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Ideas for
'Structures and Structuralism in Phil of Maths', 'Sociobiology' and 'Mathematical logic and theory of types'
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21 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10165
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'Analysis' is the theory of the real numbers [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10174
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Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10164
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Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10172
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Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
10167
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Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
10169
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Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
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10179
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There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
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10181
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Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price]
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10182
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There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
10168
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Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
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10178
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Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
10176
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Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
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10177
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Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10171
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The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
10418
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Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
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10047
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Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
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23478
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Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
21718
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Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
18126
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A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock]
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18128
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Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock]
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18124
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Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock]
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