Should we define something after we have prove that exists based on the axioms?

Question

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.

Answer 1

You are right in that definitions are just shorthands. They do not create objects. If I define a "fonum" as "any even prime number different from 2" there just is no "foonum", period. Definitions are adopted because (a) they talk about existing objects that (b) come up often enough to make the abbreviation useful, often also because (c) the defined objects have interesting/useful properties that we usefully conceptually associate to the definition.

Case in point: We define $NP$-complete problems as the hardest problems in $NP$ (essentially, problems that can be solved efficiently by guessing). Almost everybody believes $P \ne NP$ ($P$ is problems that can be solved efficiently without guesswork), so $NP$-complete problems would not be in $P$. But if $P = NP$ or not is one of the most famous open problems today. Note that $NP$-complete problems do exist, just that nobody knows if they are in $P$. If it turned out that $P = NP$, the definition would become moot, true. But the objects defined wouldn't disappear.

Answer 2

I think there are different points, where we need to go into.

First, you want to know, why the professor doesn't start with the axiom. The short answer is 1) it would take too long to derive all mathematics from the axioms and 2) they are not really useful in what you will do later on. As you said correctly: depending on what subject (even in pure mathematics) you are working on, you can use a different set of axioms. Why? Because they are easier to work with. In the end, you could deduce them from ZFC, but since we know that, everybody takes the comfort to work with a set of axioms which are easier to grasp. I want to come back to the first point. In applied sciences, the mathematics needed are often very advanced. In particular, that means that you would need a big chunk of mathematics to rigourously prove everything. But that is not practicable, because other things are more important. That is the sad side of having as much knowledge as humanity has already. Some say that Leibniz was the last human to know everything there was to know at his time, others even doubt that.

Now to your second point. You say that ZFC is all about sets, whereas functions and matrices are not. That is not true: functions can be defined through their graphs (which are sets). If you want to build matrices out of sets, you have to work a bit more. Or you see them as linear functions between some vector spaces. In this case, you "only" need to define what a vector space is and construct at least one. Or you want to define them via "tables". In this case, you need to define tuples from sets and then tables from tuples. You see that it becomes very tedious. In some way, you loose the idea behind what you are doing. That is often what you trade when becoming more rigorous.

Now to your question about summing matrices if different sizes. That is indeed possible to define such an addition. But is it useful? If it is not, why should one define it?

Finally, your question about what are the axioms. That is a very difficult question. I think that most of modern mathematics relies on ZFC (or at least ZF). However, as I said before, you may use a simplified, more adapted set of axioms, when you are working in a specific domain. One reason for this may also be that you don't need the full power of ZFC die your project. For example, abstract algebra doesn't need to know about the reals, when they work on abstract group theory. Or topologists. They don't kneed to know about vector spaces as long as they only need to prove things about abstract topology. Naturally, when you want to combine those different things, you might want to reconsider your choices of axioms to suit this new project. And in the end, one could reduce them to ZFC (except for some weird shit logicians do, but I would argue that does not concern you directly). There is no good choice for axioms. There is no valid definition for what an axiom mist be. So in the end, it is just consensus that ZFC suits everybody (most people) well enough.

If you are interested in the deeper questions about those domain dependent axioms, you might want to look into structuralism. This theory says (in very short) that mathematics are about structures. For example groups, rings, vector spaces, topological spaces etc. So, for everyone of those structures, you got axioms. And then, you mainly want to compare objects of the same structure, or see whether an object can instantiate multiple structures at the same time and what this implies.