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6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism

[structuralism with real objects or real structures]

12 ideas
There are too many mathematical objects for them all to be mental or physical [Resnik]
Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik]
Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik]
Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik]
Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Friend on Shapiro]
To see a structure in something, we must already have the idea of the structure [Brown,JR]
Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo]
'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]