Ideas of E Reck / M Price, by Theme
[American, fl. 2000, Both at the University of Chicago.]
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3. Truth / F. Semantic Truth / 2. Semantic Truth
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10170
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While true-in-a-model seems relative, true-in-all-models seems not to be
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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10166
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ZFC set theory has only 'pure' sets, without 'urelements'
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5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
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10175
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Three types of variable in second-order logic, for objects, functions, and predicates/sets
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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10165
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'Analysis' is the theory of the real numbers
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
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10174
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Mereological arithmetic needs infinite objects, and function definitions
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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10164
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Peano Arithmetic can have three second-order axioms, plus '1' and 'successor'
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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10172
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Set-theory gives a unified and an explicit basis for mathematics
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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10167
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Structuralism emerged from abstract algebra, axioms, and set theory and its structures
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
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10169
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Relativist Structuralism just stipulates one successful model as its arithmetic
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10179
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There are 'particular' structures, and 'universal' structures (what the former have in common)
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10181
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Pattern Structuralism studies what isomorphic arithmetic models have in common
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10182
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There are Formalist, Relativist, Universalist and Pattern structuralism
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
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10168
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Formalist Structuralism says the ontology is vacuous, or formal, or inference relations
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10178
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Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
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10176
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Universalist Structuralism is based on generalised if-then claims, not one particular model
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10177
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Universalist Structuralism eliminates the base element, as a variable, which is then quantified out
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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10171
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The existence of an infinite set is assumed by Relativist Structuralism
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8. Modes of Existence / E. Nominalism / 6. Mereological Nominalism
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10173
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A nominalist might avoid abstract objects by just appealing to mereological sums
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