Combining Texts
Ideas for
'Infinity: Quest to Think the Unthinkable', 'Introduction to Mathematical Philosophy' and 'Ecce Homo'
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26 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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10880
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Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]
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6. Mathematics / A. Nature of Mathematics / 2. Geometry
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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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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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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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10860
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An ordinal number is defined by the set that comes before it [Clegg]
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10861
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Beyond infinity cardinals and ordinals can come apart [Clegg]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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10854
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Transcendental numbers can't be fitted to finite equations [Clegg]
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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 / k. Imaginary numbers
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10858
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By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
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10853
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Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg]
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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 / c. Counting procedure
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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 / 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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10866
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Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
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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 / g. Continuum Hypothesis
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10869
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The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
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10862
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The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
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14431
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The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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14423
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'0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell]
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14422
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Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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14425
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A number is something which characterises collections of the same size [Russell]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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14434
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What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell]
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
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14465
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Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
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13414
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For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
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14449
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There is always something psychological about inference [Russell]
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