What is an Isomorphism and why should you care?

An Isomorphism is the mathematical concept that generalizes ‘equality’.  

‘Equality’ is just as it was presented in middle school, there is an equal sign and an expression on each side. Like so:

3x = y

This means that 3x is precisely equal to y. Anywhere you see 3x you can substitute y, and vice versa. 

But this type of equality, while intuitive, is constraining. In reality, we see things that are the ‘same’, but not precisely. You might really want to say that these things are equal, but they don’t satisfy our intuitive definition, ie. you can’t switch one out for the other willy-nilly.

For example, the Golden Gate Bridge in San Francisco and the George Washington Bridge in New York. These are both clearly bridges, so they are the ‘same’ in some sense, but not exactly. You can’t magically change the Golden Gate Bridge to the George Washington Bridge, they were specifically designed to be built in one place. 

So can we capture this equivalence? What does it actually mean to be a bridge? This is what the isomorphism helps us formalize.

While all bridges look different, they have similar parts that perform the same function. They all have buttresses connecting a road to the bank, and the road spans across a gap to the other side. There are structural members that add strength, and they distribute the loads in similar ways.

The important observation is that the essential parts are the same, as is the way those parts are connected.

Creating an ‘isomorphism’ between two objects is simply identifying the essential parts of both and drawing a one-to-one relation between them in a way that preserves the structure.

The last part is important, a pile of human bones is not isomorphic to a human, because “the knee bone is not connected to the leg bone” when they are in a pile on the floor. However, a skeleton model hanging on a rack can be isomorphic to a human, it just depends on your application. Do you care about flesh? Is that an essential part of your model?

If you don’t care about the flesh, you should switch to the simpler domain of skeletons for the analysis, whatever it may be.

Note that this definition collapses into our example of strict equality if we want. We said that 3x = y, but what does that actually mean? It means that whenever I have ‘3x’, there is a morphism that changes it to ‘y’, namely, you can just switch them out. More importantly, when we do that switch the structure is preserved, ie. the statement remains true.

Why is this useful? In mathematics, it lets you solve a complicated problem in a simpler domain. If there is an isomorphism from all the moving parts of one problem to another, you can solve the easier problem and then transform it back. This is a powerful approach when available, so it’s a happy day when a non-trivial isomorphism is discovered.

However, this technique is just as interesting to non-mathematicians. The isomorphism is a formalized analogy. Humans rely on analogies to discuss complicated ideas, so concocting a useful analogy is a valuable skill. The isomorphism is simply a more rigorous way to think about that process. 

So next time you make an analogy, try to find the isomorphism under the hood. Or if you make one up, try explicitly drawing the point-to-point map between your imagined scenario and the situation it models. 

The effort will make your ideas more compelling and less ambiguous, as well more accessible. Drawing the arrows and describing the structure will make your analogy obvious, allowing everyone to spend time solving the problem, not grokking the setup.

Of course, this sounds like overkill for most scenarios, and it is. But taking analogies seriously will make you a far better communicator. For example, here is a page from Godel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter, the undisputed king of analogy. This book won the Pulitzer Prize, mind you. 

Thanks for reading! Please leave a comment if you have feedback! I really like to hear it. -A