Some problems are easy, but most problems are hard. They exceed humans’ ability to grasp and reason about possible answers. That’s not just true of complex scientific and political problems, like making complex economic decisions or building models to address climate change. It’s also true of ordinary life. “Let’s get dinner tonight.” “Okay, but where?” Questions like these quickly descend into existential crisis without some structure. “Who am I, even?”
One way that mathematicians think about complex problems is by means of the possibility space of their possible solutions. (It’s also sometimes called a solution space, or probability space.) In mathematics, possibility spaces are used as a register or ledger of all the possible answers to a problem. For example, the possibility space for a toss of a coin is heads or tails. Of two coins: heads-heads, heads-tails, tails-heads, and tails-tails.
That’s a simple enough example, because any given subset of the possibility space can be measured and recorded. But in other cases, the possibility space might be very large, or even infinite. The forms of possible life in the universe, for example, or the possible future branches of evolution. Or the possible games of Go. Or even all the things you might do with an evening.
In these cases, not only is it difficult or impossible to diagram the possibility space completely, but also it usually doesn’t even really make sense to try. An economist might build a model of possible evenings out from, say, the net marginal benefit of a movie or a bike ride or a beef wellington in relation to the cost of those benefits, but such a practice assumes a rationalism that doesn’t exist in ordinary life.
In game design, creators often think of their work as one of creating possibility spaces for their players. In the ancient Chinese folk game Go, a set of stones, stone placement rules, and a board provide a very large possibility space for overall play. But each individual move is much more limited, reliant on the set of choices each player has made previously. Otherwise, it would be impossible for players ever to make a move. One never plays within the total mathematical possibility space of all games of Go, but within the much narrower set of possible, legal moves on a given board at a given time.
Some designers exalt the mathematical largesse of games like Go and Chess, hoping to produce the largest possibility space with the fewest number of components. But more often, what makes a game aesthetically unique is not how mathematically large or deep it is, but how interesting and unique are its components and their possible arrangements. Tetris only has seven different pieces, all of which operate the same way. The delight of Tetris comes from learning to identify and manipulate each of those pieces in various situations.
The exercise of actively and deliberately limiting a possibility space has utility well beyond science, mathematics, and game design. Every situation can be addressed more deliberately and productively by acknowledging or imposing limitations in order to produce a thinkable, actionable domain of possible action. That doesn’t mean that you have to make utility diagram every time you load the dishwasher or debate an evening out with friends. But rather, that the first step in any problem involves accepting that a wealth of existing limitations are already present, waiting to be acknowledged and activated.
When faced with large or infinite possibility spaces, scientists try to impose limits in order to create measurable, recordable work. An astrobiologist might build a possibility space of possible alien life by limiting inquiry to stars or planets of a certain size and composition, for example. When you debate a venue for an evening meal, you do likewise—even though you probably don’t think about it this way under normal circumstances: What kind of food do you feel like? How much do you want to spend? How far are you willing to travel? Fixing even one or two of them often produces a path toward progress. And it does so while avoiding the descent into an existential spiral, searching ever inward for who you really are, or what you really want as the ultimate source for human choices. In ordinary life, as much as in science, the answers are already there in the world, more than they are invented inside your head.