MATHEMATICS
Estimation, Modeling, and Accuracy
I’m currently studying both mathematics and physics in university, and I have to admit that it can be difficult to straddle the line between the two. Both are similar, yet demand different mindsets in terms of how to think about tackling a problem and actually coming up with a solution. In mathematics, not only is the right answer desirable. Every step along the way should be rigourously justified. That’s because the conclusion that one wants to get to rests on the arguments that come beforehand. Without those arguments, you don’t have anything. This is why mathematics classes require students to create proofs that carefully apply definitions. I’m not saying that there isn’t any playfulness involved, but when it comes down to making an argument, the clearer the supporting propositions, the easier it is for others to become convinced of the truth of your claim.
In physics, I’ve found that the situation is quite different. Being mathematically coherent is of course necessary within developing a theory, but the truth is that physicists are much “looser” with their mathematics, for lack of a better word. In physics, it’s often taken for granted that certain complications “are so small that they won’t make a difference”, which allows them to drop the complications. This is something that absolutely would not be allowed when proving statements in mathematics, because any weak argument is the first thing that gets attacked when someone critiques a proof. Many people think that π+e is transcendental, but since we don’t have a proof of this, it’s an unjustified belief.
The difference in physics (and science in general) is the fact that we often know what the answer should be. This makes a huge difference in terms of the way that we work through theory to get to a result. It’s a lot easier to say “these other contributions won’t have a large effect” when we know that continuing in this manner will give the observed result. Of course, it probably is true that certain contributions aren’t as important (and one can show this mathematically), but that extra work is often hand-waved away. Because of this, I’ve observed that we often will simplify matters a lot more than what I would have thought appropriate, because it gives the correct answer.
I’ve had mixed feelings about this, particularly because I’ve been on the other side in my mathematics classes, where it was necessary to go through the steps, even if something seemed obvious or didn’t make a huge difference. I often thought it was annoying (and still do, at times) when the mathematics were “simplified” in the sense that rigour was sacrificed for brevity and the final result. I wished we would rigourously justify each and every step, in order to make things mathematically correct. I also didn’t like the fact that sometimes we would “guess” results, in the sense that the best way to solve an equation was to try a solution and see what came out of it. This all seemed far removed from my studies in mathematics.
Recently though, I’ve not had a change of heart, but rather I’ve understood more of the rationale behind a lot of these decisions. As I’ve wrote about before, science is about making models of the world that both explain and predict the various features we see around us. However, in order to be mathematically tractable, simplifications and approximations are necessary. Furthermore, they aren’t fundamentally a bad thing, as long as one keeps in mind the simplifications throughout. This was the key I was missing. It’s not that we’re deliberately ignoring thorny issues, it’s that we are making a first model, which can always be refined and improved. It’s unrealistic to expect to have hyper-realistic models when first learning a subject, so these toy models with their approximations will have to do. Even if I don’t like the fact that we approximate irregular shapes as spheres, it’s done so that the problem is tractable and it doesn’t change the end result drastically.
My shift in mindset has come after really digging into some of the work of Tadashi Tokieda, who has some interesting resources from an old course available here. He is an applied mathematician who is also a great communicator. If you look at the website I linked to, you will see that he is very good at explaining things, and I particularly like how he characterizes the kind of work an applied mathematician should do. He says that an applied mathematician should be trying to do a back-of-the-envelope every day in order to increase one’s skills. The goal here isn’t to be analytically exact. Instead, it’s about probing the relationships between the items of interest. It’s about using mathematics to get to a result, without being overly worried about the formalism. That can wait for later. This has inspired me to start doing the same.
I’ve begun working on asking myself questions that delve into this sort of thing, where it’s unclear how to exactly begin, but by making approximations, a reasonable estimate can be found. It’s not easy, but it has gotten me to be more open with estimation. As the author of a book I’m reading on the subject writes, “It’s okay to say that 2 3=10.” The point isn’t to be precise. It’s to make a calculation tractable.
The more that I think about it, the more that I realize that we quickly discourage students from doing this at school. We say, “Don’t guess. Find the exact answer.” The truth is that estimation is important, and should be more frequently used. We should be able to take any kind of statement with a number attached and make sense of it. This ability is crippled when everything has to be exact. As such, I think we should be encouraging more estimation and less accuracy in order to get a foothold into a problem. Only then should we move onto refining and making a model more accurate. After all, that’s what we often do in science. We start with something we can handle, and make it more and more sophisticated.