Combining Texts
Ideas for
'Thinking About Mathematics', 'The Evolution of Co-Operation' and 'Cardinality, Counting and Equinumerosity'
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24 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
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 / b. Types of number
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 / f. Cardinal numbers
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 / 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 / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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
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 / 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 / 4. Axioms for Number / d. Peano arithmetic
17459
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Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck]
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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 / 1. Mathematical Platonism / b. Against mathematical platonism
17454
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Children can use numbers, without a concept of them as countable objects [Heck]
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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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17458
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Equinumerosity is not the same concept as one-one correspondence [Heck]
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17449
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We can understand cardinality without the idea of one-one correspondence [Heck]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
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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