Mathematical Sophistication
When I reflect on my education in science (and in physics in particular), the common theme I see is just how the amount of sophistication present in the computations and concepts I learned each year kept increasing. If there was one thing I could count on, it wasn’t learning something “new”. Instead, it was about viewing things I might have once taken for granted as a process that was much more deep than I realized.
For example, take Snell’s law. In secondary school, I learned how this phenomena worked in the sense that I could calculate the effects. I learned that Snell’s law could be written like this:
\(\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2} = \frac{n_2}{n_1}.\)
This allows one to calculate the angle of refraction for various simple systems, and this is exactly what I remember doing. Additionally, the “reason” for why this was true seemed to be something about the light “slowing down” in a different medium, but the reasoning wasn’t all that clear. In the end, it was more of a “here’s the law, now calculate it” sort of concept.
At the time, I don’t remember being bothered by this. Now though, it makes me frustrated, since what is the point of learning these ideas if one doesn’t learn why this specific result occurs? It’s something I’ve been thinking about a fair amount lately.
Fast-forward a few years, and now Snell’s law gets derived using Fermat’s principle of least time, which uses the calculus of variations, and gives one a more satisfying explanation concerning what is going on when the light rays “bend”. In this sense, the mathematics produce the result, which is better than being told the result.
Another example is one that I hadn’t thought about much until I came across it. Anyone who has gone through a class in statistics has seen how to fit a curve to a collection of data points. Usually, one is concerned only with fitting a linear curve, but sometimes we also plot quadratic curves as well (with software).
In the case of linear plots, in secondary school, the recipe went like this. First, one should plot the points on a graph. Then, one needs to carefully draw a rectangle around the data points, and then measure the dimensions of this rectangle. From there, the slope can be calculated, and then a representative point was chosen in order to find the initial value of the line. Basically, this was an exercise in graphing and drawing accuracy, not something you’d want from a mathematics class. As such, while the results were qualitatively correct, they coud differ widely from student to student.
Fast-forward a few years later once again, and the story is much different. In my introductory statistics for science class, we were given the equation that would give us the slope of our linear equation, as well as the correct point to use for the initial value. This undoubtedly produced more accurate results, but once again it lacked the motivation behind it (due to a lack of time, in this case). Thankfully, this lack of explanation was addressed in my linear algebra class, where we learned the method of least-squares. Here was finally an explanation as to how these curves were computed. In the statistics class, it was a long and complicated formula that was given. However, in linear algebra, the reasoning behind how to compute such a curve was much simpler and straightforward. In other words, it made sense as a process. Even better, this method generalizes well for other types of curve fitting, not just linear functions. As such, this explanation was much more useful than all of the other ones.
The lesson that I personally get is that, no matter the topic you’re learning, there often is another layer of understanding that can complement it. This means that I shouldn’t stop looking at concepts that I’ve seen many times just because I think they are boring! There are often new perspectives to look at the situations, and they usually come tied with more mathematical sophistication. This is something that I love to see, because it brings new viewpoints to concepts I might have though I had completely figured out. This shows me that I can always learn and understand a concept more thoroughly, and hopefully this can be good inspiration for you to seek out varied explanations of your favourite concepts.
Just because classical mechanics is, well, classical, doesn’t mean you can’t look at it in more sophisticated ways.