When one does not have well-defined research interests for PhD Programs
In mathematics it is common to have little specific idea of what you want to do in a PhD at the point of applying. Unlike in many arts subject, for example, your supervisor will largely choose the research question for you (hopefully with your input). At a graduate open day, we were told that SOPs range from 'I want to do a PhD in pure maths' to 'I want to prove the Riemann Hypothesis'. Indeed, my own loosely said 'this is what I've done, so something a bit like that might be good'. Exactly how much detail you need to give will depend on the structure of the PhD programme, and local rules/funding arrangements.

I think the thing you will need to set out clearly is why you've chosen to switch to mathematics, and how you will achieve the same standard as those who have done a maths undergrad/major.