∞/3: Epilog · Abstraction

26 June 2016

556 words

[When I gave a draft of part 3 to a friend to read, they commented on the first paragraph, “Math isn’t about giving things funny names; it’s about giving things meaningless names!” I had a thought on this topic but when I wrote part 3, but I couldn’t really make it fit and thought it wasn’t that interesting anyway, but my friend said it was, so here goes. (Parts 1, 2, 3.)]

15-year old Nino had an idea once: “Math is stupid! When solving physics problems, you always have to take the actual physical quantities, then make up weird letters to put them through the equations, and then you have to translate them back to the physical quantities. This makes it harder to see what you’re doing because, when you glance at an equation, you only see the relations between letters and not the relationships between the actual physical quantities. In the hundreds of years that science has been around, someone must’ve come up with a more intuitive way to write equations. After all, computer scientists don’t just call their variables a, b, c, x, y, z either!”

However, the meaningless symbols are actually a good thing. As I said earlier, the scientist’s job is coming up with mathematical constructs that mirror the behavior of certain aspects of reality. They use this mathematical model to look for new unexpected behaviors and run experiments to check whether these behaviors can be observed in reality. Now, since many things in the universe kinda look the same if you squint a little, it makes sense to apply the same models to them. Thus, you can solve the complicated math parts of your model once, while getting new predictions for many different experiments. For example, you only have to solve the equations for the harmonic oscillator once to model everything that sort of vibrates a little.

Or, you write an object’s kinetic energy as $E_k=m v^2/2$, and its rotation energy as $E_r = I \omega^2/2$. In both cases, you have half of some property the object has (the mass $m$ or the moment of inertia $I$), multiplied by the square of what the object is doing (How fast the object is moving $v$, or how fast it’s spinning $\omega$). So when you want to, e.g., form the time derivative of $E_k$ or $E_r$, it still has the same form ($\dot E_k = m v \dot v$ and $\dot E_r = I \omega \dot\omega$) and you can easily see that you don’t have to do the calculation twice. Thinking about formulas in this way can also help with memorization: If you take the rule, “Energy is always¹ half of a property times the square of something that changes,” what do you think the energy stored in a capacitor looks like? If you’re a person who frequently finds themselves in situations where they have to answer such questions, you may remember that the Very Important Property of a capacitor is the capacitance $C$, and a thing that changes is its voltage $V$. This suggests the stored energy would be $E_c = C V^2/2$ which, in fact, is correct.