Everything should be made as simple as possible, but not one bit simpler.
-Albert Einstein (attributed)
There’s a huge effort today to make math more accessible. I’m part of it, and so are many others I respect. However, lurking in the back of my mind is an awareness of the danger of neutering mathematics. If we oversimplify in our effort to remove the barriers to entry, if we strip away too much, we remove the beauty and the wonder as well. Math is hard, and there’s no use denying it. Anyone who does it for any length of time is going to be challenged to the absolute limit of their intellectual and creative abilities. But the challenge is part of the reward.
Einstein made the mistake of oversimplifying when he was asked to explain special relativity. “Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. THAT’S relativity.”
Simple, right? Well, it’s also totally devoid of anything that resembles the central idea of relativity. What Einstein’s describing here–that you feel things differently depending on your position–is an old idea, going back to Newton. We’ve known for a long time that there’s no way to tell how fast you’re going in an absolute way: you can only measure your speed relative to other objects around you. So if you’re on the train and drop your orange, it seems like it’s falling straight down, whereas someone standing on the ground sees you (and your orange) whip by them at high speed. There isn’t just one description of the orange’s speed, or yours, or the person off the train–you can only say how fast each one is going compared to each other or the Earth, or the train, or something else.
For years I thought that that idea was relativity (sounds right, doesn’t it?), but it’s not. Einstein’s contribution was actually based on staring long and hard at a paradoxical arrangement of truths and realizing that the only way to make sense of it was to question one of our most basic assumptions. Here are the truths he was dealing with, all of which had been scientifically observed and verified in his day:
1. Speed is relative. As long as you’re not changing speed or direction, there’s no way to know how fast you’re going except by comparing your speed to something else (and the speed of that object cannot be established without comparison to some other object as well). That’s why you feel like you’re sitting still when you’re in an airplane or train.
2. (Speed) x (time) = (distance traveled)
3. The speed of light in a vacuum is NOT relative. That is, everyone sees light as traveling the same speed.
Now these pieces are pretty simple, but you can’t get to the beauty of the idea that Einstein realized must follow until you spend some time baffled by the paradox. Do you see it? Consider the following situation: we have a race: both of us can move at the speed of light in a vacuum (maybe we’re photons). Who wins? Well, since the observed speed of light is not relative, I see you moving ahead of me at 186,000miles/sec, AND you see me moving ahead of you at 186,000 miles/sec!!
I think that sentence deserves the second exclamation point, and the more you think about it, the more you have to add. And keep thinking–what if I go almost the speed of light, and you go almost the speed of light? You’ll win the race, but if I’m going almost your speed, how can I see you as traveling 186,000miles/sec when I’m almost going the same speed as you? Shouldn’t you just be a little bit ahead of me after a second, rather than 186,000 miles?
Our intuition is checkmated by the facts. This is the kind of difficulty you have to experience to appreciate mathematical beauty. You can’t see why relativity is beautiful until you’ve, well, suffered through the apparent contradiction of facts (a young student of mine recently said, when faced with another paradox, that his “brain felt like it was twisted up.”) This is the healthy suffering that real learning requires.
Once you’ve faced the fact that things don’t seem to make sense, you eventually reach the point where you’re primed for the insight. In this case, Einstein’s observation was that there’s only one way to make sense of the facts: time doesn’t move for our racers who travel the speed of light! If time went forward, they both would lose the race, which is impossible, so time can’t move forward. As you check through all the cases and details, it starts to make sense: time must move differently for objects moving different speeds.
Now the enormity of this can hardly be stated, and once you get into it, it’s just the tip of the iceberg. It turns out that mass also changes with respect to relative speed, and the energy it takes to speed up an object to light speed is infinite. As you get deeper and deeper, you start to see that all the rules come naturally once you allow that notions we once thought of as fixed and fundamental (time, mass, etc.) can only be understood when you compare relative speeds. Light becomes the ruler by which all things in the universe are measured.
Now this is pretty hard, I admit, and I don’t think it can be made much simpler without losing its beauty. It’s not terrible, though, because I made it as simple as I could. That’s my side of the deal. But understanding it, and really thinking it through, is on you. That’s how most ideas worth thinking about tend to be.
As teachers, though, we have to be careful to balance our instinct to make math fun with a vision of math as being hard, but hard for a reason. And we can teach our students that hard is worth it.