One can draw/imagine as many unique (curved/straight) lines as he/she wants in some specified finite area (assuming that each line is unique if it doesn't overlap with another line). Then how can the number of field lines in a particular area be a fixed quantity? This statement is contradicted by the fact that a particle will experience a magnetic force for each and every point in space. This would not be possible if at some specific points there are no magnetic field lines. The surface integral approach is clearer as some limits are taken into account and also there is no such thing as 'number of lines', but I find it very confusing when people say that the strength of the magnetic field is proportional to number of field lines/area. Why is this terminology still used? Is it because we assume that no magnetic field lines exist at places where the forces are very weak?
EDIT: Then why are there gaps between the iron filing lines? Is it because of my previous statement
because we assume that no magnetic field lines exist at places where the forces are very weak
And hence the iron filings align themselves to stronger field lines. Is this a reason why this terminology is still used?