I think most people here would agree that Category Theory is part of mathematics.
The study of strongly-typed functional programming languages is really just the study of cartesian closed categories, so I think that this particular part of functional programming is legitimate mathematics. And Domain Theory is the study of the category of complete partial orders with bottom, so I would include that too.
I don't think I would extend that to untyped or dynamically-typed languages (LISP). Also, I'd probably pick a term other than "functional programming" since subfields of math are rarely named with gerunds ("strongly typed functional languages" is probably the most accurate, but a bit verbose).