How do mathematicians figure out these relationships between uhhh things...

You know nothing about mathematics. Your view of it is so ass backwards it is laughable.

Mathematicians don't just make up definitions hoping they will be useful. Mathematicians study concepts usually in the pursuit to solve a famous contemporary problem and when a certain concept become famous people try to formalize it properly and that is when all the definitions come.

Have you read the history of analysis? It was literally all just a bunch of geometric drawings in latin until someone thought "Hey, this shit is more useful than we thought. We better formalize it well."

Its so satisfactory to be non-English person and having to surpass intellect level of most of /sci.
And for you. Keep asking questions to yourself. Live alone. Isolate yourself from anything but math.

>implying mathematicians could work on undefined things

That is what mathematicians mostly did. No research starts with a new definitions. Definitions are inspired from intuition and things we "want".

I mean, analysis was one example but algebra too. We now sit comfy on the powerful definition of groups but back then when things like Lagrange's Theorem were proven, algebra was ways of permuting polynomials.

Or just read into the story of Stokes Theorem. God damn. Kid, I think you should stay in your intro to proofs 101 class for now. Don't try to pretend like you know more. I know that from your point of view it seems that mathematics is all about inventing definitions and then seeing consequences but that is just the aftermath, not the math.

>when things like Lagrange's Theorem were proven
how did he prove it without having anything defined?

>Lagrange did not prove Lagrange's theorem in its general form. He stated, in his article Réflexions sur la résolution algébrique des équations,[2] that if a polynomial in n variables has its variables permuted in all n! ways, the number of different polynomials that are obtained is always a factor of n!. (For example, if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y - z then we get a total of 3 different polynomials: x + y − z, x + z - y, and y + z − x. Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations that preserve the polynomial. (For the example of x + y − z, the subgroup H in S3 contains the identity and the transposition (xy).) So the size of H divides n!. With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.

>if a polynomial in n variables has its variables permuted in all n! ways, the number of different polynomials that are obtained is always a factor of n!
did he know what a polynomial was?

>did he know what a polynomial was?
If you ask me, probably not. At least not in the way polynomials are understood today given that back in his days, rings did not exist.

so he managed to show
>if a polynomial in n variables has its variables permuted in all n! ways, the number of different polynomials that are obtained is always a factor of n!
but if you asked him what a polynomial was you think he'd just shrug?

No, obviously he was working with polynomials as understood in his day, definition of which apparently at least included having n variables.

>With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.

Then later someone realized his work applied to other stuff they were working on.

>The "figure out" part happens once in a while when someone realizes that a math problem looks like a real-world problem and suddenly all these definitions and tools the mathematicians made up are useful.

Which is how I said it happens.

No, you are in a way implying that definitions come before discoveries, which is actually false. Discoveries come before definitions.

Even when we have formal definitions, most mathematicians still work with their intuitive notions and then only bother themselves with rigor at the time of writing. Terence Tao has written a lot about this.

Lagrange knew what a polynomial was, but it didn't really matter. He was doing something much bigger.

You're implying it makes sense to discover something about things you don't even have defined.

U=f(G,q)
I've just discovered how to unify general relativity and quantum field theory. I'll define U, f, G and q later. Do I get credit?

they're exploring it with a properly-formed and inquisitive mind, they also actually study Mathematics. Not just learn to use it's surface, they understand and explore the whole.

not that poster, but that actually is a lot what research is like. you often have collections of examples of things which have properties you like or find useful (often only semi-defined), or you have a general picture of a technique you might use to show something is true, and you have to work backwards to figure out the exact assumptions and definitions you have to invent to make the idea precise.

it's not that you are writing equations with totally undefined terms (though sometimes you are), you're not really writing equations at all for most of research. you can realize patterns like eigenvectors and eigenvalues geometrically (or vaguely algebraically) and intuitively before ever precisely saying what it means to be a linear operator.

it's part of why you see a lot of terminology and "orthography" clean up and disparity in the early stages of a theory (like how many homology theories there were before homology was axiomatized). people often, when sitting down to share their ideas, make a couple choices about how to formalize an informal idea, and that is when the equations actually enter the picture. it's why even when a bajillion new homology theories enter, or people finally abstract it to the right level, they still sort of are talking about a similar thing despite the precise methods being slightly different; because the idea isn't in the particular equations.

>Terence Tao has written a lot about this.
Can you give a link?

They saw something they thought had a connection and they wanted to prove it mathematically... so they did.

Eigenvector? Easy:
Linear matrices multiplied with a vector give you a new vector. rotating that vector differentiably, will result in a differentiable function. doing some functiond iscussion magic, you will find that the argument of the rotation and the angle of the result vector have to intersect at some point (i.e. euqal eachother). This means that for every matrix there must exist a direction for which no rotation happens. Eigenvalue can be deduced from that finding with some more playing around.

terrytao.wordpress.com/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/
terrytao.wordpress.com/career-advice/be-flexible/

>hurr durr I'm so smart
>All those grammar errors
Also you're being a total cunt for no reason all he did was ask a question, you knew literally nothing at one point too.

welcome to academia!

application and neccessity built upon previous use

hlaf the shit ou learn has no context as to why it was important, so youre memorizing sparce details of a greater machine.