For Very Large Values of Two

I was sitting today, half-asleep in a class which will bear no name, and I was struck with a halfway insightful thought.

In AP Chemistry, the course of the devil (note to Mr. Glimme if he’s reading this: I’m not kidding at all. thank you.), barely a day goes by when one student or another — often enough, me — doesn’t pose the classic question: “Why are we learning this?”

It’s the usual student’s chant, of course, but in Chem it has a special meaning. You see, the vast majority of what we learn has an unusual characteristic, which most classes try to avoid. It’s not true.

The reason is sometimes that, with our current degree of brain mass, we simply couldn’t perform the calculations to actually understand the real . . . reality. Sometimes, it’s because nobody knows the true order of things — or because (as is often the case with quantum mechanics) it theoretically can’t be known. Whatever the case, much of what we learn is wrong.

Naturally, it’s a paining truth to hear. We don’t want to be there. We most especially do not want to be there learning theories, equations, and formulas which aren’t even correct. And when we ask a question, we want a better answer than, “Nobody knows,” or “You wouldn’t get it.”

So periodically, as a kind of wave-the-dead-chicken ritual, one of us pipes up with, “What’s the point of this? Science is the way things are. Why are we learning the way things aren’t?”

Today I got it.

Let’s say that you had a sequence of numbers, and wanted to determine what value each figure increased by. The numbers were 2, 4, 6, and 8. Easy, right? It increases by two each time. Consider that the “truth.”

Now, let’s say your numbers — your “experimental results,” as it were — were less abundant. You only have 2 and 4. The “truth” is still the same, but you don’t know that.

In this case, “increasing by two” is still correct, but something else also seems to be correct — “multiplying by two.” Two times two equals four. Of course, if you had another piece of data (the next number in the sequence, 6), then you’d realize that this was wrong: 4 · 2 does not equal 6.

But for this set of data, “multiplying by two” is correct. For this limited range of reality, it provides the correct results, even though it’s technically “wrong.” Essentially, you’re getting the right answer for the wrong reason — but you know that it’s the wrong reason, so it’s okay.

This time, it only worked for the first two pieces of data. But let’s say it was correct for the first two million.

See how it just might be handy?

[Editorial Note from a later date: It has been brought to my attention that the title of this piece is a little obscure. Essentially it’s a reference to a half-serious, half-humorous mathematical joke — see this explanation.]