Combining Texts
Ideas for
'Thinking About Mathematics', 'Lectura' and 'Grundgesetze der Arithmetik 2 (Basic Laws)'
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19 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
9886
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Cardinals say how many, and reals give measurements compared to a unit quantity [Frege]
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8763
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The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
9889
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Real numbers are ratios of quantities [Frege, by Dummett]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18249
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Cauchy gave a formal definition of a converging sequence. [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764
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Categories are the best foundation for mathematics [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
10553
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A number is a class of classes of the same cardinality [Frege, by Dummett]
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10020
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Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
8762
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Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8760
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Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
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8761
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A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
8744
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Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
9887
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Formalism misunderstands applications, metatheory, and infinity [Frege, by Dummett]
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8751
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Only applicability raises arithmetic from a game to a science [Frege]
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8749
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Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
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8750
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Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
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8752
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Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8753
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Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8731
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Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
8730
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'Impredicative' definitions refer to the thing being described [Shapiro]
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