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Single Idea 10230
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
]
Full Idea
The 4-pattern is the structure common to all collections of four objects.
Gist of Idea
The 4-pattern is the structure common to all collections of four objects
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 4.2)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.115
A Reaction
This seems open to Frege's objection, that you can have four disparate abstract concepts, or four spatially scattered items of unknown pattern. It certainly isn't a visual pattern, but then if the only detectable pattern is the fourness, it is circular.
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]
|