Source: Scott Young website, Aug 2015

An analogy works by realizing that two ideas, or two parts of those ideas, are the same thing.

## The Mathematics of Analogy

There is actually a type of math which deals with these seeming-differences but internal-sameness: 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.

… 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.