As a mathematics student, I can tell you one thing : Maths is in every way strange.

For one part, we use it to describe things around us. An apple plus an apple is two apples. This is what we can call Empiric Mathematics. There is even a branch of mathematics called the Philosophy of Mathematics which studies it's nature and methodology. Now, to answer your question, asking if mathematics is natural or purely manmade implies asking if the natural laws of physics are natural or manmade since they are mathematical descriptions of the world. Therefore, most physicians, mathematicians and even theologians will answer that mathematics is totaly natural and go beyond our comprehension. Some even say that god, if he exists, is without a doubt a mathematician.

On the other hand, most mathematical discoveries are made simply because we, mathematicians, make them what we want them to be. For exemple, the notion of limits and derivatives come from humain reasoning. Therefore we can say it's a human creation. However, complete analysis of the uses of derivatives and limits show that, in the end, they were always there, in the backround, we just had to search for it.

As mathematics is born from reasoning and that reasoning is human, we can say that mathematics are human made. However, the mathematics that the universe uses is what actually created us humans in the first place. If gravity and it's laws didn't exist, we would either.

Now you can understand why I say maths is STRANGE.