# Can I say that extension is a synonym for isomorphism?

I read in Necessity and Sufficiency that

For example, in graph theory a graph G is called bipartite if it is possible to assign to each of its vertices the color black or white in such a way that every edge of G has one endpoint of each color. And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length cycles. Thus, discovering whether a graph has any odd cycles tells one whether it is bipartite and vice versa. A philosopher[5] might characterize this state of affairs thus: "Although the concepts of bipartiteness and absence of odd cycles differ in intension, they have identical extension.[6]

So, it looks like extension of equality. Isomorphism also looks like extension of equality. In programming languages and in math, I believe, we say that some objects are not equal (they are different instances) but they can be attributed to the same equivalence class. That is, they are equivalent.

Isomorphism says that you should use one equivalent object in one domain and another in the other domain. For instance, I should use 1 mile in UK and 1.609344 km in Europe, I speak about bipartite graphs in coloring and acyclic in some other context. Can I say that isomorphic objects are extensionally equivalent, that is, extension is basically an isomorphism? Both intentions and isomorphisms look like different aspects of the same entity, viewed from different angles for me.

• No. Extension is a set of objects satisfying a condition ("concept"), it is not an extension of equality. Coextensiveness (having the same extension) is an extension of equality for concepts, they may differ by only having a different intension ("meaning"). Isomorphism is an extension of equality for structures (sets with extra stuff like operations, relations, etc.) , but structures unlike concepts have no extensions, so coextensiveness and isomorphism have little to do with each other. Even structure concepts being coextensive is different from corresponding structures being isomorphic. Mar 31, 2016 at 21:11

Extension and intension are qualities of concepts, and isomorphism is a quality of objects. The extension of a concept is the set of all objects falling under that concept: the extension of the concept "red" is the set of all red things. Two concepts that have the same extension therefore have the same set of objects as their extensions. An isomorphism is a structure-preserving mapping from one object onto another. A bipartite graph is a graph with no odd-length cycles; they are (mathematically) equivalent definitions. Two definitions are mathematically equivalent if they necessarily have the same extension. Definitions are not mathematical objects, and have no concept of isomorphism.

• I asked because I do not understand why you cannot treat concepts as objects, especially after such identification seems to be the basic tool in the theory of categories, which seems to deal with isomorphisms professionally. Thanks for making this distinction explicit. Mar 31, 2016 at 20:21

Your question deal with several concepts which have to be defined first:

• An isomophism is a mathematical concept which relates two structured sets, e.g., two groups or two vector spaces. An isomorphism between two groups is a map f: (G,+) --> (H,+) which is bijective, such that f(x+y) = f(x) + f(y), and such that the same holds for the inverse map. Note that for groups the condition for the inverse map follows from the first property.

• The extension of a concept is the set of all objects to which the concept refers. E.g., the extension of the concept "human" is the set of all "humans".

• The intension of a concept is the meaning of the concept. E.g., the concept "bipartite graph" has a different intension than the concept "graph without odd-length cycles". This can be easily seen from the definition of these concepts.

According to these definitions extension is not a synonym for isomorphism.

• I have drawn common ground between the concepts. You respond like you do not notice any. Is it fair ignorance? Furthermore, I do not get why do you get such complex definition of isomorphism. Here is a simple identification of set {1,2,3} with letters {A, B, C}. Mar 31, 2016 at 21:03
• @Valentin Tihomirov Indeed, I do not see much common ground between the three concepts. In particular, there is no relation between isomorphism and the pair (extension, intension). - The term isomorphism is generally used for structure preserving maps. Hence I assumed that both sets - more complex than in your example - have the additional structure of a group. Mar 31, 2016 at 21:12