In last days I have a hard time to understand what a definition is in mathematics. Until today I thought definition had a dual role in mathematics.
Dictionary role The first role is that it mereley serves as an abbrevation. E.g. we define the sum of $3$ numbers as "$3$um". So when we say find the "$3$um of $3,4,5$" the answer is "$3$um=$12$". It actually seems like the dictionary definitions.
Creating an object The second role is that it "creates" a new object. For example, we define matrix as "a rectangular array of numbers...etc" Wikipedia. What I thought (even in Linear algebra courses) is that we actually "created" a new object and gave it a name. We can manipulate now this object based on the axioms of mathematics and discover some properties about that object we call matrix.
But the last days I thougt "All maths should be deduced from axioms and from rules of inferences". The definitions are not important. So clearly the second role doesn't make sense. But which are the axioms then? You could say ZFC but I don't think that in Newton's era (also before and after that era) mathematicians were aware of ZFC. So if definitions serve only as "abbrevations" then first we should show that the objects we want to define actually exist in our system. This also isn't happening. No one proves that an object like a function or a matrice exists when we start to talk about them. They just give the definition and this is what it bothers me the most. Should we prove that the object we gonna define exists or we just define it? The second choice gives the idea of "object creation".
Also we define some operations "subjective". E.g. why matrix addition is not defined for $3$x$3$ and $4$x$4$ matrices? E.g. I can define multiplication over matrices of all sizes. Addition will give another matrice of same size with the biggest size (in the above example $4$x$4$) where the new matrice could be constructing by just adding the fourth row and column of $4$x$4$ matrice to the $3$x$3$ matrice. That means we can define matrix addition as we want. Again, I should be able to derive all the statements in my system using only the axioms and not definitions. But how I derive theorems about functions (matrices) when the axioms are about sets (ZFC)?
Which are the axioms? Why even in different topics e.g. geometry, probability, algebra we use different list of axioms even though one "helps" the other? E.g. in probability theory we can add, substract even multiply probabilities. But in list of axioms of probability theory I don't find Peano axioms. It feels also counterintuitive when you think that when you do mathematics you simply follow the axioms. What axioms Newton followed? What axioms we follow when we are doing calculus? Even when I took a calculus/linear algebra course (Chem student) our professor's didn't even state what axioms we will use. Just from definitions and the "intuitive" axioms e.g. $a+b=b+a$ etc we derived many theorems. I insist on that because this is what makes mathematics different from empirical sciences. Its all about the axioms. It maybe sounds silly that in a Maths course for chemists the professors should state the axioms but isn't that the "heart" of maths and any other axiomatic system? Even textbooks don't refer to the axioms. They define, define and again define.
I would like to know in what axioms we derive all these theorems and also what is the role of the definition in mathematics.