2/3: Models

30 January 2016

Modified: 13 June 2016

1673 words

[Part 2 is the best part. Part 1, part 3.]

In science, we try to understand the world by building models and theories that describe it. You see an apple falling on your head, think, “oh, maybe that’s how the planets move, too”, and you write down rules that allow for the motion of planets and don’t allow for some phenomena you do not see, like things falling upward. You call the collection of those rules your model, or theory. When you have your model, you perform more experiments to test it, and every time your model’s prediction roughly matches your observations, you get more confident that your model is correct.

What does it mean for a model to be correct? This is where the trouble starts. In school we learn that classical mechanics is a pretty good approximation of reality, but quantum mechanics and relativity is the correct theory of how the universe works.¹ This framing has always bothered me: If Newtonian mechanics is wrong, why do we still use it so damn much?

Say you throw your keys out the window, and you want to calculate the path they will take to the ground as exactly as possible. So you get out your pencil and notebook and you start scribbling. Should you do your calculations relativistically? It would be more work, but you want to be really exact, so you add a bunch of γ’s everywhere and do your calculations relativistically. Then you notice that you’ve been assuming a flat earth the whole time. Oh no! All right – the earth’s a sphere, right? Let’s use that and we get an ellipse instead of a parabola for the flight path of our keychain. So – is the result more accurate than the classical, flat-earth one? Certainly not a lot more, but maybe a little? Nope. Not one bit. Why? Because the difference the air resistance makes, and the uncertainty of the direction you’re throwing in is much bigger than the difference a relativistic calculation could make.

This still doesn’t mean that objects in our everyday lives have a different nature than single electrons or supermassive black holes. It just means that, if you put enough electrons and protons and stuff together, and you don’t make them too dense or too fast, you can predict what they’re going to do by using the model of classical mechanics.

Many physics students, when they’re starting out, seem to feel like they’ve been promised something. That they’ll be led behind the curtains of reality and shown how the world really works. They seem to accept Newtonian mechanics – it works, after all. Medium sized objects, apparently, are Newtonian in nature. But it doesn’t take long for the disappointments to start. “Ideal gases don’t exist in nature, but it’s a simple model that works relatively well for lots of stuff,” they tell us. We’re not happy, but we’ll take the approximation, for now. We’re relieved when they teach us the “real gas” models, like Van der Waals gases. Then it gets worse again: “Ideal fluids are a pretty absurd approximation. There are no ideal fluids in the real world, and for most fluids, you don’t even get very good results using this model. But it’s simple, and it teaches the principles that you need to understand to work with better fluid models later.” We’re not taught the more complicated fluid models in that semester, and it leaves us with a quasy feeling. Why are we being taught a rough approximation instead of the correct model?

After a few semesters, the students get herded into a lab, to perform their first experiments themselves. Their belief is already shaken by countless lectures only teaching rough approximations instead of the real thing. But this is worse. Here, they finally see how the sausage is made. “All of physics is just estimations and approximations!” they exclaim. “Nothing here is exact!” It slowly sinks in that this is not just a rough approximation of what physicists do. Physics really is just approximations. Dutifully, the students draw error bars in their hand-crafted plots of noisy data, and wheep.

What’s important is that this isn’t a bad thing, and especially not a preventable thing. The approximations aren’t the result of laziness. The small inaccuracies in every scientific theory are the result of countless hours of patient, skillful labor. It’s awesome that we can make very accurate predictions about the behavior of gases just using pV=NT, instead of calculating the exact position and momentum of every elementary particle in our system. Because, by the way, that “system” is the entire universe. It’s super cool that we can just pretend planets are single points in space, with a mass and no size, only feeling the gravity of the sun and not each other’s, and still predict their orbits with great accuracy. Planets aren’t spheres, their orbits aren’t circles, Kepler’s Laws of Planetary Motion aren’t woven in the fabric of the universe, and yet, pretending all this is true will get us to Mars.

Since all models are wrong the scientist cannot obtain a “correct” one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity. — George E. P. Box

The goal here is not “understanding”. The goal is making good predictions. I see my fellow students understanding that ideal gases don’t exist in nature, but I don’t see them make the jump to “Van der Waals Gases don’t exist any more than ideal ones do”. I don’t see them understanding that “quantum wave functions” are a mathematical function instead of a thing, out there in the universe. The problem is that physics ventures so deep into the hidden parts of reality, that it’s no longer intuitively clear that there is a distinction between the map and the territory. They tell you about the paradox of the double slit experiment and you conclude, “electrons aren’t just particles”. They show you Schrödinger’s equation and solve it to get the wave function. “That explains it!” you think, and you conclude that electrons are wave functions.

But the universe does not run on math. The reason the universe looks so much like it’s made of math when you apply science to it is because math is really really versatile. But this doesn’t mean our Laws of Physics are more than summaries of our observations. It’s not the universe that is good at being modeled by math – it’s the math that is good at modeling anything, be it our universe or universes with different rules.

I’ve heard someone say, after reading a mechanics textbook, that they finally understand why perpetual motion is impossible. It’s because something something holonomic constraints can’t do any work because something dot product. This can’t possibly be true because that’s not the order in which things happened. First the perpetual motion machine didn’t work, then the theory was written and it was written in such a way that perpetual motion machines don’t work, and that’s why something something holonomic constraints forbids them. If the textbook had explained, in detail, why perpetual motion machines do work, that wouldn’t have made it true.

We once had a homework exercise where we were supposed to say why a particle behaved in a certain way. The obviously “correct” answer – the teacher’s password – was “because of Heisenberg’s Uncertainty Principle”. But the Uncertainty Principle just follows from Schrödinger’s equation, and we’re using that to solve all our quantum mechanics problems. So by that logic, basically everything happens because of the Heisenberg Uncertainty Principle. That can’t be right.

For example, when a pen falls off a desk, that seems to be proof that gravity exists, because gravity made it fall. But what is “gravity”? In 1500, “gravity” was the pen’s desire to go to the center of the earth; in 1700 “gravity” was a force that acted at a distance according to mathematical laws; in the 1900s “gravity” was an effect of curved space-time; and today physicists theorize that “gravity” may be a force carried by subatomic particles called “gravitons”. Gendlin views “gravity” as a concept and points out that concepts can’t make anything fall. Instead of saying that gravity causes things to fall, it would be more accurate to say that things falling cause [the different concepts of] gravity. Interaction with the world is prior to concepts about the world. (source)

It’s not the laws we have written down that tell reality what to do. It’s reality that tells us what laws to write. Writing down the law will not make reality obey it. But reality doing something unexpected will make us write a new law.

The point I’m trying to make here is, when you have electrodynamics homework to do, and taking a few shortcuts by pretending stuff doesn’t interact as much as the theory says will allow you to finish in 6 pages instead of 47, maybe you should do that. Because there is no “correct” model. You’ll never know what matter is “really” made of. All you can ask for is a good prediction.