It is. We invented the arabic numerals because they were easy to draw and we could written any numbers with them. Just like we invented higher lever computer languages instead of using assembly.
But if you're unlucky, you pick a set of axioms that makes the whole system inconsistent, which means that you can prove anything, which means that the whole system is useless. To say that it is purely arbitrary is in a sense right, but it seems to undermine the care that you have to go through in order to be reasonably sure that the system is not set up to fail.
If I have three objects and you give me two more then I'll always have five objects. You can call it cinco or 五 but there are still five of them.
Likewise, you'll always be able to determine the length of the hypotenuse of a right triangle by its two legs. No matter what system you set up, if you're cutting three boards to build a triangle the length of the big one is absolutely defined by the length of the other two, assuming Euclidian geometry. The symbols (a, b, =, c, +, superscript 2) are all totally arbitrary but if you're working with three boards there is absolutely an intrinsic correctness to the Pythagorean theorem.
There are branches of math which are just exploring the internal consistency of the system we've set up but much of physics is spent describing the real world and applying our math symbols to the universe. The amazingly cool thing is that our system is so good that we can use our abstract symbols to make predictions about physical laws and they actually come out to be true! I would argue that all of physics is basically "math that's a fundamental truth of the universe".
The fact that you count something as an object or see the world in discreet terms is also similarly arbitrary. You could see "objects" as something that's more interconnected and thus would count them differently. If we were far smaller and "looking" at things on an atomic level, putting that grouping together would not be quite as likely. And even still, you're focusing on the "discreet" positive space versus the negative space.
There are many ways to view the world that also would create its own system of abstraction and eventually "come out to be true." You're just used to one particular variety and it's all you know, so you call it the truth.
> No matter what system you set up, if you're cutting three boards to build a triangle the length of the big one is absolutely defined by the length of the other two, assuming Euclidian geometry. The symbols (a, b, =, c, +, superscript 2) are all totally arbitrary but if you're working with three boards there is absolutely an intrinsic correctness to the Pythagorean theorem.
This is only true because we are assuming Euclidean geometry, and as I had said earlier, this implies that we can logically conclude these facts because mathematics is logical and internally consistent given these axioms, so we cannot say that they are intrinsically true; that would require us to know without a doubt that the axioms are correct. Since we can use Gödel's incompleteness theorem to show that our axioms are necessarily assumed, we can say that we do not know if they are correct or not, only that assuming they are, we can make a lot of really good predictions about the world around us.
> The amazingly cool thing is that our system is so good that we can use our abstract symbols to make predictions about physical laws and they actually come out to be true!
This is definitely amazing, however it does not mean that our system is necessarily correct, in fact it is demonstrably lacking in Quantum mechanics for example, where we need to renormalize infinities, which makes almost no mathematical sense whatsoever, but we do it because our experiments tell us that if we do, we can make predictions about how things work.
> I would argue that all of physics is basically "math that's a fundamental truth of the universe".
I agree with the spirit of what you are saying here, but think I would phrase it in the following way:
I would argue that physics is basically fundamental truths about the universe that we can describe to the best of our ability using an abstract framework such as math. The fact that we can do that does not mean mathematics consists of these fundamental truths.
EDIT: Modified last paragraph to be slightly clearer.
> It is [arbitrary]. We invented the arabic numerals because...
Then it's not arbitrary; chosen at random or on a fleeting whim, without reference to a reason or system. It was invented to fill a specific need based on certain limitations.
I doubt many people would agree that you were doing maths if you made your rules and symbols up randomly.
It's true enough that mathematicians define certain things certain ways, but they generally have reasons for doing so that tie into other aspects of whatever system they're working within at the time, or with particular areas of investigation: 'If I alter this rule, or make this assumption, what does it do to the system as a whole? Does it let me find some answer more easily than another way? Does it preserve consistency/truth values? Under what conditions?'
That's far from being dependent solely on their individual whim, the decisions they make in that regard, and the answers they will get, are strongly influenced by the form the system has taken and it's uses and limitations.
Of course if you want to maintain that maths as a whole is arbitrary because you could make whatever you liked up and say you were doing maths... well, I won't argue you're not, but it seems to me you've made the objection general enough that it could safely be ignored. Anyone doing something purposeful could simply assert: 'Your's, maybe. We're trying to do our-maths-goal.' And move on.
