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Single Idea 10176

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism ]

Full Idea

Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model.

Gist of Idea

Universalist Structuralism is based on generalised if-then claims, not one particular model

Source

E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)

A Reaction

There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator.

The 12 ideas with the same theme [structuralism with real objects or real structures]:

6295 There are too many mathematical objects for them all to be mental or physical [Resnik]
6296 Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik]
6301 Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik]
6302 Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik]
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 Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
10177 Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
14089 Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo]
8700 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]