At the heart of mathematical thinking, as I see it, is the notion of abstraction. It is unfortunate that this word seems to be a turn-off for so many people. I often hear people say that they liked math until it got “too abstract”. I think statements like that come from a faulty analogy with other senses of the word “abstract”. In the art world, for example, there is a distinction between the representational and the abstract, and so it is tempting to think of “abstraction” as the opposite of “concreteness” or “reality”. From there it’s a short step to thinking of “abstraction” as the opposite of “relevance” or “importance”.
But I think this misses the point and misses it rather dramatically. So just for today, I would like you to forget the adjective “abstract” and focus instead on the verb “abstract”. I’m not talking today about abstract-ness, I’m talking about abstract-ing, an action word, about what it means to abstract a concept from a situation. I realize that the verb doesn’t get much use in general discourse, but bear with me. The reason we don’t use it much is many of us haven’t done much conscious, serious, powerful abstracting since we were very small.
So I’m not going to bury you in high-level, complicated mathematics today. Actually, I’m going to do just the opposite. I want you to think very, very hard about something really, really simple, something you probably haven’t given much real attention to in a long time. That something is the addition of whole numbers.
That’s right, this is an article about 1 + 2 = 3. Because believe it or not, 1 + 2 = 3 is a spectacularly deep mathematical discovery. Really!
Imagine yourself in the distant history of the human race. Perhaps you live in a cave, or some sort of hut. You have some words, but not a lot. You certainly don’t have math classes. And you’re about to become the world’s first mathematician, because you are thinking very hard about the apple on the giant rock that serves as your table, a short distance from two more apples, and you are about to have great insight. And the insight is not “one apple plus two apples is three apples” (or, as you would more likely verbalize it “apple and apple-apple are apple-apple-apple”). That’s easy. That’s obvious. That’s not mathematics. The heart of mathematics is realizing that “one apple plus two apples is three apples” is not about apples.
The most basic thing that just happened is you just invented numbers. You realized that “apple apple” and “wolf wolf” and “girl girl” all have something fundamental in common, something you didn’t have a word for, and you decide to call that common thing “two”.
The development of numerals…the ability to multiply and divide on paper (without counting thousands of little stones)…the ability to have any kind of accounting system… there is an enormous tree of ideas, extremely practical, indispensible ideas, which were inaccessible until we had this realization.
You might think of numbers as being not at all abstract, but remember that you’ve never in your life seen two, or touched or smelled it. You’ve seen two rocks, two people, two books, lots of things that evoke two or represent two or display two. But even something as mundane as the number two is an abstract construction of the mind.
But something else happened too, and to see what it is I want to look at a few other examples of abstraction.
- Think about the addition of whole numbers. Classify numbers based on whether they are even or odd. Notice that even + even = even, odd + odd = even, odd + even = odd, and even + odd = odd. So we can abstract numbers into “parity classes” (or whatever you’d rather call them) and we can add the resulting objects (even and odd).
- Think about the multiplication of numbers. Forget about zero, and classify the others into positive and negative. Notice that the product of any two positive numbers is positive, the product of any two negative numbers is positive, and the product of an even and an odd number is negative. The specific numbers involved don’t affect that. So we can abstract two “sign classes”, + and -, and it makes sense to talk about how they multiply.
- A non-numerical one this time. Suppose you have a piece of paper on a table. There is a dot marked in the center of the page (visible on both sides) and a dot marked in the center of the table. You get to pick up the paper, turn it or flip it or otherwise move it through space (without cutting or folding it), then put it back on the paper so that the dot on the paper lines up with the dot on the table. You might turn the paper a quarter-turn clockwise, or maybe flip it vertically, or maybe not do anything it all, just put it back how you found it. There is a natural “composition” operation on these paper “transformations”. If x is one legal thing to do the paper, and y is another, then think of xy as what happens when you pick up the paper, do y, do x, then put the paper down. (So flipping the paper vertically composed with flipping vertically is the same as doing nothing, and flipping the paper vertically, then doing a half-turn is the same as flipping the paper horizontally.
