Being close to mathematics, I see the following. There are natural parts of mathematics: positive integers, and basic operations: division (and thus multipliciation), addition (and thus subtraction). But more advanced ones are there, as well: for example, calculus, was 'discovered' for physics. However, the current mathematics includes mostly abstract parts: primes (even more: twin primes), graphs, topology, etc. While some (primes, for example) come from previously existing parts, it is not a discovery. In the nature, there is no such thing as a prime, thus we can not even discover it. On a side not: numerous theorems were later used in physics. Still, they are invented: we discovered their use later.
Math is a human construct. The only reason mathematics is admirably suited describing the physical world is that we invented it to do just that. It is a product of the human mind and we make mathematics up as we go along to suit our purposes. If the universe disappeared, there would be no mathematics in the same way that there would be no football, tennis, chess or any other set of rules with relational structures that we contrived. Mathematics is not discovered, it is invented. This is the way I see it anyway.
Math is a language of numbers. We use equations and numbers as symbols to make sense of naturally occurring phenomena. One example that I would like to bring up to address the confusion is the idea of proportions. When you measure the length of a vertical stick to the length of the shadow it creates; you can use that proportion to measure the shadow of almost any object and determine its height. While it might sound reasonable to say is that the rule the shadow proportions existed before humans, math isn't the length of the shadows to the objects, it is the symbols we use to explain the two things.
While common sciences, like biology, study a certain aspect of the world, like life, math is the study of math. It is extended logic used by humans after years of manufacturing it. Before math, people would use their fingers or stones to show numerical value. Math's invention just made the process of showing and calculating numerical value easier.
...Axioms are self-evident concepts. If something is self-evident, it need not be invented.
Here are other concepts that are axiomatic. (They require discovery, not invention):
Water is wet. Wind Blows. A day without sunshine is night. On the other hand, you have different fingers. I started out with nothing, and I still have most of it. (etc.)
In my opinion it's: Could it have been done differently, it was invented, and if not, it was discovered. America couldn't have been found somewhere else, no matter how hard someone would've tried. Thus, it was discovered. Cars can be built differently, with two wheels or with only one door (if we liked), thus, they're invented.
'Two' plus 'two' makes 'four', and even if we decided to change that, it will never equal 'five'. Therefore, math was discovered.
While its true that math is a language that allows us to predict our universe with more accuracy, things existed before they were discovered. 1+1=2. The '2' is a number and a way of expressing a value that is expressed differently in our brains but is technically the same thing. We could replace 2 and 3 and say 3 is 2 and 1+1 is 3, but no matter what we do it will remain the same. Things are added and subtracted in nature all the time. And we must define our parameters of what we mean by discovered before we ask this question. Did Pluto existed before it was "discovered" in 1930. Even if we change the value of units in math the same things will occur in nature.
My personal opinion on this is that the CONCEPT of math was invented by humans. However, the actual systems in place already were already there, so that would be a discovery. Humans have been shown to have an innate sense of math since they're infants, so this shows that math is a discovery. The universe cannot have worked without math, so we just discovered the concepts, equation, formulae, and whatnot and made it into something we can understand with more ease.
Math has always been around the language for math however has been invented. A cave man can have 14 rock in a pile he doesn't know that it is "14" per say but if someone were to take some he'd notice he had less. Therefore the actions of adding and subtracting did exist without the language. That is like someone finding a plant that plant already existed before a name was given to it by the discoverer but the name is what is made familiar to us. The language of math is what is made familiar to us.