An analogy works by realizing that two ideas, or two parts of those ideas, are the same thing.
Learning about derivatives in calculus, you may get the sneaking feeling that it reminds you of an odometer and speedometer on a car. That’s not a coincidence, the speedometer actually is the (absolute) first derivative of the amount of distance you travel.
If you learn about physics and realize that voltage in electricity is analogous with height in gravity, that’s also not a coincidence. They’re both ways of indicating potential energies.
Even if the two things being compared are totally different, an analogy seems to show that at least part of the pattern representing the ideas is the same. If I make a comparison between Viking and Greek mythology and see that Odin and Zeus are two patriarchal heads of a pantheon of other deities. They might have many other dissimilarities, but at least part of the structure of the two ideas is the same.
The Mathematics of Analogy
There is actually a type of math which deals with these seeming-differences but internal-sameness: isomorphism .
Essentially, isomorphism is asking whether two completely different descriptions of objects and their relationships, can be redescribed as being the same as something which looks completely different. Or in our case, is there some subset of the objects and relationships for two ideas which can be rearranged to actually look the same.
Unfortunately, isomorphism is a hard problem. It’s not known  if there’s a general way to quickly figure out whether two sets of objects and their relationships can be renamed to match another pattern.
However, despite this difficulty, most of the time we want to create an analogy, we don’t need a perfect match, all the time, nor do we need to work with hugely complex bundles of relationships. To make an analogy, you only need several parts of each idea to match each other.
Analogies and Learning
Analogies work so well for explaining things because your brain seems equipped to exploit this isomorphism between ideas.
Consider learning a new idea. The idea may have a lot of details to get right. Those details also have relationships with the other details. Learning the idea involves practicing those details and their connections until you can recall the idea easily.
Now, if I told you that this new idea is actually isomorphic with another idea you’ve already practice the job becomes a lot easier. You can simply pair the features of the idea with each other and then you can exploit the fact that you already understand the connections and relationships in the first idea.
Most of the time this happens, I think, it’s so banal as to not even merit attention. When I’m learning capital cities of countries, and someone tells me that Paris is the capital of France, I’m exploiting the fact that I already have this mastered knowledge of other countries having capital cities, and that the capital is usually the seat of the government, it is often, but not always, the largest city in the country, etc.
The only time this mental feat is even noteworthy is when the isomorphism is rather clever. Because it may not seem obvious that complex numbers and two dimensional vectors are actually the same in many ways, learning of this isomorphism can be quite useful in helping to demystify the notoriously hard concept of an imaginary number.
I think people sometimes make the mistake of thinking analogy works, as a process, only when it’s of this rather clever type. Most analogous reasoning is rather obvious, but it is still an essential component of learning new ideas.
My original ideas on holistic learning , put a lot of emphasis on finding these isomorphisms between ideas. If you can find an analogy, you can learn a complicated idea much faster because you exploit the fact that you’ve already mastered the pattern. Given that many ideas are at least partially isomorphic to one another, this creates a powerful principle for learning quickly.
I’ve since grown a bit more skeptical of hunting for analogies as being the primary way to learn something. Mostly because finding isomorphisms is a hard problem that depends, in no small part, to already somewhat understanding the idea. Finding these isomorphisms becomes a task for the master of a subject, not the complete beginner.
Instead, I feel the foundation for learning has to be based on active practice. Sometimes raw practice will lead to this epiphanies that two ideas are actually related. Sometimes that insight will come much later, once you’ve already practiced the ideas somewhat.
However, I still believe that analogous reasoning is an important part of learning, even if it’s not always an easily-controllable process. Finding isomorphisms between sets of knowledge is a potent tool in understanding.
Which Subjects Have the Most Analogies?
Perfect analogies occur when the two ideas in question share an identical deep structure. In other words, they are completely isomorphic to one another.
This rarely happens by coincidence. In fact, the more complicated an idea is, the less likely you’d have a perfect isomorphism between ideas if the two ideas were really unrelated. The speedometer/odometer relationship and first order derivatives aren’t coincidentally the same, they’re actually the same.
Math, is very often the link that connects the two sets of ideas. Because math is pure abstraction—stripping away the details to have only the unlabelled relationships themselves exposed—it is very often the analogy that connects the two sets of ideas.
I might even go so far as to say that all true analogies (meaning complex sets of ideas which are completely isomorphic) are mathematical. Math, then, might be simply the description of what things in the world really have the same logical relationships to one another.
Math can be about calculation. But it can also be an inventory of all the basic types of patterns and logical relationships which can exist in the world for there to be analogies between them.
Things Which Look Different Being the Same
The fact that isomorphism is probably a hard, theoretical problem, means that we may never get to a point where all the possible analogies of different ideas are known. But that also makes it special when you do discover that two things you were looking at which seemed different, are actually the same.
I’ve spoken about examples from math and physics, but I think this equally applies to all aspects of life. Many of the blog articles I’ve written which have superficially different titles, concepts and definitions, are about the same ideas. In many cases, even I haven’t realized that certain seemingly different ideas are expressing the same point.
A lot of learning is pegged as the act of seeking out more facts and data. But I think it’s equally a task of seeing how things which look different are actually the same. That connection between ideas mattering as much as the ideas themselves.