The legal transformations of the paper are naturally divided into two categories. There are rotations, in which the paper is turned around the dot, and reflections, in which the paper is flipped over (and the top of the paper becomes the bottom). Convince yourself that any combination of rotations and reflections, always keeping that dot in the same place, ends up equivalent to just a single rotation or reflection. (Doing nothing is considered an especially simple kind of rotation.)
Now it turns out that a rotation followed by another rotation is the same as a rotation. A reflection followed by another reflection is the same as a rotation. A rotation followed by a reflection, or a reflection followed by a rotation, has the same net effect as a reflection. So once again we can abstract from our main objects (transformations of a piece of paper) two categories (rotation and reflection), and then we canlook at how the operation affects the categories.
If you look closely, there is another level of abstraction here. All three of these examples give us two objects with an operation for combining them. And all three configurations are the same, except for what words we use for things. (Hint for the confused: “odd” corresponds to “negative” . . . do these correspond to “rotation” or “reflection”?) In a meaningul sense, odd and even numbers add the same way negative and positive numbers multiply.
Let’s back up a little bit — I don’t want to give you the impression that all mathematical ideas are the same as all other mathematical ideas. Here is an example that works differently.
- Think about the how odd and even numbers multiply. This makes perfectly-good sense, since odd x even = even x odd = even, odd x odd = odd, and even x even = even.
Again we have two objects (odd and even) and an operation for combining them. But I’ll let you work out why this is fundamentally different than the examples we’ve seen thus far; it’s not just the same pattern with different names for things.
And sometimes abstraction doesn’t work out as nicely as in these examples. It’s not true that any way of classifying numbers (or other things) into categories will give a viable abstraction.
- Consider integers, positive, negative, and zero, and how they behave under addition. What is positive plus negative? Well, it might be positive (+6 + -3 = 3) or it might be negative (+1 + -3 = -2) or it might be zero (+2 + -2 = 0). There is no sensible way to abstract the addition operation to these three categories.
So the fact that we can treat “odd” and “even” as addable objects was not obvious or guaranteed after all, and the fact that it works tells mathematicians something deep about addition. It’s mathematically interesting when certain kinds of abstraction is meaningful.
So there really is more going on than realizing that “apple apple” and “fish fish” and “tree tree” have something in common and naming that something “two”. We’re realizing that that something-in-common is compatible with combining-piles-of-things, and that’s not automatic. If you put a pile of two somethings together with a pile of three somethings, you get a pile of five whatever it was. It makes sense to abstract piles of objects into something called numbers and to abstract combining piles into one big pile into addition of numbers.
There is something subtle going on here. Abstraction gives rises to new concepts — we take “rock rock” and “cat cat” and “apple apple”, from that manufacture the idea of “two”, which gets a brand new word and a brand new numeral — so it looks like the purpose of abstraction is to create new ideas, to create new objects. And in some sense it is, but we are not making up new ideas for the sake of making life more complicated. We are reorganizing ideas in order to foreground what is most important and disregard the information that doesn’t actually matter. This is an important function in problem solving.
When we are young and learning new ways to think comes naturally and fast, we intuitively know this. If you ask a child “If you have four apples and I give you three more apples, how many apples will you have?”, she’ll probably say “seven”. Maybe she’ll say “I don’t know”, or count wrong and say “six” or “eight”, but probably “seven”.
What she won’t say is “Red apples or green?”. Because it’s a stupid question.
And it’s a stupid question shes never ask, because she intuits that this is a question about numbers. She abstracts the question you mean, “what’s 4+3?”, from the question about fruit you actually say. She is being a little mathematician by avoiding that question. She is not distracted by the thousand thoughts there are to think about apples that won’t help with the question.
Another thing she won’t do is run to the kitchen to find some apples to count. Even if she needs to count, she’s more likely to count with something immediately available, say, on her fingers. And again she is displaying sophisticated mathematician-ness. Because you didn’t even mention fingers in your question. This is important. Even though the question (about apples) is completely real-world, and the method of solution (counting on fingers) could not be more tangible, there is an essential abstraction at work explaining why the two have anything to do with each other.
Despite the fact that the question uses the word three times, even as children we all had that first crucial spark of mathematicianly insight — that the question is not about apples.