Mathematics is hard for mathematicians to understand too

I love math but the symbology and notations get in my way. 2 ideas:

1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable

2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..

The upside of a "dry and formal" presentation is that it removes any ambiguity about what exactly you're discussing, and how a given argument is supposed to flow. Some steps may be skipped, but at least the overall structure will be clear enough. None of that is guaranteed when dealing with an "intuitive" presentation, especially when people tend to differ about what the "right" intuition of something ought to be. That can be even more frustrating, precisely when there's insufficient "dry and formal" rigor to pin everything down.

Mathematics is hard when there is not much time invested in processing the core idea.

For example, Dvoretzky-Rogers theorem in isolation is hard to understand.

While more applications of it appear While more generalizations of it appear While more alternative proofs of it appear

it gets more clear. So, it takes time for something to become digestible, but the effort spent gives the real insights.

Last but not least is the presentation of this theorem. Some authors are cryptic, others refactor the proof in discrete steps or find similarities with other proofs.

Yes it is hard but part of the work of the mathematician is to make it easier for the others.

Exactly like in code. There is a lower bound in hardness, but this is not an excuse to keep it harder than that.

In this modern era of easily accessible knowledge, how gate keepy is it though? It's inscrutable at first glance, but ChatGPT is more than happy to explain what the hell ℵ₀, ℵ₁, ♯, ♭, or Σ mean, and you can ask it to read the arxiv pdf and have it explain it to you.

As someone who has always struggled with mathematics at the calculational level, but who really enjoys theorems and proofs (abstract mathematics), here are some things that help me.

1. Study predicate logic, then study it again, and again, and again. The better and more ingrained predicate logic becomes in your brain the easier mathematics becomes.

2. Once you become comfortable with predicate logic, look into set theory and model theory and understand both of these well. Understand the precise definition of "theory" wrt to model theory. If you do this, you'll have learned the rules that unify nearly all of mathematics and you'll also understand how to "plug" models into theories to try and better understand them.

3. Close reading. If you've ever played magic the gathering, mathematics is the same thing--words are defined and used in the same way in which they are in games. You need to suspend all the temptation to read in meanings that aren't there. You need to read slowly. I've often only come upon a key insight about a particular object and an accurate understanding only after rereading a passage like 50 times. If the author didn't make a certain statement, they didn't make that statement, even if it seems "obvious" you need to follow the logical chain of reasoning to make sure.

4. Translate into natural english. A lot of math books will have whole sections of proofs and /or exercises with little to no corresponding natural language "explainer" of the symbolic statements. One thing that helps me tremendously is to try and frame any proof or theorem or collection of these in terms of the linguistic names for various definitions etc. and to try and summarize a body of proofs into helpful statements. For example "groups are all about inverses and how they allow us to "reverse" compositions of (associative) operations--this is the essence of "solvability"". This summary statement about groups helps set up a framing for me whenever I go and read a proof involving groups. The framing helps tremendously because it can serve as a foil too—i.e. if some surprising theorem contravene's the summary "oh, maybe groups aren't just about inversions" that allows for an intellectual development and expansion that I find more intuitive. I sometimes think of myself as a scientist examining a world of abstract creatures (the various models (individuals) of a particular theory (species))

5. Contextualize. Nearly all of mathematics grew out of certain lines of investigation, and often out of concrete technical needs. Understanding this history is a surprisingly effective way to make many initially mysterious aspects of a theory more obvious, more concrete, and more related to other bits of knowledge about the world, which really helps bolster understanding.