Isomorphism: An Underutilized Concept
Today’s subject is a one-off I’ve been meaning to write for all of 2021. Being in the thick of consumerism season (i.e. the month of December) makes this an especially appropriate time to share it. Also, I have the day off to stretch some of my neurons that have been sidelined for a bit, so here we are. The word ‘isomorphism’ belongs originally (modulo Greek etymology) to the language of mathematics, but the concept that the word represents and whose definition it makes precise is a useful one in many other contexts. One angle from which to approach what this word means is from the consideration of the question: “Under what conditions are two things the same?” This is such a compelling question even from the surface of how it’s asked, because in some sense, for there to be two things to begin with means that they are distinct and not the same. This is a useful and legitimate notion, but if we stop there, we stand to miss out on a great deal of rewarding nuance. Because if we stick with figuring out why there’s more to that question, eventually we see that there are situations when it’s useful to consider two things that are distinct physical or conceptual entities to be the same.
I’ll ground this idea right away with some reflections on consumer goods. One of my personal favorite examples of this is yogurt. In our multicultural world, there are many, many different types of yogurt, and there have been for a very long time. But since I was a child, yogurt in North American grocery stores has proliferated to occupy entire walls of large stores, where before you had a small set of shelves. At first you encounter this wall and are astounded that there are so many yogurt options available to your whim of the moment. You could get kefir, Polish yogurt, Icelandic yogurt, Greek yogurt, yogurt with chocolate crunchies, whole fat, 0% fat, fruit-flavored, with granola, lactose-free, organic, and often combinations of these (That’s how the number of options really starts to blow up fast.). Yogurt is just a droplet in the oceans of options in consumer goods that are available.
Maybe some people relish contemplating this apparent vastness of options in all contexts. At the same time, it’s clear that many of us don’t enjoy the size of many problems we encounter (as measured by the number of options to be selected from in making a decision). So there’s an underlying aversion to many necessary decision-based interactions with the world. There can also be fear of missing out alongside this aversion, which might lead us to think something along the lines of: “Well, if I must choose, then I better make the best possible choice!” Here, there’s often an implicit assumption that finding the highest-ranked option, according to our own internal ranking system, always results in making the best choice. As a “consolation punishment”, we can end up holding ourselves hostage to a situation to which society has already held us hostage! It’s no wonder shopping can make us cranky, dismayed, or angry.
None of these thoughts is new. What I am going to share is an insight and hack I’ve gained from mathematical training that I haven’t seen before in articles or books on the now well-trodden topic of choice dilemmas. I’m going to throw some abstraction and symbols at you now, but I believe in you! You’ve got this. Let’s say we have a set of objects Set 1 and a set of objects Set 2. We’ve already established that there’s a sense in which Set 1 and Set 2 are not the same. But what if we discover the following about Set 1 and Set 2—that for every object in Set 1, there is a way to uniquely match it to an object in Set 2, and for every object in Set 2, there’s an object in Set 1 that matches up with it. We also discover that we can switch the set names in that sentence and the sentence remains true. And one last, joyful discovery: We learn that if we combine two objects in Set 1 and match the result to an object in Set 2 by the same matching scheme as before, we get the same object in Set 2 as though we had matched each one individually first and then combined the results. More concisely and precisely, if f is the name of our special matching scheme, or “bijection”, then
f(object 1 * object 2) = f(object 1) ** f(object 2)
where * is the operation of combining objects in Set 1 and ** is the operation of combining objects in Set 2.
Right, so back to yogurt. Let’s say that our Set 1 is the set of yogurt varieties we’ve bought and enjoyed before, and Set 2 comprises all the varieties we haven’t tried before and therefore might contain something even better than everything in Set 1. So each set object is a yogurt container with a label on it. Now suppose that the operation * for our Set 1 of known-and-loved yogurts is mixing of yogurts, and so is ** but for yogurts in Set 2. In both Set 1 and Set 2 yogurts are considered equal if you can’t taste a difference after mixing them. f is the correspondence of yogurt types between brands. For example, if I have a brand of 2% vanilla in Set 1, it matches with the 2% vanilla of a different brand in Set 2. Now, our symbolic sentence from above means this:
f(object 1 * object 2) = f(object 1) ** f(object 2)
<–>
type of (yogurt 1 mixed with yogurt 2) tastes the same as the type of (yogurt 1) mixed with the type of (yogurt 2)
[Mathy aside: If we want to preserve full mathematical rigor, then we had better be careful about details like our sets of yogurts having the same number of elements, since we’re talking about finite sets. But since this degree of formalism already borders on overkill for our purposes, we can settle for not an isomorphism (requires a bijection) but for a homomorphism (which just requires the above equality statement about combinations of objects in the sets).]
The key observation I draw your attention to is that we’ve found some meaningful property of yogurts for which brand is completely irrelevant. So, maybe we can ignore it and focus on more important stuff, like whether two given yogurts are actually interchangeable in terms of taste. Brand is just one of possibly limitless examples of data that can be fruitfully ignored in certain situations. Let’s think about something that should be as straightforward as khaki pants. As topological structures, all pants are basically the same. Now, their geometry can differ meaningfully, and fit is important. But if you know your size and filter options through that criterion, you can still be left with many at least superficially different options: number of pockets, color, brand, fabric composition. (And don’t get me started about all the apps and software packages out there that are also functionally iso/homomorphic and differ only in their presentation, leading to lots of shallow debates about which one is better.) Your power is that you get to decide when two things are interchangeable and therefore one can be ignored, thus reducing the total number of objects under consideration, sometimes dramatically. The savings can really add up when options need to be considered carefully.
Reducing the size of a decision problem doesn’t address essential questions such as when you maximize decision outcome utility by optimizing or satisficing among options. But what it can do is make both approaches more tractable, increasing the odds that you’ll be able to do either well as the situation requires. The lesson that I’ve internalized is that forgetting about certain information that makes options technically distinct reduces their number in a lot of my quotidian decision problems, which means that I can get away with spending less time and mental energy on them. This in turn means that I have more of those resources for what’s meaningful and rewarding to me. You, too, can practice this thinking technique in the holidays and beyond to make more space for peace and contentment.