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

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?

• "Preservation of operations" and "bijective relationship" are pretty ubiquitous in mathematics, it is impossible to understand even what a structure itself is without operations and relations. It is a set with operations and/or relations. Homomorphisms are maps that preserve those, i.e. map things that are in a relation in the source set into things that are in the corresponding relation in the target set. Isomorphisms are homomorphisms that have inverses, also homomorphisms, so they go boes ways basically amount to relabeling elements of the source into elements of the target, or vice versa. Apr 27, 2020 at 17:20

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.

• OP can't grok bijection but you're telling him to study category theory? Oh well you got a checkmark so if he's happy I am, I guess. Apr 27, 2020 at 21:33
• I would put the last paragraph at the beginning - it's really important to emphasize the incompleteness of your (quite good) summary. Apr 28, 2020 at 0:27
• @NoahSchweber. We certainly don't do that for other domains here. People freely discuss Kant, though the Critique of Pure Reason is about the same thickness as Hungerford or Chang and Keisler, and has just as much detail. Math is not that special. Apr 28, 2020 at 17:04
• @hide_in_plain_sight I don't understand your point. "People freely discuss Kant" of course - but not without having read some Kant, or currently reading Kant and having issues, or in some way actually engaging with Kant. It doesn't make sense to try to study the philosophy of X without actually also studying a bit of X. Math is indeed not special. Apr 28, 2020 at 17:34
• I have definitely read plenty of Algebra and Logic texts. I do not presume the person to whom I am giving an answer about Kant has anything more than the most basic understanding -- they often do not understand the basic vocabulary. And in that case, I don't expect them to go read his definitions. I give a coverage with the same level of rigor I have used here -- which is not much. Apr 28, 2020 at 17:37

"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.

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.)

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.