, in place of each element n, substitute a series Mn, of the type a, and let S be the series formed of all the series Mn, taken in the following order:
1) any two elements of S, which belong to the same series M, are to preserve the order they had in Mn
Two elements which belong to different series Mn and Mn’ are to have the order which n and n’ have in N. then the type of S depends upon an a and B and is defined to be their product aB, when a is the multiplicand and B the multiplicator
2w is the type of series defined as: e1,f1,e2,f2,e3,f3…..ev,fv
Which is a progression so that 2. w = w but w.2 is the type
e1,e2,e3…..ev…..; f1,f2,f3…..fv
Which is a combination of two progressions, but not a single progression
The term ordinal number is preserved for well-ordered series, i.e. such as have the following two properties:
(1) there is in the series F a first term
(2) if F’ is a part of F, and if F possesses one or more terms which come after all the terms of F’, then there is a term f’ of F, which immediately follows F’, so that there is no term of F before f’, and after all terms of F:
The definition of a progression is relative to some one – one aliorelative P.
When P generates a progression, this progression with respect to P, and its types, considered as generated by P, is denoted by w. thus, the whole series of negative and positive integers is of the type +w +w
Cantor, ‘Mathematische Annalen’ vol xlvi
…but other types of order, as we have seen, have very little resemblance to numbers.
What may be called ‘relation-arithmetic’
If PQ be two relations such that there is a one-one relation S whose domain is the field of P and which is such that Q = SPS, the P and Q are said to be ‘like’, the class of relations like P, denoted by λP, is called P’s ‘relation-number’.
If the fields of P and Q have no common terms, P and Q is defined to be P or Q or the relation which holds between ant term of the field P and any term of the field Q, and between no other terms. Thus P and Q is not equal to Q and P. again, λP and λQ is defined as λ(PandQ)
Let P be such a relation (an aliorelative whose field is composed of relations whose fields are mutually exclusive) p. its field, so that p is a class of relations. The SpP is to denote either one of the relations of the class p, or the relation of any term belonging to the field of some relation Q of the class p to the term belonging to the field of another relation R (of the class p) to which Q has the relation P. ( if P be a serial relation, and pa a class of serial relations, SpP will be the generating relation of the sum of the various series generated by terms of p taken in the order generated by P) we may define of the relation- members of the two various terms of p as the relation number of S If all the terms of p have the same relation number, say a, and if B be the relation number of P, axB will be defined to be the relation number of SpP. Proceeding in this way, it is easy to prove generally, the three formal laws which hold of well – ordered series namely:
(a +B) +y = a + (B +y)
A(B+y) = aB + ay
A(B)y = a(By)
0 is a cardinal number -> the number of numbers up to and including a finite number n is n+1.
N = a finite number
N + 1 = a new finite number different from all its predecessors.
Therefore finite cardinals form a progression and therefore the ordinal number w and the cardinal number a0, exist ( in the mathematical sense)
Hence, by more re-arrangement of the series of finite cardinals we obtain all ordinals of Cantor’s second class.
XXXIX The Infinitesimal Calculus
Traditional name for the differential and the integral calculus
The philosophical theory of the calculus: in a somewhat ‘disgraceful’ condition. – BR
Leibniz – thinking of Dynamics – his belief in the actual infinitesimal hindered him from discovering that the Calculus rests on the doctrine of limits..
In this respect (given proofs) Newton is to be preferred
Newton’s ‘Lemmas’ 327
Cohen’s ‘Princip der Infinitesimal methode’
The constructivist theory (as it results from modern mathematics)
Couterat, ‘De L’Infini Mathematique’
The differential co-efficient depends essentially upon the notion of a continuous function of a continuous variable.
Dini –
X . f(x) fx Î 1 -1 in C(int)
Fx int C fx Î1,1 PII comp of ~n
F(def. > (f)int a betB(a) some € rn.
F(x) contin(x = a)in a ¹ 0(o)<
E = int Î ¹ 0 for (x)(d)n-e ¹ f(a + d) – f(a) – s
Fx ∞ in x = a for V(F)(a0)
Lim < > (a) = f(a).
XI The Infinitesimal and the Improper Infinite
Cantor’s ‘Uneigentlich – Unendliches’
Axiom of Archimedes
If P, Q ba2n ^ a2m2 = NwR + 2m2 (if P< Ef(n) π nP>Q
(1) = 2x > (1 +1) (ζs) ^ 2(0) wζn Æ
(2) P(m) 1 n^ ~ a(f) ζ (n,c,d)
P~a(m) 2k Mm
(2x,n^c) = f(a) . f(b)
W .w.2 1 (1) f2 1d . 1yRb
= P2 . 2Pr
– QED
Staudt’s quadrilateral p.333
Stolz
duBois, Raymond, Stolz
Cantor
Axiom of Archimedes
Cantor’s viscious circle
XLI Philosophical Arguments Concerning the Infinitesimal
What can be mathematically demonstrated is true – BR
Cohen
dx . dy dy/dx – not a fraction
F(x) similar to Newton’s y1
Leibniz employed dy/dx because he believed in infinitisimals
Cohen dx and dy treated as separate entities
Space and motion
Epistemology – intuitions as well as categories
The notion of the method of limits
finis
tomreading 
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