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Single Idea 9977
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
]
Full Idea
It is impossible that the ordinals should be, as Dedekind suggests, nothing but the terms of such relations as constitute a progression. If they are anything at all, they must be intrinsically something.
Gist of Idea
Ordinals can't be defined just by progression; they have intrinsic qualities
Source
Bertrand Russell (The Principles of Mathematics [1903], §242)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.249
A Reaction
This is the obvious platonist response to the incipient doctrine of structuralism. We have a chicken-and-egg problem. Bricks need intrinsic properties to make a structure. A structure isomorphic to numbers is not thereby the numbers.
The
23 ideas
with the same theme
[objections to structuralism about mathematics]:
8639
|
If numbers are supposed to be patterns, each number can have many patterns
[Frege]
|
9977
|
Ordinals can't be defined just by progression; they have intrinsic qualities
[Russell]
|
9627
|
Different versions of set theory result in different underlying structures for numbers
[Zermelo, by Brown,JR]
|
9828
|
Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
[Dummett]
|
9829
|
The identity of a number may be fixed by something outside structure - by counting
[Dummett]
|
9192
|
The number 4 has different positions in the naturals and the wholes, with the same structure
[Dummett]
|
18116
|
Numbers can't be positions, if nothing decides what position a given number has
[Bostock]
|
18117
|
Structuralism falsely assumes relations to other numbers are numbers' only properties
[Bostock]
|
10815
|
We don't need 'abstract structures' to have structural truths about successor functions
[Lewis]
|
10628
|
The structural view of numbers doesn't fit their usage outside arithmetical contexts
[Hale/Wright]
|
10629
|
If structures are relative, this undermines truth-value and objectivity
[Hale/Wright]
|
18500
|
How could structures be mathematical truthmakers? Maths is just true, without truthmakers
[Heil]
|
10274
|
Does someone using small numbers really need to know the infinite structure of arithmetic?
[Shapiro]
|
10186
|
If set theory is used to define 'structure', we can't define set theory structurally
[Burgess]
|
10187
|
Abstract algebra concerns relations between models, not common features of all the models
[Burgess]
|
10188
|
How can mathematical relations be either internal, or external, or intrinsic?
[Burgess]
|
9628
|
Sets seem basic to mathematics, but they don't suit structuralism
[Brown,JR]
|
10171
|
The existence of an infinite set is assumed by Relativist Structuralism
[Reck/Price]
|
8926
|
For mathematical objects to be positions, positions themselves must exist first
[MacBride]
|
14083
|
Structuralism is right about algebra, but wrong about sets
[Linnebo]
|
14090
|
In mathematical structuralism the small depends on the large, which is the opposite of physical structures
[Linnebo]
|
14505
|
Some questions concern mathematical entities, rather than whole structures
[Koslicki]
|
17932
|
If 'in re' structures relies on the world, does the world contain rich enough structures?
[Colyvan]
|