# Cut the enemy

Created 2022-04-01, Edited 2022-04-01

The primary thing when you take a sword in your hands is your intention to cut the enemy, whatever the means. Whenever you parry, hit, spring, strike or touch the enemy's cutting sword, you must cut the enemy in the same movement. It is essential to attain this. If you think only of hitting, springing, striking or touching the enemy, you will not be able actually to cut him. Miyamoto Musashi, The Book of Five Rings

I've been thinking about this quote in the context of learning recently, allow me to rephrase it

The primary thing when you take a book in your hands is your intention to learn, whatever the means. Whenever you do an exercise, read a chapter, or take notes, you must build understanding in the same movement. It is essential to attain this. If you think only of finishing exercises, reading chapters or passing exams you will not be able to truly learn.

To expand: I've noticed I sometimes have a mindless attitude when studying to "mindlessly" read or do exercises, when I should be using them as a bridge to the underlying principles. Put another way, intuition isn't optional.

Math example: Say you learn the statement and proof of the mean value theorem, the learning isn't done! You still need to meditate on when it will be useful and what it means (pun intended).

We could view the mean value theorem as a bridge between $f$ and the derivative $f'$, crossing this bridge has a cost: We won't know exactly where we end up, i.e. we find a $c \in [a,b]$ with

$f'(c) = \frac{f(a) - f(b)}{a-b}$

This gives a good intuition for when it will be useful, whenever we have a condition in function-land we want to translate into derivative-land. Sometimes finding the right bridge won't be so obvious, but once you know you've got to travel between function and derivative land half the battle is won.

Are we done? No! We still haven't thought about how this connects to everything. What are some special cases? Well, if $[a,b]$ is tiny we can use continuity to state $|f(a)-f(b)|$ is small. In general, every theorem you learn or exercise you solve should get you closer to understanding the true names of the theorems involved.