Combining Philosophers
Ideas for H.Putnam/P.Oppenheim, Donald C. Williams and Bertrand Russell
expand these ideas
|
start again
|
choose
another area for these philosophers
display all the ideas for this combination of philosophers
37 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
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]
|
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
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]
|
14145
|
For Cantor ordinals are types of order, not numbers [Russell]
|
14141
|
Ordinals are defined through mathematical induction [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
14134
|
Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell]
|
7556
|
A collection is infinite if you can remove some terms without diminishing its number [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]
|