The fact that 1 plus 1 equals 2, or that there's an infinite number of primes, are truths about reality that held even before mathematicians knew about them. As such, they're discoveries but they were made using techniques invented by mathematicians. For example, according to Pythagora's theorem, the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. This is true for all right-angled triangles on a level surface, so it's a discovery.
Showing it is true, however, requires the invention of a proof. And over the centuries, mathematicians have devised hundreds of different techniques capable of proving the theorem. In short, maths is both invented and discovered. source : (BBC Science Magazine)
