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] |
9908 | The job is done by the whole system of numbers, so numbers are not objects [Benacerraf] |
9907 | If any recursive sequence will explain ordinals, then it seems to be the structure which matters [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] |