Question: What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered? 

Is this even a serious philosophical question or just a meaningless/tautological linguistic ambiguity?

Answer: 
My elementary math lecturer likes to say

> 
> God created the number 0, and the [successor](http://en.wikipedia.org/wiki/Successor_function). The rest was invented by mankind.
> 
> 
> 

I think there is some truth in this quote, even if you don't believe in God. So to answer your question: I'd say that the very basis of math was discovered, but most of the sophisticated math was invented.
From a Neo-Intuitionist perspective, to the degree mathematics is invented, it is still discovered.

Did we invent, or discover the consonant 't'? We discovered that our mouths reasonably make that sound, across a wide swath of our species. But we decided that this was an important thing, and in so doing, we invented the idea of 't'. We invented a consonant by discovering a fact about ourselves.

From this perspective, mathematics is a set of ideas to which humans are naturally attracted in a given way. But those ideas themselves are a product of the human mind, the way the consonant 't' is a natural product of the human vocal apparatus. Those ideas arise out of individual humans, who can be considered to invent them. (Someone first uttered the sound of t. Someone first asked if -1 has a square root, or whether infinity comes in various sizes.)

But mathematics chooses out the ones that feel a given way and isolates those that appeal broadly to a given emotional reaction. In that sense it is a branch of psychology which discovers things about human thought.

It elaborates those ideas to a degree that makes it seem like it is creating things, but really, it is exploring our shared fund of ideas for ones that seem purely symbolic and not worthy of questioning, and sees how their consequences fit together.