The imaginary mathematical world
Even though mathematicians today no longer believe that there are actually two separate worlds- the natural world and the world of perfect forms- they still use this idea when doing math. Today they think of it as there being a real world (in which we all exist) and an imaginary mathematical world (where everything is shaped perfectly).
When today's mathematicians want to switch from talking about the real world to talking about the imaginary mathematical world, they will say that they are "describing the situation mathematically". This could be translated as "describing what the situation would be like in the imaginary math world". Mathematicians and scientists perform this switch from the real world to the mathematical world all the time, without even noticing, and with practice you will also become adept at it.
Here's an example of describing something in the real world and then describing it mathematically:
Consider two communities, Baker's Lake and Umingmaktok and the relatively straight path through the air that an airplane could take to get from one to the other.
A mathematician would describe this mathematically (describe what it would be like in the imaginary math world) by saying the following:
There are two points in space, one labelled Baker's Lake and one labelled Umingmaktok. They are joined by a line, labelled Airplane Flight Path.
The mathematician would make a diagram that looked like this:
In the next section, on algebra, we will return over and over again to this idea of a mathematical world populated with mathematical objects like lines and numbers. In the section on graphing we'll see how geometric shapes are matched up with other types of objects in the mathematical world to help us solve real world problems.