I don’t like arithmetic, maybe in part because I’m not especially good at it. But what I dislike more than the feeling of doing arithmetic is the fact that so many people think math is nothing but arithmetic.
So let’s start with arithmetic — because if I’m trying to undermine the view of math as a mule train, there’s no better place to start than the place where the tethers feel tightest.
Suppose someone asked you to compute 997 + 998 + 3 + 2. How might you do it? You could use the standard one-size-fits-all procedure for adding up a list of nonnegative integers illustrated below (I’m omitting the cross-outs and carries).
But if the niceness of the final answer leads you to suspect that there’s a slicker way, you’re right: 997 + 998 + 3 + 2 equals (997 + 3) + (998 + 2), which equals 1000 + 1000, which equals 2000. This alternative path will lead you to the right answer because addition satisfies laws: the commutative law, which guarantees that changing the order of the terms doesn’t change the sum (so that 997 + 998 + 3 + 2 equals 997 + 3 + 998 + 2) and the associative law1, which guarantees that how you group the terms doesn’t change the sum (so that 997 + 3 + 998 + 2 equals (997 + 3) + (998 + 2)).
Unlike human laws, which constrain the behavior of people, number laws constrain the behavior of numbers and thereby free people to solve problems in flexible ways.
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