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Full Idea
Of the equivalence relationships which occur between patterns, congruence is the strongest, equivalence the next, and mutual occurrence the weakest. None of these is identity, which would require the same position.
Clarification
'Congruence' is identity of shape
Gist of Idea
Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest
Source
Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.3)
Book Ref
Resnik,Michael D.: 'Mathematics as a Science of Patterns' [OUP 1999], p.209
A Reaction
This gives some indication of how an account of mathematics as a science of patterns might be built up. Presumably the recognition of these 'degrees of strength' cannot be straightforward observation, but will need an a priori component?
| 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] |