Combining Texts
Ideas for
'Law and Causality', 'Foundations without Foundationalism' and 'Logicism, Some Considerations (PhD)'
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7 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
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13412
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Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
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13413
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We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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13641
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Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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13411
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If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
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13676
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Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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13677
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Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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13652
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The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
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