The perceived beauty and interconnectedness of mathematics often leads many to conclude that math is wholly apart from man, and that advances in mathematics are in fact discoveries rather than creations. However, mathematics is simply one of many logic structures used to explain the world around us (i.e. science), meaning its creation is the result of human cooperation and world building (refer to Peter Beger's Sacred Canopy). Let us take a quick thought experiment to evaluate this statement: what is mathematics without humans studying it? Does math still exist then? A quick parallel to help us understand this concept is the difference between science and nature. Barring philosophical questions, we know that nature would exist without humans, but science would not. This is because science is our understanding of reality, it is not reality itself. Thus goes the saying, "Newton's Laws never moved a billiard ball."
Examining a brief history of mathematics we can see that mathematics has always come about as an explanation of the world we see. Imagine the earliest forms of algebra: I have 3 apples, and after picking 2 more, I have 5 now. The question would be: why do I have 5 now? Why not 4 or 6 or something else? The answer we know now is that this a physical manifestation of addition, by adding 2 I must increase my original number by 2. Thus we have addition as an explanation of physical events. Next we can examine geometry: how come the distance across this round stone seems to be proportional to the distance around it? I measure this stone and I can find this number that relates the two, which I call pi. Again I can see this physical phenomena in nature and create mathematics to try to understand it. Astronomy and the rest of the sciences all logically fall in suit of this idea, with people wanting explanations to why certain "stars" were at certain locations during the year, etc. In fact, in the case of sciences, the mathematics and the explanation of the phenomena are heavily interwoven to the point where they are interchangeable and complementary.
Since our conclusion is that mathematics does not exist without those studying it and therefore man made, we must lastly address the reason we do not "discover" mathematics. Let us bring up another question: does the mathematics exist before it is known? Since we've concluded that mathematics is an explanation, like science, let us draw another comparison between the two. Quantum mechanics, the funky physics of the small, was obviously not always known in the past and was formulated by scientists in the early 20th century. However, the reality of how these particles behaved was always true in nature, regardless of our understanding of it. Essentially what this means is that nature is right regardless if science is. With the case of mathematics, there is no equivalent world where mathematics exists regardless of humans. Any math we "see" is superimposed on the world; it is our metric to quantify reality. We yearn to explain that which is around us, the goal of academia. Meanwhile, reality simply is. Mathematics is mental; it is a framework we use to understand, and like science, it is not reality itself.
I have another discussion on this topic that responds to a few of the criticisms of this argument, many of which were my own (I am not personally convicted either way but I want to facilitate conversation on this topic) I take a slightly modified position, of which I posted here.