I can't talk too much about biostatistics since I'm not a biostatistician, but I do know something about the relationship between physics, computer science, and mathematics and the dialog goes in both directions. In order to do anything with computational physics you have to develop some new math.Well, of course there is an obvious difference between fields like algebraic topology and fields like biostatistics. Someone in algebraic topology is seeking to answer questions of mathematics. Someone in biostatistics is seeking to answer questions of biology, and using mathematics to do it.
And I think that's a problem. The trouble is that (especially with undergraduates and high school students) the question they are asking and the assumptions that are making aren't the ones that the *should* be asking and making.When someone asks about a mathematics PhD, he almost surely is not thinking of biostatistics.
If the OP knew about the Harvard programs in biostatistics and applied mathematics, and didn't want to apply, that's fine. However, my guess is that the OP didn't know that those programs existed. One problem with physics and mathematics is that people have a highly distorted idea of what physicists and mathematicians do because they have stereotypes that involved zooming in on one aspect of mathematics. Getting rid of those stereotypes means questioning those assumptions.
Sometimes you have figure out what the real question is. I can't say for certain who the original poster is, but the question that he is asking is pretty typical of people in high school, and usually when that question is asked, the asker is really asking something else like "what should I do with my life?" "do I have any hope of being a mathematician?"All this other stuff is just not relevant to the question, regardless of your philosophical views on what is mathematics.