Is it possible to explain the difference between isomorphism and homomorphism without resorting to advanced mathematical concepts?

Question

I am reading and trying to understand in particular the structuralist set-theoretic conception that is based on the semantic view, and they talk about the relationship between the model and its target system, a relationship that could be isomorphic or homomorphic, among others. I have googled these words and could not find a place that explains conceptually without resorting to advanced mathematical concepts, such as "vector space", "preservation of operations", "bijective relationship", etc. Is it possible to explain these concepts without being introduced to that field of mathematics, in order to understand this branch of the philosophy of science?

Answer 1

You can get a high-level view from 'Universal Algebra' or its modern equivalent 'Category Theory'.

A homomorphism is a relationship between structures of the same kind that does not create any more structure among the elements it maps to. Any structure in the image was already in the original.

Most of these will, however, remove some structure. The removed structure is sometimes called the 'kernel' of the homomorphism. The original is then a sort of 'combination' closely related to the product of the image and the kernel. So this approach to things decomposes complex structures into 'simple' structures. (As in the classification of simple groups, one of the major results in algebra in the 1970-90's.) Then as we work on the simplified domains, many of the things we find true of the homomorphic images can be lifted back to say meaningful things about the things of which they are images.

Isomorphisms, being bidirectional homomorphisms, can't remove any structure either way, so the related structures are logically equivalent.

Applying this to set theories, homomorphisms can be used embed an entire set theory into a simpler model whose elements you can fully define, in such a way that if you can get a contradiction to some assertion in the smaller image, of which you have a detailed model, it will be false in the original, too.

There are also predictable relationships between the collections of similar structures (Categories) if their homomorphisms are related in predictable ways. Natural mappings between different categories can transfer observations from one domain of math to another in fields like algebraic topology. So we can, for instance, say things about the relationships between the category of models of set theory and the category of languages that describe them.

None of this means anything, without a lot of technical details. But it explains the motivation for choosing the approach. And hopefully, it can motivate you to slog through the numerous, though often obvious, details involved.

Answer 2

"Is it possible to explain these concepts without being introduced to that field of mathematics, in order to understand this branch of the philosophy of science?"

No, there really isn't. If you want to study the philosophy of a subject, you should know the basics of that subject in the first place. A decent knowledge of some basic algebraic structures and naive set theory is absolutely essential to actually understanding the ideas of structural set theory (and other foundational programs) at more than a metaphorical level.

And to be honest it's not that much material: a first semester abstract algebra class will introduce the basic concepts, and a first semester mathematical logic class will develop the more foundational framework they live in. Of course more is better - to really understand what's going on you should be comfortable with a fairly wide range of mathematical topics - but the above is a bare minimum for following the basic debates.

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That said, hide_in_plain_sight's answer does do a very good job of giving an informal summary of the topic. (But as they say at the end, it shouldn't be taken as a stopping point but rather motivation to learn exactly those details you ask about avoiding.)

Answer 3

The original Greek meanings of the terms are quite illuminating in this case:

isos + morphos ⇒ of identical structure,

homos + morphos ⇒ common in structure.

Let us try an extra-mathematical perspective. We may think homomorphism and isomorphism (their brethren mappings, automorphism etc. as well) as a collection of objects related by a kernel idea bringing about a structure.

In the case of isomorphism, the emphasis is on the preservation of objects indiscernibly (hence, 1-to-1 correspondence of objects). In optics for example, the real and virtual images are isomorphic, each one of the points on the real image is preserved on the virtual one.

As for homomorphism, the emphasis is on structural relation. Though, homomorphism is mathematically a weaker form of isomorphism, we can view it as a mapping with an additional feature. Within reasonable bounds of imperfection, we can design or observe homomorphisms. An example is a mapping from a natural language into its formal grammar, with grammatical classification imposed (consider Montague's Universal Grammar). Usually, translation and interpretation among natural and formal languages are homomorphisms.

To be sure, the gist of the matter is not in naming a correspondence schema in various areas of erudition. Such a schema is rightly to be called homomorphism or isomorphism only if we could apply the relevant mathematical facts about them, even though not with mathematical exactitude.