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Single Idea 10177
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
]
Full Idea
Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out.
Gist of Idea
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)
A Reaction
I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra.
The
12 ideas
with the same theme
[structuralism with real objects or real structures]:
6295
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There are too many mathematical objects for them all to be mental or physical
[Resnik]
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6296
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Maths is pattern recognition and representation, and its truth and proofs are based on these
[Resnik]
|
6301
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Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest
[Resnik]
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6302
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Structuralism must explain why a triangle is a whole, and not a random set of points
[Resnik]
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10220
|
Because one structure exemplifies several systems, a structure is a one-over-many
[Shapiro]
|
10223
|
There is no 'structure of all structures', just as there is no set of all sets
[Shapiro]
|
8703
|
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics
[Shapiro, by Friend]
|
9625
|
To see a structure in something, we must already have the idea of the structure
[Brown,JR]
|
10176
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Universalist Structuralism is based on generalised if-then claims, not one particular model
[Reck/Price]
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10177
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Universalist Structuralism eliminates the base element, as a variable, which is then quantified out
[Reck/Price]
|
14089
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Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure
[Linnebo]
|
8700
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'In re' structuralism says that the process of abstraction is pattern-spotting
[Friend]
|