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Single Idea 10190

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism ]

Full Idea

From the axiomatic point of view, mathematics appears as a storehouse of abstract forms - the mathematical structures.

Gist of Idea

From the axiomatic point of view, mathematics is a storehouse of abstract structures

Source

Nicholas Bourbaki (The Architecture of Mathematics [1950], 221-32), quoted by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.79

Book Ref

-: 'Bulletin of Symbolic Logic' [-], p.79

A Reaction

This seems to be the culmination of the structuralist view that developed from Dedekind and Hilbert, and was further developed by philosophers in the 1990s.

The 34 ideas with the same theme [general ideas concerning the structuralist approach]:

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]
23448 Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo]
8701 The number 8 in isolation from the other numbers is of no interest [Friend]
8702 In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
17931 Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan]