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Single Idea 10167
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
]
Full Idea
Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures.
Clarification
[Bourbaki is a notably unusual mathematician - worth investigating!]
Gist of Idea
Structuralism emerged from abstract algebra, axioms, and set theory and its structures
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)
A Reaction
In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent.
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
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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
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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]
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