Much of our lives is about problems and solutions. Faced with a barrier, we find a way to knock it down. Presented with a challenge, we work to overcome it. Our collective problems bring us together, and our collective solutions make us safer, stronger, and happier.
These problems come in many shapes and sizes: math problems, career problems, logistical problems, emotional problems, physical problems. Rather than maintain special problem solving techniques for individual problem types, we can expand our methods into a global group, and learn from one type what may be helpful for another. Sticking with math – a common language and underlying framework of nature and intellect – we can relate our methods for solving math problems to the rest of the world around (and above us).
There is an innate simplicity to many math problems: there are knowns and there are unknowns. The solutions often reside in the application of methods and operators to the knowns to determine one (or all) of the unknowns. Therefore, the first step is often determining what is known and what is unknown. Although, this notion has been most popularly represented by set theory (a foundational system of mathematics that deals with collections of objects), this concept has been the spark for other applied methods and disciplines through which many more complex problems are tackled in today’s society.
In game theory, a player’s strategy can be represented by differentiating the sets of moves that could make positive gains versus those that could make negative gains, given the possible situations at each stage of the game. Closely related is decision theory, where we look for the pros and cons, uncertainties, and rationalities behind potential decisions to determine an optimal course of action. In chaos theory, we define initial conditions and explore how the behaviors of some dynamical systems change as those knowns vary or as unknowns are introduced into the system.
To delve deeper into the questions of known-unknown identification, set theory, and related applied methods, we can think about a problem that began on day one, has no end in sight, yet has made incredible progress over centuries in terms of approaching a solution: what’s above us? What’s with the sky, the planets, the stars, the universe – the aether that surrounds us?
Is the total set of knowns and unknowns about the universe infinite? Does a new known always present us with a new unknown? Is the same true for every problem, or just some? For which types of problems might this be true? Are we better at approaching a solution collectively or as individuals? How can this be determined at the onset of a problem? If the set of unknowns has no limit or boundary, is the solution intelligently impossible? Does a single element of randomness deny a complete solution from every being possible? Are we better off existing without a solution? Or would we be complete with a world of all knowns?
I fundamentally believe that we find meaning in life through the unknowns, not the knowns. The set of unknowns is infinite, and it is our drive to understand unknowns and, in general, the curiosity into the mysterious world that provides completeness. The knowns give safety, guidance, comfort, and pleasure.
For all problems we face, and as with the aether abound, we can continue to move forward, learn what we know, and question that which we don’t know. We can start with sets – knowns and unknowns – and move from there. Problem solving can be simple, if you start simple. As for the things we don’t know we don’t know – the unknown unknowns – well, we better stay curious with the mysterious, and just be happy for that.