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Single Idea 14153
[filed under theme 6. Mathematics / A. Nature of Mathematics / 2. Geometry
]
Full Idea
The approach to practical geometry of the empiricists, notably Mill, was to show that no other set of premisses would give results consistent with experience.
Gist of Idea
In geometry, empiricists aimed at premisses consistent with experience
Source
Bertrand Russell (The Principles of Mathematics [1903], §353)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.373
A Reaction
The modern phrase might be that geometry just needs to be 'empirically adequate'. The empiricists are faced with the possibility of more than one successful set of premisses, and the idealist don't know how to demonstrate truth.
Related Idea
Idea 14152
In geometry, Kant and idealists aimed at the certainty of the premisses [Russell]
The
92 ideas
from 'The Principles of Mathematics'
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]
|
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]
|
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]
|
14118
|
We can define one-to-one without mentioning unity
[Russell]
|
14117
|
Numbers are properties of classes
[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]
|
14141
|
Ordinals are defined through mathematical induction
[Russell]
|
14142
|
Ordinals are types of series of terms in a row, rather than the 'nth' instance
[Russell]
|
14139
|
Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic
[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]
|
14168
|
Occupying a place and change are prior to motion, so motion is just occupying places at continuous times
[Russell]
|
14169
|
The 'universe' can mean what exists now, what always has or will exist
[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]
|