Another answer is that every variable in a formula can be considered a “dimension.” A formula like y = 3x — 5 is two-dimensional, and you can plot it on a two-dimensional sheet of paper (it’s a line). A formula like z = 3y + 2x +7 is three dimensional (it describes a plane). A formula like w = 3z — 2y + 4x +2 is four dimensional. You can’t plot it in three dimensions. The best you can do is plot the planes corresponding to different values of x, y and z to show the behavior of the formula. Nevertheless, it’s perfectly possible to have such a formula describing a physical system.

A detailed weather map is at least four-dimensional. It will have geography (2 dimensions) and symbols at stations showing wind direction and speed (2 more) and cloud cover. It may also have contours showing barometric pressure. So that’s six dimensions. If you animate the map, time becomes a seventh dimension.

In relativity, you have the spatial dimensions x, y and z, plus ct (speed of light times time) and ct in many formulas plays the same role as x, y, or z. So time is not a yardstick dimension (I’d like some 3.8 nanosecond wing nuts, please) but mathematically it’s a dimension.