Combining Philosophers
Ideas for Hermarchus, Bill Brewer and Stewart Shapiro
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18 ideas
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
10200
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We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
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10210
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If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
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10215
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Platonists claim we can state the essence of a number without reference to the others [Shapiro]
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10233
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Platonism must accept that the Peano Axioms could all be false [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
10244
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Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
10280
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A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
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
13625
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Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
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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
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 / a. Constructivism
10254
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Can the ideal constructor also destroy objects? [Shapiro]
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10255
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Presumably nothing can block a possible dynamic operation? [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
13663
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Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
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8730
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'Impredicative' definitions refer to the thing being described [Shapiro]
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