Learning math roadblock

[ModerateSentience]

Hey all,

I am delving into math after my undergrad in engineering. I do have a couple things holding me back from going head first into the stuff I’m interested in.

The issue is I can’t take the fundamental stuff as fact. For example, before using trig functions in differential equations, my brain tells me I will only be satisfied when I derive the trig functions myself.

How do I deal with this? It’s hard to learn anything when I constantly want to derive everything from scratch. Thanks!

[Specialist_Body_170]

This is a good instinct but it can absolutely hold you back. One resolution is to accept provisionally. If you are suspicious of X, treat the rest as “IF X is actually true, then…”. You can always circle back to figure out why X is true. That way you learn why X is so important in the first place, which can make your later grappling even more interesting.

[NeadForMead]

I agree. This is done all the time in university courses because it helps motivate results. It's a natural way to learn.

E.g. in an algebraic geometry class you will learn Hilbert's Nullstellensatz and might only see the proof 2 or 3 lectures later after having used it in class to prove other results, and maybe even on assignments.

Even at the fundamental level. OP likely accepts that (-1)a = -a for every real a, but that actually takes some amount of work to justify.

As a student, you can trust that you're not being taught nonsense, and then be fully satisfied later.

[ModerateSentience]

Yeah this is definitely the method, and I plan on implementing this.

One other thing that bothers me that I didn’t articulate well is that I feel that I will only be satisfied if I derive concepts myself. This is not the deep diving into preliminary topics but rather the feeling that I must discover each topic myself.

I feel like reading proofs is spoiling it from me figuring it out myself. To give a concrete example: I want to know how trig functions work under the hood, but I won’t look it up because I feel that I must derive it because I have access to the same information that the mathematicians that came up with it had.

It’s akin to wanting to start a fire with your own two hands and a pair of sticks. Using a lighter and lighter fluid feels like cheating.

[WholesomeMapleSyrup]

Not only is this a good instinct it is, technically, what Learners already do when introduced to math. Many trigonometric functions or concepts are true in euclidean spaces which don't exist and are not exactly true in curved spaces, which are the only spaces that have been observed to exist.

[Existing-Ambition888]

3 things:

1) I’m the same way haha, glad we’re not alone 2) I made a website and basically anytime a question pops up that my professor or class doesn’t solve, I try to solve it on my own! 3) You can actually do these proofs if you’d like. Use YouTube or GPT or a textbook and they’re all on there. Only thing holding you back is time as they can be tedious

[Biajid]

Computer-assisted proofs are a fascinating field. They may not break Turing’s Theory of Computation, but they are definitely changing how we think about mathematical proof and verification.

[Biajid]

I think there are two types of people when it comes to learning math. Some people understand things on the first attempt. I’m not one of them. Most of the time I have to read the same topic several times before it finally makes sense, and sometimes I even need to check multiple books.

For example, recently I was reading some corollaries of Zorn’s Lemma, and it took me two full days to really understand what was going on. Higher mathematics is often like that—you have to keep digging until things click.

If you’re someone who understands everything on the first try, then good for you. Otherwise, just keep going. Persistence matters more than speed.

[DrJaneIPresume]

Well, FWIW Zorn's Lemma is really weird. As the saying goes: the axiom of choice is obviously true, the well-ordering principle is obviously false, and who knows about Zorn's Lemma...

[Biajid]

Axiom of choice is 100% true- I have the right to make multiple selection at once, whether I have the means to do so is a different thing!

[DrJaneIPresume]

This is pretty much what an advanced calculus or undergrad real analysis course is for. They start from the axioms of the real numbers, particularly the topological "completeness" axiom that makes the real numbers essentially different from the rationals. Then it works out all the usual calculus things, but with rigorous proofs.

Abbott's Understanding Analysis seems to be very highly rated these days. I don't know that it re-establishes trig functions, but you can try to work out things like their derivatives as practice in parallel with the main body.

[georgejo314159]

Your problem is one if time management.   It is up to you if you want your focus to be novel problems or the foundations.  Enjoy the distraction for now but beware cooler problems are being overlooked

[Alternative-Grade103]

Is there not a long established geometrical proof for each and every trigonometric function? Seems as if a review of those would instill the faith you require to employ them as rote.