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Full Idea
Mathematicians tend to regard the differences between isomorphic mathematical structures as unimportant.
Gist of Idea
Differences between isomorphic structures seem unimportant
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.69
A Reaction
This seems to be a pointer towards Structuralism as the underlying story in mathematics. The intrinsic character of so-called 'objects' seems unimportant. How theories map onto one another (and onto the world?) is all that matters?
| 14207 | If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth [Putnam] |
| 14212 | A consistent theory just needs one model; isomorphic versions will do too, and large domains provide those [Lewis] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
| 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
| 10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 10884 | A theory is 'categorical' if it has just one model up to isomorphism [Horsten] |
| 10758 | If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg] |