If I multiply two numbers, the result is a number. If the two numbers represent speed and time, the new number represents distance. If the two numbers represent the width and height of a box, the new number represents the area of the box. Are the numbers that represent distance and area different kinds of numbers? Why or why not?

A mathematical expression, 3 times 4, produces a mathematical object, in this case, the number 12.

However, if the numbers 3 and 4 represent width and height in the physical world, then the 12 has a meaning in the real world that is different than if the numbers 3 and 4 represented speed and time.

The realm of mathematics is abstract. The realm of the real world is not.