
What do the following three things have in common: a loaf of bread, a pie, and a golf ball?
I encountered this question when I was about 12 years old, on the bus coming home from school. For a few weeks, the radio station that our bus driver liked to have on ran a contest called Tribond, where the host would list three things, and the caller would have to figure out what those three things had in common in a certain amount of time. Whenever we heard that it was on, a few of my classmates and I would listen in on the question and, as one does, try to get the answer before the contestant. This question, though, stumped all of us. The caller couldn’t crack it either, and the host didn’t reveal the answer once the time was up. My friends went back to chatting amongst themselves, but the unsettled question left me feeling a little, well, unsettled. It wasn’t until after a slightly dejected walk home and 10 minutes spent pacing in the living room that I came to a realization and rushed to the phone to call my friends. “It’s ‘slice’!,” I exclaimed. “They can all be sliced!”
Tribond, as well as other far more popular word games like Wordle and Wordscapes are really exercises in a sort of abstract thinking. Here, and for the rest of this post, by “abstraction” I roughly mean “the process of forgetting about presently irrelevant details of a thing to arrive at an underlying idea.” In the case of each of the word games, we entirely ignore the meaning of the words and focus only on their literal structure (ignoring semantics in favour of syntax, if you want to use the technical terms). In the Tribond question, it doesn’t matter what “slice” means in relation to each of the three objects; only that it is valid in the English language to apply the verb “slice” to each of them. For Wordle and Wordscapes, the meanings of the words involved never come into it; all that matters is that the strings of letters we come up with are valid English words.
My general wheelhouse, category theory, which falls under the regrettably named umbrella of “pure” mathematics, or “math for math’s sake,” applies this general principle of abstraction to things like quantity, shape, and space, and tries to push it as far as it will go. And while I wouldn’t broadly advocate for engagement in this extreme level of abstraction, I do think that, at a more moderate level, it is a very useful tool for learning about and navigating the world.
Why do I think this? Because I am of the opinion that the universe, even just the planet on which we find ourselves, is more complicated than our poor human brains can handle without taking steps to simplify it. By filtering out the “noise” of a scenario and recognizing what lies at its core, abstraction allows us to draw otherwise obscure connections and potentially unify our understanding of many things under a single idea, effectively saving “storage space” in our brains.
This is not to say that there aren’t potential negatives to abstraction as well. It can be misused or abused as pedantry, presenting conclusions which omit relevant aspects of a situation. Practicing abstraction can also mean thinking in ways where your intuition no longer applies. This is difficult, and can be discouraging, unsettling, even frightening, since it can feel as though you’ve set foot in another world with no way to ground yourself to reality.
The important thing to remember, though, is that intuition can change with time and practice. You have likely already developed a very deep intuition for at least one completely abstract concept: numbers. Suppose you have three apples in one hand and three bananas in the other (and, if necessary, that you have hands big enough to do so). In essence, the number “three” is the commonality between what is in each of your hands after you have forgotten about their “appleness” and “banananess,” respectively. But hardly anyone thinks of numbers this way, because we have become so accustomed to working on the level of numbers that the abstraction happens almost instinctively.
If we can all master numbers, what’s to stop us from diving deeper? What might “diving deeper” even look like? In my next post, I’ll introduce you to my main area of interest, “the mathematics of mathematics”: category theory.
Photo by Firkin via Openclipart