Combining Texts
Ideas for
'Thinking About Mathematics', 'Definitiones cogitationesque metaphysicae' and 'Structuralism and the Notion of Dependence'
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11 ideas
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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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 / f. Zermelo numbers
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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
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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 / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
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14085
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'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo]
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14084
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Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo]
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14086
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'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo]
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14087
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'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
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14089
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Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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14083
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Structuralism is right about algebra, but wrong about sets [Linnebo]
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14090
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In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo]
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