Full Idea
The central dogma of the axiomatic method is this: isomorphic structures are mathematically indistinguishable in their essential properties.
Gist of Idea
Axiomatiation relies on isomorphic structures being essentially the same
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.406-2)
Book Reference
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.406
A Reaction
Hence it is not that we have to settle for the success of a system 'up to isomorphism', since that was the original aim. The structures must differ in their non-essential properties, or they would be the same system.