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Single Idea 18246
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
]
Full Idea
Dedekind's demonstrations nowhere - not even where he comes to cardinals - involve any property distinguishing numbers from other progressions.
Gist of Idea
Dedekind failed to distinguish the numbers from other progressions
Source
comment on Bertrand Russell (The Principles of Mathematics [1903], p.249) by Stewart Shapiro - Philosophy of Mathematics 5.4
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.175
A Reaction
Shapiro notes that his sounds like Frege's Julius Caesar problem, of ensuring that your definition really does capture a number. Russell is objecting to mathematical structuralism.
The
92 ideas
from 'The Principles of Mathematics'
15895
|
Russell discovered the paradox suggested by Burali-Forti's work
[Russell, by Lavine]
|
18254
|
Russell's approach had to treat real 5/8 as different from rational 5/8
[Russell, by Dummett]
|
7530
|
Russell tried to replace Peano's Postulates with the simple idea of 'class'
[Russell, by Monk]
|
11010
|
Being is what belongs to every possible object of thought
[Russell]
|
11849
|
It at least makes sense to say two objects have all their properties in common
[Wittgenstein on Russell]
|
19164
|
If propositions are facts, then false and true propositions are indistinguishable
[Davidson on Russell]
|
15894
|
Russell invented the naïve set theory usually attributed to Cantor
[Russell, by Lavine]
|
18722
|
Negations are not just reversals of truth-value, since that can happen without negation
[Wittgenstein on Russell]
|
14102
|
What is true or false is not mental, and is best called 'propositions'
[Russell]
|
14103
|
Pure mathematics is the class of propositions of the form 'p implies q'
[Russell]
|
14104
|
Constants are absolutely definite and unambiguous
[Russell]
|
14105
|
There seem to be eight or nine logical constants
[Russell]
|
14106
|
Implication cannot be defined
[Russell]
|
14107
|
Terms are identical if they belong to all the same classes
[Russell]
|
14108
|
It would be circular to use 'if' and 'then' to define material implication
[Russell]
|
14109
|
The study of grammar is underestimated in philosophy
[Russell]
|
7781
|
I call an object of thought a 'term'. This is a wide concept implying unity and existence.
[Russell]
|
14110
|
Proposition contain entities indicated by words, rather than the words themselves
[Russell]
|
14111
|
A proposition is a unity, and analysis destroys it
[Russell]
|
14112
|
A set has some sort of unity, but not enough to be a 'whole'
[Russell]
|
14113
|
The null class is a fiction
[Russell]
|
14114
|
Variables don't stand alone, but exist as parts of propositional functions
[Russell]
|
14115
|
Definition by analysis into constituents is useless, because it neglects the whole
[Russell]
|
14116
|
Numbers were once defined on the basis of 1, but neglected infinities and +
[Russell]
|
14117
|
Numbers are properties of classes
[Russell]
|
14118
|
We can define one-to-one without mentioning unity
[Russell]
|
14119
|
We do not currently know whether, of two infinite numbers, one must be greater than the other
[Russell]
|
14120
|
Counting explains none of the real problems about the foundations of arithmetic
[Russell]
|
14121
|
The part-whole relation is ultimate and indefinable
[Russell]
|
14122
|
Analysis gives us nothing but the truth - but never the whole truth
[Russell]
|
14123
|
Some quantities can't be measured, and some non-quantities are measurable
[Russell]
|
10583
|
Abstraction principles identify a common property, which is some third term with the right relation
[Russell]
|
10582
|
The principle of Abstraction says a symmetrical, transitive relation analyses into an identity
[Russell]
|
10584
|
A certain type of property occurs if and only if there is an equivalence relation
[Russell]
|
14124
|
Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater
[Russell]
|
14125
|
Finite numbers, unlike infinite numbers, obey mathematical induction
[Russell]
|
14126
|
Order rests on 'between' and 'separation'
[Russell]
|
14127
|
Order depends on transitive asymmetrical relations
[Russell]
|
10586
|
'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness
[Russell]
|
10585
|
Symmetrical and transitive relations are formally like equality
[Russell]
|
14128
|
Some claim priority for the ordinals over cardinals, but there is no logical priority between them
[Russell]
|
14129
|
Ordinals presuppose two relations, where cardinals only presuppose one
[Russell]
|
9977
|
Ordinals can't be defined just by progression; they have intrinsic qualities
[Russell]
|
14132
|
