Coming full circle
Experts often end up where they started as beginners.
If you’ve never seen the word valet, you might pronounce it like VAL-it. If you realize the word has a French origin, you would pronounce it val-A. But the preferred pronunciation is actually VAL-it.
Beginning musicians play by ear, to the extent that they can play at all. Then they learn to read music. Eventually, maybe years later, they realize that music really is about what you hear and not what you see.
Beginning computer science students think that computer science is all about programming. Then they learn that computer science is actually about computation in the abstract and not about something so vulgar as a computer. But eventually they come back down to earth and realize that 99.44% of computer science is ultimately motivated by the desire to get computers to do things.
In a beginning physics class, an instructor will ask students to assume a pulley has no mass and most students will simply comply. A few brighter students may snicker, knowing that pulleys really do have mass and that some day they’ll be able to handle problems with realistic pulleys. In a more advanced class, it’s the weaker students who snicker at massless pulleys. The better students understand a reference to a massless pulley to mean that in the current problem, the rotational inertia of the pulley can safely be ignored, simplifying the calculations without significantly changing the result. Similar remarks hold for frictionless planes and infinite capacitors as idealizations. Novices accept them uncritically, sophomores sneer at them, and experts understand their uses and limitations.
Here’s an example from math. Freshmen can look at a Dirac function δ(x) without blinking. They accept the explanation that it’s infinite at the origin, zero everywhere else, and integrates to 1. Then when they become more sophisticated, they realize this explanation is nonsense. But if they keep going, they’ll learn the theory that makes sense of things like δ(x). They’ll realize that the freshman explanation, while incomplete, is sometimes a reasonable intuitive guide to how δ(x) behaves. They’ll also know when such intuition leads you astray.
In each of these examples, the experts don’t exactly return to the beginning. They come to appreciate their initial ideas in a more nuanced way.
When two cars are competing for the same lane, the unexpected happens.