Math! What is it Good For?

In my last post, I explored the idea of a system and what it means to have a system boundary. An even more mathematical and abstract but similar idea is the idea of a set. Like a system, a set is defined by identifying what is in the set versus “not in the set.” What makes sets different is that they can have anything in them. You can have a set of systems, a set of foods, even a set of sets!

Some of the intuition we have for how to define a system carries over, but a set is a more all-purpose holder of… “things.” And before I lose everyone because you don’t know why a set could be important for problem solving, a set could contain all the business problems you care about, all the constraints you need to consider, all the solutions you could choose, and all the results of an action. Since it can hold anything, it is a useful way to ask yourself “Am I including everything that belongs in the set, and only the things that belong in the set?”

You already use the idea of a set to help you make decisions. My go-to example of choosing dinner could be treated as a set-defining problem. First lets define the goal-set:

Dinner tonight

=

the set of dinner options we choose.

In this case, the set should probably be a single element, I.e., one thing for dinner. But as soon as we start talking about the characteristics of a set, my brain at least immediately jumps to “oh, but are there interesting options I just ruled out with the way I said that?” Specifically, should dinner tonight actually be just a single element?

If you’re cooking at home, you usually have more than one component of dinner. If you decided to make tacos but someone said they want burritos, it can be a really trivial amount of extra work to have both. On the other hand, going to a restaurant can look like a single element in the set, but restaurants usually have many options you could pick once you are there.

By defining the mathematical properties of our goal-set and asking if it should be a single element or not, we already have some interesting questions about “what makes a good solution?” While there may be other interesting questions at this point, I will go to the next step and think about what set we are choosing from to get the set of dinner tonight:

The set of dinner options we choose

(is a subset of)

The set of dinner options we don’t rule out

(is a subset of)

All possible dinner options.

Figuring out the most useful sets to consider is an art. It is why last week’s article on defining system boundaries came first. In the series of sets I captured above, I have chosen what I think of as our key decision points. First we have everything that counts as dinner, then we have the things we are ok with, then we have the dinner we choose. As I write that, again there are a jumble of priorities, questions, and implied decisions in my framing. Some of the most relevant:

• The set of all possible dinner options probably should rule out things that aren’t dinner (My son a few nights ago was trying to guess where we were eating out. He misremembered the restaurant’s name so he said “It starts with a T!” I joked: TJ Maxx?).

• The set of all possible dinner options probably should not rule out breakfast for dinner or cake for dinner. It could… but in my mind that would be something that happens in the next step.

• The set of all possible dinner options might imply specifically dinner for tonight, in which case it should rule out things like that favorite restaurant in a foreign country.

• The set of all possible dinner options also can rule out things that are impossible. Like if your group does not include an expert chef, you would not include expert chef quality meals but cheaper because they are made at home.

• The set of dinner options we don’t rule out should be definitively smaller than the set of all possible dinner options. But ideally, we want to have more than 1 choice at that point. This is a judgement thing, but I find there’s a risk of ruling out everything, in which case with this set framing it is clear you will be unable to choose dinner other than the empty set (i.e., nothing).

• We could skip the middle subset entirely in our math if we wanted. But from a decision making point of view, I think it is an important way-point from “everything possible” to “the one thing we choose.”

• Part of why I like to include “the set of dinner options we don’t rule out” is because if you find yourself having ruled out everything, it is pretty clear with this framing what mistake you made. Since we know there are many elements of the set of all possible options, and you want the set of dinner options you choose to have at least one element, you have made a mistake when you ruled things out. This kind of set math, keeping track of the goal set and what it is a subset of can make those mistakes more explicit.

• Etc.

That was a lot, but just like when we defined our set as having a single element we choose for dinner, the 3 sets above give a pretty clear picture of the decision making process. It also has a million little system boundaries you would have to define if you actually wanted to create a list of everything in the set. And herein is another useful aspect of sets: they can be composed of other sets. So rather than making a list of every restaurant, you could just write:

The set of all possible dinner options

=

The set of restaurants we could eat at

+

the set of alternatives to restaurants.

Or another useful breakdown:

The set of restaurants we could eat at

=

The set of restaurants we like

+

The set of restaurants we might like if we tried them

+

All other restaurants not included above.

Defining these sets lets us think about the abstract characteristics of our options. It can make ruling things out much easier since you can avoid listing each and every option (no one was actually going to do that anyway). But you can also rule things in more easily.

And this is the real power of using sets in the context of problem solving. Once you have these categories and their relationships defined, you can pick any individual option and identify which sets it belongs to. If we decide that while restaurants are a possibility, we want to rule them out for dinner tonight, suddenly there is a whole set of elements (dinner options) that we don’t have to consider. The decisions we make about what characteristics we want for our dinner have a direct impact on our sets.

I will have more to say on this topic, but here seems like a good stopping point for this week.