Combining Texts
Ideas for
'The Intrinsic Quality of Experience', 'Infinity: Quest to Think the Unthinkable' and 'Cardinality, Counting and Equinumerosity'
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16 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 / a. Numbers
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17453
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The meaning of a number isn't just the numerals leading up to it [Heck]
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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 / f. Cardinal numbers
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17457
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A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck]
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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 / 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 / c. Counting procedure
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17448
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In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck]
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17455
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Is counting basically mindless, and independent of the cardinality involved? [Heck]
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17456
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Counting is the assignment of successively larger cardinal numbers to collections [Heck]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
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17450
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Understanding 'just as many' needn't involve grasping one-one correspondence [Heck]
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17451
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We can know 'just as many' without the concepts of equinumerosity or numbers [Heck]
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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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