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Full Idea
I argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides the connection between the mathematical proof and mathematical truth.
Gist of Idea
Maths is pattern recognition and representation, and its truth and proofs are based on these
Source
Michael D. Resnik (Maths as a Science of Patterns [1997], One.1)
Book Ref
Resnik,Michael D.: 'Mathematics as a Science of Patterns' [OUP 1999], p.9
A Reaction
The suggestion that patterns are at the basis of the ontology of mathematics is the most illuminating thought I have encountered in the area. It immediately opens up the possibility of maths being an entirely empirical subject.
| 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] |