Identity Elements

In math, there’s the concept of an identity element. Basically, it’s an object that, when used in an operation on another object, leaves the latter object unchanged. In subtraction and addition, for example, the identity element is 0, as adding 0 to anything leaves does not alter the original number. In multiplication it’s 1, as  x*1=x for all x, and with matrices it would be the identity matrix. In logic (and I hadn’t thought about this before doing a little research) an identity element is “truth” with the “and” operation, and “falsity” with the “or” operation, which is really interesting.

Identity elements can even be nontrivial. For example, a matrix that rotates a vector 360 degrees about the origin in R^2 is an identity element even though the general transformation that it performs (a rotation) can substantially change a vector. To be clear, a rotation of 360 degrees would just turn out to be the identity matrix:

but I think being an identity element is a property an object has, rather than the object somehow being identical to the identity element. Both the rotation matrix and the “standard” identity matrix have a similar form that contains certain properties, but they are distinct. The rotation matrix describes a rotation of x degrees about the origin, while the identity matrix as it stands has no geometric interpretation.

There are tons of identity elements spanning all of mathematics and I don’t know much about them. Yet, the idea is interesting to me. Intuitively, most operations have some type of effect on an object. It seems like applying an operation in a substantive sense to any thing is guaranteed to engender some type of change, or else you haven’t really applied an operation. Yet, if we generalize the idea of identity elements beyond math and into day-to-day living, they’re much more common than you’d think.

Let’s think of a non-mathematical identity element in a personal context as an object that invokes no change in an individual (person) when used with an operation. For example, a bowl of cereal when paired with the operation “eating” can be an identity element, as eating a bowl of cereal may satiate hunger, but doesn’t alter the individual in a substantial way. However, the object “God” when paired with the operation “contemplating” in most cases does not qualify as an identity element. Seriously reflecting on the divine is likely to change a person.

I’ll be the first to say generalizing from numbers to people is a terrible move. People are messy and complicated and don’t have any of the nice clean properties that numbers have. Is the element “work” when coupled with the operation “going to” count as an identity element? Heading to the office on the average day doesn’t change you, but the accumulation of each day over many years will certainly make you different. It’s a classic example of the sorites problem, but I’m going to table the issue for now. Even if we don’t have a clear idea of when a person is different in an important way from who they were a little bit ago, we can still draw parallels between identity elements in math and elements that might have a similar property when applied to people. Broadly speaking, some element operation combos leave people the same more often than not and others don’t.

Acknowledging the existence of identity-type elements that affect you (the person) is interesting. It means you’re aware that some actions elements and operations have the potential to change you in a certain way, and it’s incumbent on you to identify which changes are desirable and seek out the corresponding action. I’d hope most people like themselves and are comfortable with who they are, but I think an essential part of something important and true about living well is deliberately trying to be a better person. I also believe that “being a better person” doesn’t mean cultivating a specific set of skills like kindness or empathy that are exercised when needed, but is a result of fundamental changes that happen to you and not necessarily what you can and can’t do. If you believe this, then non-identity element-operation pairs are essential. They’re how you realize change.

I see this idea (if correct) as a tool of aspiration. If we’re seriously committed to “being a better person,” then we better take stock of all of the operations and elements around us and see whether they are nudging us in a direction we want to go. It forces us to consider what things will change us, and choose things that will. How will majoring in statistics vs engineering change you? Anthropology vs English? Living in one area vs another? How would you feel if some of these are identity elements after all, and leave you the same?

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