Faces are attributes of polyhedra, so it doesn't make sense to ask how many faces a cone has.
Are there traditional scholars that use faces attached to cones? How do different countries deal with the subject?
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This isn't a bad question, it's a good opportunity to discuss how mathematical definitions are made and their relative merits.
In convex geometry, a face of a convex body $K$ is the intersection $K \cap H$ where $H$ is a supporting hyperplane of $K$. (See, for example, here.) So a cone has infinitely many faces -- the base disc, and also the infinitely many line segments joining the cone point to each of the points on the perimeter of the base disc.
I'm sure there are different definitions in other fields.
This answer explains why this question can't be answered (but it does not fit inside a comment section).
First of all: what is a cone?
According to this Wikipedia-page, a cone has a circular ground surface, which is in most mathematics handbooks pure nonsense:
Please clearly define what you mean by a cone.
Next: what is a face? This Wikipedia-page mentions different definitions of that word, but the geometrical definition is not present.
So, a face: is this a subset of a plane, where every two points can be connected by a line, never leaving the face? Or do you have a better definition?
Please, get your definitions straight and re-phrase the question.
AT the end of the day, it just depends on how you want to define things and becomes a bit of a semantics game. (We used to have a guy who was obsessing over squares versus rectangles. We also get the area of a circle versus area enclosed by a circle kvetchers.)
Personally, I think of a cone as a physical object, with a volume and mass. Probably made out of marble or concrete. So I would say it has two faces. The bottom flat face and the curved upper face. (If you think of it sitting nicely, pointing up.) And yes, it still has faces even if I'm considering the idealization (maybe a figure of one).
Also, I would not assume that some sort of conic section, mathy view of a cone is the only way to think about it. Look at a general engineering reference instead (e.g. Lindberg EIT reference manual...and it's got a base surface and a volume.)