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### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

#### [general ideas concerning the structuralist approach]

33 ideas
 9793 Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
 8924 Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride]
 10180 Mathematicians do not study objects, but relations between objects [Poincaré]
 14434 What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell]
 10190 From the axiomatic point of view, mathematics is a storehouse of abstract structures [Bourbaki]
 10242 I apply structuralism to concrete and abstract objects indiscriminately [Quine]
 13415 An adequate account of a number must relate it to its series [Benacerraf]
 9907 If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf]
 9908 The job is done by the whole system of numbers, so numbers are not objects [Benacerraf]
 9909 The number 3 defines the role of being third in a progression [Benacerraf]
 9911 Number words no more have referents than do the parts of a ruler [Benacerraf]
 8925 Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf]
 9938 How can numbers be objects if order is their only property? [Benacerraf, by Putnam]
 15515 To be a structuralist, you quantify over relations [Lewis]
 8921 Structuralism is now common, studying relations, with no regard for what the objects might be [Hellman]
 9966 The subject-matter of (pure) mathematics is abstract structure [Jubien]
 6300 Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik]
 6303 Sets are positions in patterns [Resnik]
 10218 Baseball positions and chess pieces depend entirely on context [Shapiro]
 10224 The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
 10228 Could infinite structures be apprehended by pattern recognition? [Shapiro]
 10230 The 4-pattern is the structure common to all collections of four objects [Shapiro]
 10249 The main mathematical structures are algebraic, ordered, and topological [Shapiro]
 10273 Some structures are exemplified by both abstract and concrete [Shapiro]
 10276 Mathematical structures are defined by axioms, or in set theory [Shapiro]
 10184 Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure [Burgess]
 10189 There is no one relation for the real number 2, as relations differ in different models [Burgess]
 8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
 8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
 10167 Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
 8702 In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
 8701 The number 8 in isolation from the other numbers is of no interest [Friend]
 17931 Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan]