Combining Texts
Ideas for
'Metaphysics: the logical approach', 'Foundations without Foundationalism' and '30: Book of Amos'
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12 ideas
6. Mathematics / A. Nature of Mathematics / 2. Geometry
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3332
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Greeks saw the science of proportion as the link between geometry and arithmetic [Benardete,JA]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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3330
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Negatives, rationals, irrationals and imaginaries are all postulated to solve baffling equations [Benardete,JA]
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3337
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Natural numbers are seen in terms of either their ordinality (Peano), or cardinality (set theory) [Benardete,JA]
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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 / 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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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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13657
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First-order arithmetic can't even represent basic number theory [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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13656
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Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
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13664
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Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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13625
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Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
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13663
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Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
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