Combining Philosophers
Ideas for Cleanthes, Bertrand Russell and Stuart Glennan
expand these ideas
|
start again
|
choose
another area for these philosophers
display all the ideas for this combination of philosophers
80 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10059
|
In mathematic we are ignorant of both subject-matter and truth [Russell]
|
6. Mathematics / A. Nature of Mathematics / 2. Geometry
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]
|
14151
|
Pure geometry is deductive, and neutral over what exists [Russell]
|
14153
|
In geometry, empiricists aimed at premisses consistent with experience [Russell]
|
14155
|
Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG]
|
14442
|
If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
18254
|
Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett]
|
14144
|
Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell]
|
14438
|
New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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]
|
14132
|
Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell]
|
13510
|
Could a number just be something which occurs in a progression? [Russell, by Hart,WD]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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]
|
14145
|
For Cantor ordinals are types of order, not numbers [Russell]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
14146
|
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
14135
|
Real numbers are a class of rational numbers (and so not really numbers at all) [Russell]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
14436
|
A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
14439
|
A complex number is simply an ordered couple of real numbers [Russell]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
14421
|
Discovering that 1 is a number was difficult [Russell]
|
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
14123
|
Some quantities can't be measured, and some non-quantities are measurable [Russell]
|
14158
|
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
14120
|
Counting explains none of the real problems about the foundations of arithmetic [Russell]
|
14424
|
Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell]
|
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
14118
|
We can define one-to-one without mentioning unity [Russell]
|
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
14441
|
The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell]
|
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
14119
|
We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
|
14133
|
There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
|
14420
|
Infinity and continuity used to be philosophy, but are now mathematics [Russell]
|
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
7556
|
A collection is infinite if you can remove some terms without diminishing its number [Russell]
|
14134
|
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
14143
|
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
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]
|
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
17627
|
It seems absurd to prove 2+2=4, where the conclusion is more certain than premises [Russell]
|
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
10052
|
Geometry is united by the intuitive axioms of projective geometry [Russell, by Musgrave]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
14431
|
The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
|
14124
|
Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
7530
|
Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk]
|
18246
|
Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell]
|
14422
|
Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
|
14423
|
'0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14147
|
Denying mathematical induction gave us the transfinite [Russell]
|
14125
|
Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
14116
|
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
14117
|
Numbers are properties of classes [Russell]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
14425
|
A number is something which characterises collections of the same size [Russell]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
14434
|
What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9977
|
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
14162
|
Mathematics doesn't care whether its entities exist [Russell]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17628
|
Arithmetic was probably inferred from relationships between physical objects [Russell]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
5399
|
Maths is not known by induction, because further instances are not needed to support it [Russell]
|
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
14465
|
Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
13414
|
For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf]
|
14103
|
Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
|
6108
|
Maths can be deduced from logical axioms and the logic of relations [Russell]
|
6423
|
We tried to define all of pure maths using logical premisses and concepts [Russell]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
21555
|
For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [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]
|
10418
|
Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
|
10047
|
Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
|
23478
|
Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell]
|
23457
|
Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell]
|
21556
|
Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
21718
|
Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B]
|
6. Mathematics / C. Sources of Mathematics / 7. Formalism
21570
|
Numbers are just verbal conveniences, which can be analysed away [Russell]
|
6424
|
Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess [Russell]
|
6425
|
Formalism can't apply numbers to reality, so it is an evasion [Russell]
|
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
6104
|
Numbers are classes of classes, and hence fictions of fictions [Russell]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
6426
|
Intuitionism says propositions are only true or false if there is a method of showing it [Russell]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
21559
|
We need rules for deciding which norms are predicative (unless none of them are) [Russell]
|
18126
|
A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock]
|
18128
|
Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock]
|
18124
|
Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock]
|
21568
|
A one-variable function is only 'predicative' if it is one order above its arguments [Russell]
|
21558
|
'Predicative' norms are those which define a class [Russell]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
14449
|
There is always something psychological about inference [Russell]
|