Combining Philosophers
Ideas for Herodotus, R Kaplan / E Kaplan and Brian Clegg
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13 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 / 3. Nature of Numbers / b. Types of number
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15712
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1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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10861
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Beyond infinity cardinals and ordinals can come apart [Clegg]
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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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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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15711
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The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan]
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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 / 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 / 4. Using Numbers / f. Arithmetic
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15714
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'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan]
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15715
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'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan]
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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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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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