Is there a formal system of expressing the context for mathematical proofs/problems Is there a formal way we can explain ideas in a mathematical problem/proof that are contextual and to the issue at hand?
For example if I have a mechanics problem I may want to be specific that the symbol $t$ is a variable, the symbol $m$ is a parameter that allows me to explore different 'contexts' and the symbol $u_0$ is a constant such that $u_0=10$?
If I have a proof that I want to show that for any real number it's additive inverse is gained by subtracting it from zero.
So for any real number $x$, in this case $x$ is a variable varying over all reals.
$0-x=-x$
Is there any formal system or name for these 'scenarios' that allow us to speak about them more concretely
For example we could see a 'physical formula' as a statement that can be true in each 'scenario', if we could formulate it formally. Formulas have an often limited context given and we just infer the relationship described will hold in many cases.
 A: Yes, but not only one. There are many ways and different ideas on this subject, of formal languages. This is the keyword. What you are referring to, describing problems and proofs in a formal way, is formal languages. Depending on the task at hand, "context" as you call it, you can choose such a language to formalize it.
I will give a simple example. Let's say we want a formal language for the addition of natural numbers. A language requires symbols, which are then combined together to form sentences, which are then evaluated as to whether they hold or not. Three steps all in all. Bear with me.
Our symbols are going to be: Variables ${x,y,z,...}$ and the symbols for addition $+$, and equality $=$.
We are going to combine these symbols as follows: All variables are words, and if $s_1,s_2$ are words, so is $s_1 + s_2$. Finally, if $w_1,w_2$ are words, $w_1=w_2$ is a sentence. For example, the expressions $(x+y)=z$ and $(q + w) + e = d$ are sentences, while $x=+a$ is not. Note that my use of parentheses just denotes the order in which the words are formed.
Now we need a mechanism dictating how a sentence is going to be evaluated to be true or false.
$\bullet W=W$ is true for all words W.
$\bullet$ If $A=B$ is true, then $B=A$ is true.
$\bullet$ If $A=B$ and $B=C$ are true, so is $A=C$.
$\bullet$ $(a+b)+c = a+(b+c)$ is true for all words $a,b,c$.
$\bullet$ $a+b=b+a$ is true for all words $a,b$.
$\bullet$ If $A=B$ is true and $A$ is a word occurring inside the true sentence $W=Q$, the sentence $(W=Q)[A\rightarrow B]$ is also true, where this symbolism denotes the substitution of $A$ for $B$ inside the sentence $W=Q$.
How do we know that our language behaves as we want it to? We can prove that actually, by assigning a natural number for every variable, and seeing which equalities hold regardless of the assignment we choose, we end up with the set of true sentences.
A: Yes. In fact, there are multiple such systems. The prevailing one is set theory, specifically Zermelo–Fraenkel set theory.
It's a very complicated topic, so I can't explain all of it here, but I can talk about some of the basics.
The symbols $\forall$ and $\exists$ are used to mean "for all" and "there exists". Your statement about the additive inverse is represented as $(\forall x)(0 - x = -x)$, and "there exists a number greater than 10" is $(\exists x)(x > 10)$.
Another important tool is the $\in$ symbol, which means "in" or "is a member of". If I wanted to state the solutions to $x^2 - 4 \le 0$, I could write that as $x \in [-2, 2]$, meaning that "x is a member of the interval [-2, 2]".
Intervals are the types of sets that appear most commonly in ordinary mathematics, but there are also special symbols for some specific sets, like $\mathbb{Z}$ for "the set of all integers", or $\mathbb{R}$ for "the set of all real numbers".
With the help of some other symbols like $\cup$ (union of sets) and $\subset$ (subset), as well as logical connectives like $\lnot$ (not), $\lor$ (or), and $\land$ (and), we can represent many different sets and statements about sets.
These tools make up the foundations of modern mathematics. I've only really gone over the tip of the iceberg here, so I highly recommend that you do additional research on this topic if you are interested.
