To describe something, a person uses words. If I have seen a pineapple before, and you have seen a pineapple before, and we both know the word “pineapple,” we can use the word “pineapple” to communicate. But if you’ve never seen such a fruit, I have to be creative to get you to understand. I might describe its stringy chewiness, bright yellow color, and piquant aroma; I might draw parallels to your other experiences, perhaps compare it to other foods. That way, we could communicate, even though you don’t have a previous concept of “pineapple.”
So here I am, every week in this blog, trying to explain math ideas in creative ways — such that, even if you’ve never savored a good piece of it, you can appreciate math’s unique flavor. This particular post will try to explain, in the best words I can find, what the experience of math is like for me.
What it’s not
- Anything like what you did in middle school.
- Adding, subtracting, multiplying, or other arithmetic operations. Or, manipulating algebraic symbols, remembering formulas, or drawing accurate pictures. In my experience, these are tools — useful, but not the heart of the experience. They are there to facilitate the more interesting ideas that come later.
- Getting the “right answer.” That framework is pretty much irrelevant.
What it is (sort of)
Thinking about math involves multiple steps, much the way that baking a cake involves measuring, mixing, and applying heat.
Math starts with a translation step. When reading mathematics, you have to peel the words and symbols off the page, and turn them into a concept in your head.
If I read, for example, about an “open sphere S in R^4,” I first translate that symbolic language to semi-English words: “the interior of a four-dimensional sphere.” Then I mentally review what “interior,” “sphere,” and “four dimensions” mean. (I’m not going to explain what these are here — I’m just bringing them to show that I translate symbolic statements into ideas I can understand.)
This translation process gives me a kind of mental picture of the sphere and its properties. I then hold that picture in mind and move on to the next phrase. A slightly sped-up version of this happens when I am sitting in class: when my professor refers to a difficult concept, I’ll pull up associations and mental pictures to make sense of what he or she is saying.
All of this translation can be difficult sometimes. It’s not to figure out anything deep — it’s just for seeing what exactly is being referred to. But when I encounter this type of challenge, even though it happens before I encounter the actual math, I’ll keep working at it, because I know that the concepts will be interesting once I get ahold of them.
Once the concept is in my head, there is … well … there’s the math.
Math requires you to take a concept and stretch it, turn it over, invert it, or break it into pieces. It involves taking a logical argument and deciding whether it really makes sense, creatively employing tools to solve a problem, and often inventing tools on your own. More than almost anything else I know, it really works the mind.
The world of the concrete, of grass, clouds, chairs, and pencils, falls away. In its place, I consider an abstract scene: the grand, sweeping architecture of calculus; the straight, clean spaces of linear algebra; the chopped-up landscape of discrete mathematics.
The experience is completely absorbing in a way that I find almost meditative. My mind has no space for other thoughts. Everything else must quiet. With intense focus and hard thinking, I can often see the elegance of a mathematical pattern: something beautiful, outside of the self, that isn’t like anything else I know.
I can’t explain what math looks like from the inside, if you’ve never seen it, any more than you can explain the flavor of a pineapple to someone who has never tasted an acidic fruit. But I can say that pineapples are tingly, bright, sweet and little sour; and I can say that understanding math abstractly is like coming upon a mountain or a waterfall after a long hike. And that if the moment of understanding is like coming upon a waterfall, then, the experience of thinking is like the exhilaration of maneuvering through a dense growth of forest.
For me, climbing that path is almost as thrilling as reaching the destination.