I’m looking for advice to understand why math is the way it is

Math is a myriad of different fields and topics. You can’t understand all of math there is waaaay too much. Just focus on the specific topics you need/want for what you are going to study in university
>I’m starting to think math is a huge myriad of different fields and topics.

Correct you are!

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>I’ve tried to look on the internet for the way how math is formed and its philosophy but so far all I got was maybe very complicated math laws and topics, and logic stuff.

There are two very different meanings behind the phrase *how math is formed*.

The first one is learning the basic tools and disciplines, such as arithmetic, basics of Euclidean geometry, basics of combinatorics, trigonometry, etc. Then you progress towards being more rigorous in exploring the concepts you're already pretty familiar with (calculus and linear algebra are rather good polygons for doing that). This is how you gain what is commonly referred to as "mathematical maturity".

Now that you have gained sizable experience you can start looking at the second meaning of *how math is formed*, i.e., can we set up some kind of a simple^(\*) framework within which we will be able to build up everything we are already familiar with. This is where the study of fields like formal mathematical logic and set theory fit in.

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(^(\*)) "simple" in this context means "with a small number of assumed objects and a small number of assumed statements about those objects".

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>I thought math is used for science and calculations

It is used for that too, but there is so much more than mere calculation.

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>now I have absolutely no idea what to even study to truly understand math

I hope that the few paragraphs I wrote above are helpful in some way.

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>maybe I will get really lost when i study even more complicated math in university and beyond.

As long as you don't let yourself get scared off, you won't.

Also, what kind of mathematics will you encounter at the university level highly depends on what kind of studies you end up choosing to do.

If you end up studying law, you will get exposed to very little (if any) advanced mathematics.

If you decide to study economics or some branch of engineering you will learn a lot about the fields which are immediately applicable to your future profession (in most cases, this will be some form of advanced calculus topics, plus probability and statistics).

If you end up being a physicist (especially theoretical physicist), you will often have to go beyond just calculus and into fields like topology, differential geometry, and group theory, but you will most likely not have to bother with the foundational theories like set theory.

And, of course, if you end up studying mathematics, you will be exposed to a wide variety of mathematical topics from abstract to applied, and from foundational to high-level.

(Of course, the above list is not supposed to be an exhaustive list of all the possible things one could decide to study. It's just a list of a few examples to illustrate how vastly different your experience with university-level mathematics can be, depending on what you choose to study.)
This is such a profound question that I couldn't resist voting it up.

Richard Courant's classic book, called *What is Mathematics?* is, very good, but it doesn't actually answer the title question. (It's really just a survey of a few branches of math.)

Eugene Wigner's 1960 paper, *The Unreasonable Effectiveness of Mathematics in the Natural Sciences,* comes closer to the point, but Wigner is interested in a different (but related) question. He doesn't come up with a definitive answer to either one, and the paper is fairly hard reading.

I've tried myself over the years to come up with a pithy explanation of what makes one topic math and another topic not math. The best I've been able to come up with is that mathematics is *the study of the formal consequences of formal rules*. Formality means that you are working with strings of symbols, with rules for generating the strings that are like the rules of a game. The exact rules for generating the strings differ from one branch of math to another, but the game mechanic is always pretty much the same. Although it can be tedious to work out the details, *every* branch of mathematics can be approached in this way.

I don't want to bore you by going into more details, especially because it doesn't help answer your main question, which is what to study next in order to get closer to mathematical enlightenment. Here the answer is pretty simple: the standard university math curriculum looks very much the same no matter what institution you attend, and it got that way through a long process of trial and error that paralleled the development of the field itself. Your undergraduate mathematical life is pretty much on rails for at least the first two years, and you just don't need to worry about the issue. You'll have three or four terms of calculus and differential equations, and then your attention will split into two streams: analysis and algebra. In analysis you'll learn the very careful lawyerly underpinnings of calculus -- what are real numbers, really? What is a function? What are the rigorous definitions of derivatives and integrals, and which functions can and can't be differentiated and integrated? On that branch, you then jump to calculus on manifolds and the theory of functions of a complex variable.

On the algebra branch you probably start with linear algebra, a particular type of higher arithmetic, and then proceed to abstract algebra, which presents many other kinds of higher arithmetic and gives the unifying theory of those different systems.

To satisfy your own curiosity about the nature of mathematics as a whole, I would recommend that around the time they start letting you choose electives (probably third year) you take a class in computation theory and one in mathematical logic. After you've managed to swallow the Halting Problem and Gödel's Incompleteness Theorems, you will be *much* closer to knowing the answer to your big questions.
Most generally, math is the study of formally-defined relationships.  What this means is that you start with some set of statements, and from there you deduce other things that are a necessary consequence of those statements.  For example, you can define 2 as the general idea of a set with two elements, 4 as the general idea of a set with four elements, + as the operation of combining two sets into one sets containing the elements of both, and = as the statement that two sets have the same size, and with those definitions (and more rigor than I have time for right now), we can deduce that 2 + 2 = 4 by necessity.  2 + 2 = 4 is *always* true with these definitions (assuming I made them correctly, anyway).

This is distinct from, say, physics, where the laws of physics must be discovered by experiment.  You measure stuff and then you can make a law about it, and assuming your law is right, you can use math to work out the consequences of this law assuming other laws, but the laws themselves come from observation.  Math would be exactly the same regardless of the universe's laws.  The only things that would change are the definitions, but if two sets of aliens come up with the same definitions, they will be able to derive the same math results, even if they're in different universes with different laws.  If you define 2, 4, +, and = the same way, you will *always* get 2 + 2 = 4 no matter how alien you are.
I really enjoyed the BBC series "The Story of Maths" with Marcus du Sautoy and it might answer some of your questions.  It covers how math developed in different cultures and how all of that knowledge has come together.  I'm not saying binge it - it is a little dry but it really helped me put things into place within my head.
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Math is such a nebulous field that there is a time period in history that we can consider "the last time you could realistically know all known math."

These days, it's more a category of something than it is a specific subject, so "math" doesn't correspond to "english" or "chemistry," it corresponds to "natural languages" or "natural sciences."
>I thought math is used for science and calculations

When you look into the history of math, many of its topics were not used for science, but were more like a hobby for people who had other jobs. For example mathematicians would challenge each other to find roots of polynomials, and people found a formula for the roots of a degree four polynomial, but they kept it to themselves to keep an advantage in those competitions.

Applied mathematics used to mean calculating integrals and solving differential equations. But with quantum physics, abstract stuff like group theory became applied mathematics and with the widespread use of computers stuff like cryptography became applied mathematics.
In addition to some of the other reading suggestions, you might find it useful to read *The Mathematical Experience* by Davis and Hersh.

It’s a book about the philosophy of mathematics: what it is and how it is done. I’m fairly certain you’ll be able to follow it given your stated experience, but keep in mind it is **not** a math book; it’s a philosophy book.
This might not be exactly what you are after but years ago I picked up Karl Popper’s (I think it was called) Philosophy of Science from my university library and found it very interesting.
Math is used for calculations and science but that's NOT what math is. A lot of years ago Katy Perry asked Neil Degrasse if math is related to science and she got such a hate for that question, since everyone thought "well yes of course". In my opinion the question was fair and justified. Science anf math have a one-side relationship. Science needs math but math doesn't care about science.

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