What is the definition of vertical in math?

Answer 1

The definition of vertical is dependent of the scale. On a small scale, it means going straight up, at right angles to the [levelled] surface of the earth at that point. Alternatively, going up the page.

Because the earth is approximately spherical, the vertical direction at he equator is different from the vertical direction at the North Pole. On that scale, one could consider the line joining the South Pole to the North Pole as the vertical. But that definition fails when you consider the earth over its orbit. So you could define it in terms of the North and South poles of the sun. But the sun itself is not stationary and so on.