Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 I - Isomorphism


Isomorphism: From the Greek: "Having the same form". As spelled out with some modern mathematics: A 1-1 mapping between two sets such that some (structural) relations - the forms - are preserved.

The mathematical statement just given has the following import: Let A1..An and B1...Bn be sets,; let f be a 1-1 function from Ai to Bi for 1<=i<=n; let S be a relation of the elements of A1..An and R a relation between the elements of B1..Bn. Then using Cartesian Products from set-theory and first-order predicate logic:

       f is an isomorphism from A1*..*An to B1*..*Bn for S and R
  IFF S(a1 .. an) iff R(f(a1) .. f(an))

In words: IFF a1 .. an stand in relation S precisely if f(a1) .. f(am), that is the values of a1 .. an of f, in that order, stand in relation R.

Often the relations S and R are taken to be the same: Then the same relation - such as a shape, form, curve - obtains in B and in A for the elements f maps from A to B.
Then again, this may not hold, or may hold only in fairly fanciful ways, such as the curves and bends in a line on a map that is supposed to represent the curves and bends in a river.

There are many variants, applications and precifisications in various fields of mathematics of the ntionn of isomorphism and also outside it.



See also: Function, Information, Map, Relation, Representing, Structure


Carnap, Hinton, Hawkins, Muller, Russell, Stegmuller, Wittgenstein

 Original: Feb 8, 2012                                                Last edited: 09 February 2012.   Top