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.
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