Combining Philosophers
Ideas for Rescher,N/Oppenheim,P, Bertrand Russell and Jonathan Glover
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37 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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10059
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In mathematic we are ignorant of both subject-matter and truth [Russell]
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6. Mathematics / A. Nature of Mathematics / 2. Geometry
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14152
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In geometry, Kant and idealists aimed at the certainty of the premisses [Russell]
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14154
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Geometry throws no light on the nature of actual space [Russell]
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14151
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Pure geometry is deductive, and neutral over what exists [Russell]
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14153
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In geometry, empiricists aimed at premisses consistent with experience [Russell]
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14155
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Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG]
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14442
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If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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18254
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Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett]
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14144
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Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell]
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14438
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New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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14128
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Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell]
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14129
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Ordinals presuppose two relations, where cardinals only presuppose one [Russell]
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14132
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Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell]
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13510
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Could a number just be something which occurs in a progression? [Russell, by Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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14141
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Ordinals are defined through mathematical induction [Russell]
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14142
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Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell]
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14139
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Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell]
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14145
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For Cantor ordinals are types of order, not numbers [Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
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14146
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We aren't sure if one cardinal number is always bigger than another [Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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14135
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Real numbers are a class of rational numbers (and so not really numbers at all) [Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
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14436
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A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / j. Complex numbers
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14439
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A complex number is simply an ordered couple of real numbers [Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
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14421
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Discovering that 1 is a number was difficult [Russell]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
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14123
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Some quantities can't be measured, and some non-quantities are measurable [Russell]
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14158
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Quantity is not part of mathematics, where it is replaced by order [Russell]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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14120
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Counting explains none of the real problems about the foundations of arithmetic [Russell]
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14424
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Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
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14118
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We can define one-to-one without mentioning unity [Russell]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
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14441
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The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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14119
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We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
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14133
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There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
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14420
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Infinity and continuity used to be philosophy, but are now mathematics [Russell]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
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7556
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A collection is infinite if you can remove some terms without diminishing its number [Russell]
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14134
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Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
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14143
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ω names the whole series, or the generating relation of the series of ordinal numbers [Russell]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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14138
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You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell]
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14140
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For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell]
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