Do numbers exist?

"People always think that math is just a resource, like someone goes into a mine with a pickaxe and discovers math for us, when in reality it's the accumulation of brilliant thoughts, thought by brilliant people."
 -slightly paraphrasing my MTH 255 professor.

I've talked to this professor about his philosophy of math a bit during an appointment. I think any discussion about someone's philosophy of math boils down to the question "Do numbers exist?" and I liked numberphile's description of some roads one may take to answer that question. 

 For now, I'm a fictionalist. I don't see much reason to say math is true. Fictionalism is also nice because it reminds us that just because something works and is useful does not necessarily mean it's true. Many scientific theories that worked have been superseded by theories that work better. The best part about being a fictionalist is that it annoys mathematicians that would rather not call their work fiction.