Definition
An isomorphism between two systems is a one-to-one correspondence between their elements that preserves the operations and relations of both. If f maps system A to system B, then for any operation op on A and elements x, y of A, f(x op y) = f(x) op' f(y) where op' is the corresponding operation on B. The two systems are the same up to relabeling.
In Gödel, Escher, Bach Hofstadter argues that meaning emerges through isomorphism: when a formal system's symbols can be mapped onto an external domain in a way that preserves theoremhood, the symbols inherit meaning from the mapping. The pq-system's strings become arithmetic equations because the rules of pq mirror the laws of addition.
Why it matters
How it works
To establish an isomorphism, list the elements and operations of each system. Find a bijection between elements that turns each operation in one system into the corresponding operation in the other. The bijection must commute with every operation: applying the operation first and then the bijection gives the same result as applying the bijection first and then the corresponding operation.
Once an isomorphism is in hand, anything provable in one system has a counterpart provable in the other, with the same proof modulo renaming. This is why mathematicians work hard to find isomorphisms — they let a hard problem in one domain be solved by translating it to an easier domain. The shift from groups to permutations, from Boolean algebras to set theory, from physical electrical circuits to abstract linear algebra — each is isomorphism-mediated.