How to Learn Advanced Mathematics Without Heading to University – Part 3

That's sounds typical for a graduate course in the US and atypical for an undergraduate, even an advanced undergraduate course. Most assign problems from texts and will have one or two texts that are required or possibly strongly suggested to have. On the other hand, many students don't really read the texts except for the questions. In graduate courses there are usually just a few suggested books you could use if you wanted to.

> it is essential that we study topics such as Measure Theory and Linear Functional Analysis

I love this sentence. I'm not sure it is true, but it is nonetheless a great sentence.

I'm glad you had success but...let's not measure your outlier experience with the rest of the world. Especially in Adv Mathematics.

I'd point you and other HNs to Srinivasa Ramanujan. He is self taught but...he was wrong [1]. He had a brilliant mind but...due to being self taught, he made some critical mistakes.

Being self taught can easily lead the learner to some critical mistakes. Eventually, they may be corrected (and at what 'cost' does this mistake cause an organization or business or those involved) but it's more efficient of someone's time to just learn from another. I'm not saying everyone needs a University Degree. I'm saying that everyone needs a teacher. Everyone. Why? Because instead of 'the blind leading the blind' (you as a 'blind' teacher, leading you as a 'blind' learner). You have the efficiency of being led by a mentor of some kind that can steer you away from faulty concepts that may come in.

It's great that we now have more free/cheap materials than ever before at our disposal but without a mentor or some kind of peer-review, we could be misapplying concepts.

Also, to comment on something you specifically said:

> Because in my companies view, being able to teach yourself all that math is much more impressive than being taught from a University.

Yes, it's 'impressive' but...most don't learn this way. Which is way it's 'impressive'. Also, being self-taught, how do you truly verify what you understand mathematically is accurate and solid? [2] You might be and I'm not going to fault you but learning concepts is one thing but applying them is even more challenging. It's one thing to be 'impressive', it's a whole other thing to have mastery over a topic. And I'm a firm believer mastery is mostly achieved with peer/mentor feedback.

I applaud you but let's not steer others to just teach themselves, without help from others. Let's encourage self taught and peer feedback. It's not one or the other, it's both.

[1] - https://www.youtube.com/watch?v=jcKRGpMiVTw

[2] - I searched for 30 minutes to find this article, that I read, that stated the current environment of Mathematical Research [3]. Namely, it stated that a lot of research is being published that is NOT peer-reviewed because there isn't enough skilled* Mathematicians to review the work. That it's a 'dirty little secret' in the industry that "known" Mathematicians would get a pass (published w/o review) but many others trying new groundbreaking ideas couldn't get their research peer-reviewed. And with the given University culture to publish NEW research and not review, it's understandable how this environment was created. Namely, Einstein gets the fame but it took numerous people to peer-review his work before it was accepted.

[3] - I know this article exists. It's one of the reasons why I'm becoming a Mathematician. I read it in the past 2-3 years. It was a major site (NewScientist or something that focuses on emerging research). If you can find it, I'd be very grateful. I'm now* using Zotero to save all my findings, so hopefully when I quote something I'll have a source. ;)

*(edited) - original said 'not'. I meant 'now I'm using Zotero'. ;). original said 'skill', I meant 'skilled'

Unless you work on graphics, physics, signal analysis, sound, trading, data science, computer vision, machine learning, etc... it's hard as a software engineer to be exposed to math past the basics, meaning that you can survive without having to go beyond arithmetic.

You might still get some exposure to discrete mathematics once in a while. Statistics is always there to help you, some people avoid it, some others embrace it.

As I a software developer with no degree (working since 2009 in 3 countries), I can share my experience of attempts to learn advanced math.

In 2010, I was very interested in foundations of mathematics, an extremely abstract math branches:

https://en.wikipedia.org/wiki/Foundations_of_mathematics

In particular I spent huge amount of time on:

https://en.wikipedia.org/wiki/Nicolas_Bourbaki (Set theory)

https://en.wikipedia.org/wiki/Principia_Mathematica

https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic

http://www.jhtm.nl/tudelft/tw3520/Introduction_to_Mathematic...

What attracted me is that these books doesn't require any specific knowledge of classical math. I.e. they are self-contained.

It was fun and ... the experience to delve into highly abstract view on entire math.

The big problem is that while I read that for more than a year, I had no experience in problem solving and just ignored exercises (thinking that concept is everything). As a result of that, my entire knowledge is completely evaporated and I literally can't solve any of exercises.

After that year, I dropped math till recently.

Now, I have completely different approach. I learning elementary olympiad style math and most importantly solving problems all the time. Currently, I'm into series of books:

https://www.artofproblemsolving.com/store

These books made for math olympiad preparation. While I solving exercises, I feel how solid my knowledge is.

So if you want to learn advanced mathematics, learn elementary olympiad-style math first. It will give you solid background to start learning advanced math (not just knowledge background but most importantly problem solving skills).