I've recently started my second semester studying Mathematics at an undergraduate level. I'm attempting to find a practical framework through which I can process new mathematical concepts in a way that doesn't lean too far into intuition (which sacrifices an ability to rigorously manipulate definitions and theorems) but also doesn't lean too far into formalism (which sacrifices my ability to genuinely understand how a creative or nice proof can be found).
This post discusses my issue to some extent but it centers on self-studying. In that kind of situation you can explore freely, starting at an idea you'd like to understand and working backwards to build up the prerequisites. But in a formal learning environment, that's not usually possible - no matter how much I may want to understand the Banach-Tarski Paradox or Gödel or whatever, I will need to prioritise my course.
An example of my struggle with formalism is this question I got for Analysis homework last week.
State a countable set $E\subseteq\mathbb{R}$ with uncountably many limit points. Prove that it has uncountably many limit points.
Immediately my intuition led me to $\mathbb{Q}$ as my answer (since you can get arbitrarily close to any real number $x$). I then thought I could prove this by taking a sequence $p_n=(\lfloor10^n\cdot x\rfloor)/10^n$ which truncates it and using the theorem that a sequence converges implies the point of convergence is a limit point. But I found manipulating the floor function to prove $p_n\to x$ was difficult. My next approach involved a kind of bisection algorithm; $\mathbb{Q}$ is unbounded, so $x$ is between two rationals, and then I found the halfway point between those two and used that as a new endpoint. (IE: $\frac{21}{7}<\pi<\frac{22}{7}, \frac{1}{2}(\frac{21}{7}+\frac{22}{7})=\frac{43}{14}<\pi<\frac{22}{7},...$) But I wasn't sure how I would incorporate this into the definition of convergence. My formalism kind of let down what I thought shouldn't be too hard of a proof - I just kind of stared at the symbols for a while, unable to translate my abstract approach to a solution.
Conversely, I struggle with the intuition for a lot of questions; I don't really understand the Lagrange multiplier method for finding extrema of functions, but when studying multivariable calculus last semester I could apply it every time algorithmically and get most questions right. As soon as I was hit with a question where the Lagrangian had three constraints instead of one, I got stuck. If I had intuitively understood why the Lagrangian works then this would have been quite a simple problem.
My current ideas that I've been incorporating is to
- Try and rewrite a proof in natural language after seeing it (so that I understand what steps are being taken)
- Find a simpler case in which I can do the proof and see if it generalises
- Reread the definitions and prerequisites I think I need for the proof and write them in the margin as like a "mental toolbox"
- Honestly? Sleeping on it - it's worked a few times, sometimes I'm just too tired
I'd appreciate any constructive criticism on my methods, any advice on how to develop new ones, or any general improvements to the formulation of my question. I've tried to follow these two guides for this question because I realise this is very subjective. Keep in mind I don't want an explanation or hint on how to solve my Analysis question or grasping the Lagrangian - just approaches I can generalise to my entire studies.