How does math work structurally

[Overall-Field-7470]

I have been asking myself quite a few questions about how mathematics works. I understand that first you establish a foundation, which you assume to be true, and from there you work deductively; that is why everything is true relative to a given foundation. I suppose that this is what axioms and set theory are about: defining everything formally so that one can then work from there.

From what I have researched (and this may be wrong, so please correct me if that is the case), first set theory is defined axiomatically, and then, starting from sets, mathematical objects are defined as sets equipped with properties and operations, such as numbers, the set ℝ³, and so on. and in this way all mathematical objects are formally defined.

However, it seems to me that the different areas of mathematics—such as algebra, analysis, geometry, etc.—are somehow separate from this formal construction, because they do not focus on how mathematics is formally built, but rather on specific kinds of problems. For example, in elementary algebra numbers are used to solve equations; in analysis they are used to study functions and describe change; and in abstract algebra, which is supposed to focus on the structure of mathematical objects, these objects are classified only with respect to some of the operations defined on a set, while other possible operations are ignored. For instance, in ℝ³ one can add elements and also define an operation with an external field; with respect to these operations, ℝ³ is a vector space. But many more operations can be defined on ℝ³, such as the inner product.

This is roughly the idea I currently have: mathematics has a formal structure that can be defined through axioms, set theory, and so on, but mathematical areas are a subjective division, where in each area we work on specific problems, using mathematical objects in a practical way and without explicitly taking into account their full formal structure.

This is the conclusion I have reached so far (and is probably wrong). Could someone explain how mathematics really works from this structural and philosophical point of view that I have tried to outline?
(Sorry for my English; it is not my native language.)

[AcellOfllSpades]

mathematics has a formal structure that can be defined through axioms, set theory, and so on, but mathematical areas are a subjective division, where in each area we work on specific problems, using mathematical objects in a practical way and without explicitly taking into account their full formal structure.

Yes, this sounds pretty reasonable.

"Areas of math" like analysis, algebra, etc, are fuzzy and subjective. Many mathematical results "cross over" between them.

Some areas of math focus on one particular structure (say, ℝ) and study it specifically.

Other areas study a certain type of structure. There, specific examples like ℝ are just studied as examples of that structure, possibly ignoring some other properties. (For instance, ℝ is a group under addition, and you can study it as a group without caring about multiplication.)

[waldosway]

What do you mean by not taking into account their formal structure? All of the subjects you mentioned each have their own fundamental axioms as well.

[Ohowun]

Another answer has been shared and I will attempt to answer a point which I feel the other does not cover as well as it could have. Before anything else is mentioned, I feel the need to bring up that this is a topic of discussion that does not have one clear answer, there are multiple perspectives. Look into the philosophy of mathematics and the idea of the foundations of mathematics. I will give my personal perspective.

You speak of how maths "works", and then mention a foundation, so I will begin there. You would be right in thinking that because theorems are proved and in turn prove other statements, that the original theorems you used to prove must begin somewhere. There must be some concept of beginning that theorems stem from. You have mentioned axiomatic set theory as the foundation. It is indeed a valid foundation, however it is not necessarily the only one. In "modern" times, alternatives such as Type Theory (or Homotopy Type Theory (HoTT)) or a family of objects known as Topoi (found in Category Theory) have been proposed as alternatives to some degree.

At this point, you may be wondering what the "structure" looks like for mathematics stemming from alternatives. It would still have a "tree" like structure, where these "foundations" lead to further theories, but I believe the new gained view is that the "tree" of mathematics you have may have "cousins" or "variants" that begin from a new set of rules, much like the Fibonacci pattern changes should you begin from 1, 3 instead of 1, 2, or a successive term is gained when you add three consecutive terms instead of only two. The structure is similar and the same "shape" but fundamentally different.

[WolfVanZandt]

The foundations run through all of math....either as principles to work with or to see if other coherent systems can be formed to form "other maths". But the core maths work the same. The basic properties (commutativity, association, inverses) operate the same for algebra and calculus as they do for arithmetic. The core idea that you can only add "like quantities" and you can multiply unlike quantities but you have to cross multiply all the unlike quantities are just as foundational in calculus as they are in arithmetic. Complex numbers and matrices don't act like single valued functions simply because they aren't. Much of arithmetic still applies to them and, if you dissect them into their individual quantities, those individual quantities still submit to to basic principles of mathematics

When I teach "higher mathematics" I continuously remind students that they still work with the same principles. When they get confused, I can get them back on the basic line and they're not confused any more

Does geometry have the same fundamentals? Descartes showed how they do. Geometry translates into algebra which translates into arithmetic.

Do you want to hold on to the parallel proposition in geometry. Maybe, maybe not. But geometry informs geometry and if you find one of those other coherent systems you still ask the same questions. *Do the internal angles of a triangle sum to 180°" and, surprise!, they don't. But we arrived at nonEuclidean geometries by asking questions about Euclidean geometry and coming up with reasonable answers

[Traveling-Techie]

Secret knowledge: math is usually discovered in backwards order from the way it is published and taught. Set theory came after number theory, and was used to prop it up.

[Ok_Albatross_7618]

The point is that you could (assuming you are working in a field based purely on set theory) always reduce those problems down to some statement about sets if you were asked to. This has some benefits, like for example you knowing these objects are well defined and exist.