Combining Texts
Ideas for
'Understanding and Essence', 'Infinity: Quest to Think the Unthinkable' and 'Replies on 'Limits of Abstraction''
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12 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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
10573
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Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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
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
10575
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Why should a Dedekind cut correspond to a number? [Fine,K]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
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
10574
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Unless we know whether 0 is identical with the null set, we create confusions [Fine,K]
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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 / 5. The Infinite / a. The Infinite
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
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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