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

[general ideas concerning the structuralist approach]

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