## Studying physics at the three levels of mathematical reasoning

The previous two posts were groundwork for this one.

In order to learn physics at a high level, you have to be able to understand the mathematical objects of each problem at all three levels.

Example: A meter stick is travelling parallel to its length at a speed of 0.6c with respect to you. How long does it appear to be?

You need to have the particular object image pre-developed to perceive that this is a length contraction problem (concept image). You have to remember what the conditions are for using the simple Lorentz-Fitzgerald formula rather than the more general case (advanced mathematical thinking). You have to be able to remember the formula for gamma (concept image). Finally, you have to know the algebra necessary to isolate the correct variable, and the arithmetic to compute the answer (fundamentals).

This may just be a Chinese room exercise for you, but fundamentals are still the most important part of solving any problem. If you don’t have them, you’d have to invent them on the spot for every little thing and that would take forever. Every little part of every little exercise becomes a monolithic effort of creative problem solving. Most problems can just be solved by chaining algorithms together and applying just a tiny bit of perception (that is, the concept image) in order to pick which algorithm to apply.

Now, onward to application.

1. Drilling fundamentals

This is the easiest to do, and the most often neglected in Western countries. Somewhere along the way, we as a culture acquired a distaste for rote memorization, and somehow this extended into a distaste for mechanical practice and drill. This is beyond silliness. Drilling fundamentals is the only thing you absolutely must do to master any trade, profession, .

It is like martial arts. Better to throw 1000 jabs every day and know nothing else than to be able to do all the fanciest flying kicks, but have a slow jab. The neglect of fundamentals is the cause of much uselessness and unnecessary suffering in this world.

In practice, this means reviewing old math textbooks and solving easy problems from your childhood. That means cross-multiplication and completing the square and applying the chain rule to functions of two or more variables, depending on your current level. You don’t need to reread your old books, but make sure you can still do all of the exercises without exerting yourself. This should be fun because it’s so easy, and it ought to be possible to do almost all of them very quickly in your head.

If you find weaknesses, that is better than not finding them. But there’s no payoff in this as such, and you might as well be running a cash register unless we inject a little substance into the math.

2. Exploring concepts

Believe it or not, this is accomplished by two activities that couldn’t be more diametrically opposed: 1) reading about physics while completely ignoring equations, for the purpose of really understanding the real blood and soil of the subject matter, and 2) memorizing equations and using them to solve lots of physics exercises mechanically.

Remember the problem above? This corresponds to the two parts that comprise the vast bulk of undergraduate physics tests and homework assignments. You have to get your hands dirty until you’re up to your elbows in tools and techniques that no one else in your class has heard of. Build familiarity through immersion.

This is sufficient for a person to perform lab experiments and actually know what they’re doing and what the results mean. But knowing why

3. Theoretical physics

When the problem becomes one of theoretical models and postulates, derivation, or proof, the game changes a bit. This is a creative exercise, and as in all creative arts the most important component is drilling fundamentals mechanically. But it also requires perfect knowledge of the object definitions all the way to the bottom, if need be. A person who has such knowledge in combination with well-drilled fundamentals perceives every proof as trivial and easy. He applies no effort, merely watching as the equations rearrange themselves before his eyes.

This is where physics is more fun than work. Thus, it is full of poseurs who do not have the necessary fundamentals.

There is another level of physics that goes beyond mathematical reasoning, which I’ll have to explain another time because I’m getting kicked out of here.