This question is purely philosophical and as philosophical questions usually go, it is not well-defined. So, let us try to do that first. What does it mean for something to be invented? In the case of physical objects there is not really much confusion. The telephone is considered a human invention because it did not exist in any physical form before its invention and all its components are man-made. On the other hand, America was discovered by Columbus because it had existed for a long time before Columbus landed there.
In the case of Mathematics, we are in a spot because Mathematics is abstract. It need not have any physical manifestation to be valid. In fact, development of an axiomatic description of mathematical concepts (like counting and sets) is considered to be a milestone in the evolution of modern mathematics. One argument in favour of treating it as an invention is as follows: Mathematics is built upon self-obvious truths or axioms. Mathematics simply draws logical consequences of these axioms. Hence, all the mathematical tools that we have---numbers, logic---are purely a construct of the human mind. Further, discovery implies prior existence. In our case, the question is who could have thought of these concepts before humans did. (This question is not precise because we do not have a concrete definition of thinking beyond the human brain.)
To consider Mathematics a discovery calls for first proving prior
existence. Note that it is not important to know who the
(if there was one). Hence, we can skip getting into a religious terrain.
Now, despite the possibility of having totally arbitrary axioms, they
do have to be logically consistent and most of mathematics actually
makes use of axioms that
make sense. For example, Peano's axioms
that are the basis of our usual number system are very intuitive. What
intuitive and what is not is determined to a large part by our
experiences in our surroundings (call it nature). For such mathematical
concepts that are influenced by natural experience, it is perhaps
reasonable to call them discoveries.
- Mathematical Universe Hypothesis : our physical world is an abstract mathematical model.
- Is Mathematics independent from reality?