How Much Is Too Much When It comes to understanding from first principles?

Question

My attempt to take math more seriously awoken a dormant obsession of trying to deconstruct everything to it's bare bones, and of not being satisfied with superficial understanding. As a rule of thumb i don't understand anything until i can prove it from scratch, however even though im still in highschool, i've found this incredibly difficult and time taxing, and I've been wondering if it's even possible in my case, maybe im just not smart enough for it. Also, this method of 'learning' or whatever you might call it, requires me to go on very long tangents when studying of doing math, for example while studying for a chapter on algebraic structures i've had to learn some linear algebra in order to understand the properties of matrix multiplication. And while studying calculus i had to prove every trigonometric identity i came across. This may not sound like alot but given the time constraint i have on exams it's been impossible to stay on schedule.Thoughts?

Answer 1

Let's say you are trying to understand Theorem X. There is an important method of understanding which I like to call "suspension of disbelief", or as my student calls it, "not falling down a rabbit hole". It goes like this.

At the top level of your understanding is a very short proof of Theorem X. What this proof consists of is the "ideas" of the proof, an "outline" of the proof, perhaps citing one or two other theorems that are used in the proof. Here is an example:

Theorem: $\cos(3\theta) = \cos^3(\theta) - 3 \cos(\theta) \sin^2(\theta)$.

and now for

The Proof (outline): This follows using algebra from the standard identities for $\cos(\alpha \pm \beta)$ and $\sin(\alpha \pm \beta)$. QED

Now, your time is valuable to you. Perhaps you want to KNOW EVERY DETAIL of how to use those standard identities to prove the $\cos(3\theta)$ identity. But, you also need to move on to other things.

So, you make a rational decision to suspend your disbelief in that proof outline. You decide to accept that proof outline because you have faith in your own ability to fill in the details at any time, by remembering the standard identities and doing the algebra. And now you can move on to the next theorem.

Suspension of disbelief is an important skill, it keeps you from falling down a rabbit hole and making no progress.

Answer 2

It's not that you're not smart enough. Forcing mathematical enlightenment to be linear seems to be the problem. Wandering around into other topics while you are progressing in a certain direction is a great idea, but....

If you are not keeping a journal, start keeping a journal. A computer journal would be best because then you can insert new entries into the appropriate place more easily. I take along a pad of newsprint (cheap paper) with me everywhere I go so that I can diddle and doodle with a current problem whenever inspiration or whim occurs to me.

Think about that scene in the movie, Patton, where an entire brigade could not cross a bridge because two donkeys refused to move. Patton shot both of them between the eyes and had them thrown off of the bridge.

If your compulsion to prove everything is holding back your college education, shoot that bugger between the eyes and throw it off of the bridge. With a journal, you can record your problems and you can address them at your whim. You will be their master instead of the other way around.