To the above comment, wouldn't you need the anti-matter equivalent of an apple in order for them to cancel out? Matter can't be obliterated unless you flung the apples at each other at speeds that accelerated electrons reach to smash each other. Sorry, I'm just trolling right there, but science itself is discovered, however, methods & processes are invented, i.e, you invented a way to smash the fuck out of apples to literally defy physics.
In response to if mathematics was invented or discovered it is all relative towards what you are defining when you say "mathematics", since many people look at the macroscopic view of math. "The abstract science of number, quantity, and space" is what mathematics is defined as in the dictionary, so in essence, as with any science, it is discovered. However, methods in which certain processes are simplified are invented. When you look at something as simple integration, it is an INVENTION. It never existed in any way or form other than adding a bunch of rectangles under a curve together. Sir Isaac Newton developed, and almost literally, invented procedural integration in order to prove a scientific theory, which was discovered.
To prevent any back lash, think of the Fibonacci sequence. The Fibonacci sequence is the sequence that goes 1,1,2,3,5,8,13,21,34,55, etc(I can't write the subscript summation equation for it on here, but you basically add the previous number to the current number, so 1+2=3, 3+2=5, and so on). MOST flowers have a quantitative factor the applies to the numbers in the Fibonacci sequence, such as the number of petals(hence the reason that four-leaf clovers are so rare and are only result of mutation). It provides access to calculating consecutive number ratios in the sequence in order to achieve a closer number to the golden mean. The Fibonacci sequence also exists in the amount of branches a tree grows per interval of time at the same time, as well as applying to the 137.5 degree start of the Fibonacci spiral in shell formation, flower formation, etc(Google Fibonacci spiral).
Now, sure as hell, the sequence that Leonardo Fibonacci thought up of was not something that did not exist in nature; it existed prior to his thought process of it, therefore he discovered it. However, future mathematicians developed methods in which the Fibonacci sequence can be reasoned and understood such as the process of adjacent squares in the Fibonacci spiral, or the summations and inventions that resulted from it. That is the product of inventions of process/innovative methods of attaining "x" when using "y".
That being said, it is a very abstract science, and science is, for the most part, pre-existing. However, the methods for using discoveries to create something and processing information to simplify or recreate a discovery is an invention in of itself.
Innovation is invention, new observation is discovery.