Properties of numbers don't rely on progressions, so cardinals may be more basic
[Russell]
|
14133
|
There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal)
[Russell]
|
14134
|
Infinite numbers are distinguished by disobeying induction, and the part equalling the whole
[Russell]
|
14135
|
Real numbers are a class of rational numbers (and so not really numbers at all)
[Russell]
|
14137
|
'Any' is better than 'all' where infinite classes are concerned
[Russell]
|
14138
|
You can't get a new transfinite cardinal from an old one just by adding finite numbers to it
[Russell]
|
14140
|
For every transfinite cardinal there is an infinite collection of transfinite ordinals
[Russell]
|
14139
|
Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic
[Russell]
|
14142
|
Ordinals are types of series of terms in a row, rather than the 'nth' instance
[Russell]
|
14141
|
Ordinals are defined through mathematical induction
[Russell]
|
14143
|
ω names the whole series, or the generating relation of the series of ordinal numbers
[Russell]
|
14144
|
Ordinals result from likeness among relations, as cardinals from similarity among classes
[Russell]
|
14145
|
For Cantor ordinals are types of order, not numbers
[Russell]
|
14146
|
We aren't sure if one cardinal number is always bigger than another
[Russell]
|
14147
|
Denying mathematical induction gave us the transfinite
[Russell]
|
14149
|
The Achilles Paradox concerns the one-one correlation of infinite classes
[Russell]
|
14148
|
Infinite regresses have propositions made of propositions etc, with the key term reappearing
[Russell]
|
14151
|
Pure geometry is deductive, and neutral over what exists
[Russell]
|
14153
|
In geometry, empiricists aimed at premisses consistent with experience
[Russell]
|
14152
|
In geometry, Kant and idealists aimed at the certainty of the premisses
[Russell]
|
14154
|
Geometry throws no light on the nature of actual space
[Russell]
|
14155
|
Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective)
[Russell, by PG]
|
14156
|
Mathematicians don't distinguish between instants of time and points on a line
[Russell]
|
14158
|
Quantity is not part of mathematics, where it is replaced by order
[Russell]
|
14159
|
In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives
[Russell]
|
14160
|
Space is the extension of 'point', and aggregates of points seem necessary for geometry
[Russell]
|
14161
|
Many things have being (as topics of propositions), but may not have actual existence
[Russell]
|
22303
|
It makes no sense to say that a true proposition could have been false
[Russell]
|
14162
|
Mathematics doesn't care whether its entities exist
[Russell]
|
14163
|
Four classes of terms: instants, points, terms at instants only, and terms at instants and points
[Russell]
|
14166
|
Unities are only in propositions or concepts, and nothing that exists has unity
[Russell]
|
14164
|
The only unities are simples, or wholes composed of parts
[Russell]
|
14165
|
Analysis falsifies, if when the parts are broken down they are not equivalent to their sum
[Russell]
|
14167
|
The only classes are things, predicates and relations
[Russell]
|
14169
|
The 'universe' can mean what exists now, what always has or will exist
[Russell]
|
14168
|
Occupying a place and change are prior to motion, so motion is just occupying places at continuous times
[Russell]
|
14170
|
Change is obscured by substance, a thing's nature, subject-predicate form, and by essences
[Russell]
|
14171
|
Force is supposed to cause acceleration, but acceleration is a mathematical fiction
[Russell]
|
14172
|
Moments and points seem to imply other moments and points, but don't cause them
[Russell]
|
14173
|
What exists has causal relations, but non-existent things may also have them
[Russell]
|
14174
|
The laws of motion and gravitation are just parts of the definition of a kind of matter
[Russell]
|
14175
|
We can drop 'cause', and just make inferences between facts
[Russell]
|
14176
|
"The death of Caesar is true" is not the same proposition as "Caesar died"
[Russell]
|
21555
|
For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x
[Russell]
|
18002
|
As well as a truth value, propositions have a range of significance for their variables
[Russell]
|
18003
|
In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless
[Russell, by Magidor]
|
21341
|
Philosophers of logic and maths insisted that a vocabulary of relations was essential
[Russell, by Heil]
|
18246
|
Dedekind failed to distinguish the numbers from other progressions
[Shapiro on Russell]
|
19157
|
Russell said the proposition must explain its own unity - or else objective truth is impossible
[Russell, by Davidson]
|