Galois Theory and Languages

So, I've been thinking about mathematical metaphors for philosophy again, and this time Galois theory has come up. But first, the philosophical problem: when I speak in different languages, am I saying the same thing? Is there actually an English statement that corresponds to a French one? Are there propositions embedded in human sentences? If the mathematical analogy which I propose works, than there exists a complex yet satisfying answer to the problem.

Abstract algebra is about mathematical convenience. Mathematicians were tired of proving a theorem in numbers, then in geometry, then in permutations, and so on, when the theorem in question is practically the same thing in each case. So, they abstracted out the common features of the systems which were actually what ground the systems, and gave names to the sets of features. Two systems in particular are "groups" and "fields" (for those interested, the set of integers under addition is a group, and the set of rational numbers under addition and multiplication is a field). Galois theory states that there are relations between groups and fields such that you can match them up.

Now, groups and fields are not reducible to each other. One could perhaps say that groups are building blocks for fields, but there are interactions within fields which are not directly reducible to their group properties. So they are different, and one is not saying the same thing about groups as one is about fields. However, there is a direct connection between them as Galois theory points out, such that the structure of individual groups is the "same" (in a precise mathematical sense) as that of individual fields. So they are saying the same things as well. Further, the interplay of difference and similarity is what Galois Theory uses to establish its unique points: for example, that there is no general equation for solving fifth-degree polynomials or higher.

To bring this to play with languages: English and Japanese are just saying the same things; they have different webs of meaning, and this is important. But that doesn't mean that there are no structural isomorphisms at all. In addition, bringing the two together doesn't merely increase our understanding of our native tongue, but actually produces new thoughts.