Ideas from 'Structures and Structuralism in Phil of Maths' by E Reck / M Price [2000], by Theme Structure
[found in 'Synthese' (ed/tr ) [ ,]].
Click on the Idea Number for the full details 
back to texts

expand these ideas
3. Truth / F. Semantic Truth / 2. Semantic Truth
10170

While trueinamodel seems relative, trueinallmodels seems not to be

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10166

ZFC set theory has only 'pure' sets, without 'urelements'

5. Theory of Logic / G. Quantification / 5. SecondOrder Quantification
10175

Three types of variable in secondorder logic, for objects, functions, and predicates/sets

6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
10165

'Analysis' is the theory of the real numbers

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
10174

Mereological arithmetic needs infinite objects, and function definitions

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2ndorder
10164

Peano Arithmetic can have three secondorder axioms, plus '1' and 'successor'

6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
10172

Settheory gives a unified and an explicit basis for mathematics

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
10167

Structuralism emerged from abstract algebra, axioms, and set theory and its structures

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / b. Varieties of structuralism
10169

Relativist Structuralism just stipulates one successful model as its arithmetic

10179

There are 'particular' structures, and 'universal' structures (what the former have in common)

10181

Pattern Structuralism studies what isomorphic arithmetic models have in common

10182

There are Formalist, Relativist, Universalist and Pattern structuralism

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / c. Nominalist structuralism
10168

Formalist Structuralism says the ontology is vacuous, or formal, or inference relations

10178

Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism
10176

Universalist Structuralism is based on generalised ifthen claims, not one particular model

10177

Universalist Structuralism eliminates the base element, as a variable, which is then quantified out

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
10171

The existence of an infinite set is assumed by Relativist Structuralism

8. Modes of Existence / E. Nominalism / 6. Mereological Nominalism
10173

A nominalist might avoid abstract objects by just appealing to mereological sums
