Combining Texts
Ideas for
'Structures and Structuralism in Phil of Maths', 'The Essential Child' and 'The Theory of Logical Types'
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17 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
23457
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Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell]
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21556
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Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
21568
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A one-variable function is only 'predicative' if it is one order above its arguments [Russell]
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