What do you mean something other than mathematics? It's just a language problem nothing else. Just because the English language lacks the descriptive power it does not mean those objects are "mathematical", just as a round ball is not "English" in it's nature just because you can describe it's properties and behavior using that language. A quantum field is just that, a field, it exists just as much as a magnetic field exists and that you can observe directly. It is an area of space where a specific "force" you might say has impact on objects that enter that area of space. It's properties are measurable and they produce consequences in the world and it has a precise description using the language of mathematics