Combining Texts
Ideas for
'Mahaprajnaparamitashastra', 'What Required for Foundation for Maths?' and 'Principia Mathematica'
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24 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
17784
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Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18248
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A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
17782
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Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
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17781
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Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
17799
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Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
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17797
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Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17775
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If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
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17776
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The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
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17777
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Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
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17804
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Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17792
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1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17793
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It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / a. Defining numbers
18152
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Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Russell/Whitehead, by Bostock]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17794
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Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
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17802
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We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
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17805
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Set theory is not just another axiomatised part of mathematics [Mayberry]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
10025
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Russell and Whitehead took arithmetic to be higher-order logic [Russell/Whitehead, by Hodes]
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8683
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Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend]
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10037
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'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
10093
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The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman]
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8691
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The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
10305
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In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
8684
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Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Russell/Whitehead, by Friend]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
8746
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To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro]
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