IF the earth were flat, then a world map on an equally flat paper would involve no more mathematics than a simple ratio (e.g., one inch equals one thousand miles) - and every world map would look the same even if they were larger or smaller. If the earth were flat then a world map would be little more than a floor plan drawn to a reduced scale. The math required would be extremely simple.
But because the earth is round, flat world maps involve very complicated math, depending on the projection chosen. There are well over a hundred different choices of map projection - some appropriate for world maps and some appropriate only for maps no larger than a state or province because of the distortions.
One way of demonstrating that the earth is not flat is that the actual distances between distant cities, altho apparently measurable on flat maps, turn out to be very different from the actual distance covered by airplanes or ships making the actual trip.