G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be beautiful, but unlike music, visual art or literature, much of the beauty, particularly at a higher level, is only available to the initiated. And while many great scientific advances have been built on mathematical discoveries, many mathematicians make no pretense of caring about any practical use their work may have. So when the taxpayer or a private organization provides grants for mathematical research, what are they expecting to get in return?
I'm asking not just so I can write more honest research proposals, or in case it comes up in an argument, but so I have an answer for myself.
Best Answer
I will suggest a "proof by contradiction" here.
First, let us make a very quick (and poor) statistical analysis on the amount spent on Mathematics research. Let us say that the research agencies spend 2% of money on mathematics*. Excluding the other basic sciences, this means that more than 90% of research is being spent on "helping deactivating a gene that kills hundreds of people" (to quote some comment above). Of these 2%, a small portion produce more applied results, that maybe will be used in high-standard technology in a soon future. The other portion provides comprehension of a very specific field, helping bulding an environment for eventual disruptive breakthroughs ("on the shoulder of giants", right?).
Hence, as an investor, I would have no doubt in putting 2% of my money in mathematics. In return, I get some quick and applicable results that may lead to patents or products - (e.g. statistical models, industrial operations research models, etc). Eventually I get big breakthroughs with various applications, which come with the bonus of helping improving human comprehension of the world's very basic and profound questions (e.g. Complexity Theory, Population Dynamics, Information Theory).
Of course I could take the 2% of pure research and redirect it to more applied research. But, is it worth it? Is it worth NOT funding mathematics and loosing all the potential advances of science? And here is, as promised, the "proof by contradiction". NOT funding mathematics is dangerous, and may block very profound advances. Ergo, agencies should give money to research in mathematics.
Furthermore, although I am an applied mathematician, I completely disagree that you would get more "useful" results in redirecting all the efforts to the problems that come from real world applications (medicine, engineer, etc..). This can give you quicker results. This probably can help you downloading a video 1.3x faster. This probably can help a doctor with a better MRI equipment. Nevertheless, the real breakthroughs occur when one investigates more profound questions.
*Here in Brazil 2% is a very good approximation for the percentage of math scholarships, in comparison to scholarships granted to all areas. If anyone has more accurate information on the US agencies (for example, this NSF link is useful), I would be happy to know.