Update from since I first posted this question: just keep sticking with it. I think that I am through the thick of it now and am finally doing ok. I really want to add an answer to this question myself though when I get the time.

TLDR; To summarize, the emphasis in most real math courses seems to be on proof-based things about seemingly esoteric or abstract structures which is, sadly, making me lose my love for math and motivation to do it. Moreover, this exercise doesn't seem to even have bearing on mathematical discovers yet it is all I can see I look deeper into math classes. I have not had the opportunity, to grasp the functional connection between axioms and their use in math. And finally, it may be helpful to know that I know solidly up the basics of matrices, multivariable calculus, and the basic ideas in differential equations.

I have heard that most mathematicians have discredited "game formalism". As per here, it seems like in the working world they don't really care much about foundations and axioms either: Why did we settle for ZFC? . I love math and am interested in studying some more advanced topics, yet this seems to be all that we learn about. If mathematicians do not use axioms to solve problems, then why does every class seem to be about knowing the definitions of structures and then proving things about them? I have not taken a course in real analysis or abstract algebra, but I have dipped my toes in enough to know what it's about.

When I hear people say things like "in math we invent rules and see what is true" or "it's true because that's what we define it to be", I just lose morale. I do not understand how manipulating symbols according to pointless rules is any sort of "game", because it is not fun in any way. It does not reveal truths about the universe as the type of math I know and love.

If this is what advanced math is, then I don't want to become a mathematician. **Please, what is the purpose of learning so much axiomatic math if the working mathematicians does not use it? Isn't that an indicator that it shouldn't be taught?**

And then there is the conflicting trio of reasons for axiomatic math which I have never resolved:

- It seems like the axiomatization of things has come about through a messy process of taking into account and adjusting for many esoteric examples instead of one crystallized concept. I don't see how I can learn what these concepts are without a ridiculous search through historical documents which would take a lifetime of work. And it feels like you have to know literally everything about a subject too to include all those little examples, which is hopeless. Having to blindly take by faith that something is useful takes all the fun out of math, and it turns into grunt memorization.
- If axiomatization of something is not just a more compact and reasonable way of saying what you know, as I have talked about above, but instead a way to capture abstraction that can apply to literally everything, then why do the definitions of things generally not transfer smoothly into the real world? Why do they not have a concise "essence" or "concept" in any lingo beyond math? My interest in this was first sparked when a friend explained that anything, not just "mathematical' things but things like colors could be viewed as vectors. The idea of the inverse of a function is very nice for example. I spent a year pursuing crystallized, applicable English definitions of mathematical objects, but to no avail. I wasted an entire year of my life, when I could have been learning other math, and do not want wish to do it again. Here is one such question I asked during that period of my life: What do Monic and epic morphisms imply?
- If axioms are not made for everything, but just a few specific mathematical objects, then once we see the abstract connection between between those few structures it would seem simpler to solve them in each individually. If it is only 2 or 3 things that satisfy a given set of axioms, why do we talk about it otherwise? Is 50% of advanced math really just learning
*how to save time in one or two examples?*I don't think so.

I believe I have the wrong core philosophy on axioms and their purpose, but being a young isolated self-learner, I lack the subtle ways axioms are utilized which can only be picked up through a classroom environment. If a teacher says "let this be true" vs "we arrive at this because" vs "we'll give this a name because it comes up a lot" vs "this equations represents the geometric object X" vs "we say this because the notation is efficient", are saying very different things with very different implications despite introducing the same object, with the same definition. But again, I lack the correct contextual attitude towards axioms as the core of math.

To summarize, the emphasis in most real math courses seems to be on proof-based things about seemingly esoteric or abstract structures which is, sadly, making me lose my love for math and motivation to do it. Moreover, this exercise doesn't seem to even have bearing on mathematical discovers yet it is all I can see I look deeper into math classes. I have not had the opportunity, to grasp the functional connection between axioms and their use in math. And finally, it may be helpful to know that I know solidly up the basics of matrices, multivariable calculus, and the basic ideas in differential equations.

I have tried so hard for so long, but I think I may not carry on with math without this.

Thank you, so, so, much