We say “theoretical physics” and we say “pure mathematics”. Why is it correct to say theoretical physics but it’s not correct to say theoretical mathematics?
The answer is clear, although it might not be that clear for some people!
We have theoretical physics because we have experimental physics too! The first study things based on reasoning, modeling and theorizing while the later study physics through experiments. Theoretical physics and experimental physics have two different approaches but they focus on the same direction.
We don’t say theoretical mathematics because we don’t have experimental mathematics. We have pure mathematics and applied mathematics. Both have theoretical essence. Both work on topics through reasoning, modeling and theorizing. The difference is that pure mathematics focus on math’s subjects for the sake of mathematics in which, applied mathematics focus on applications of each math’s theory outside of math land.
I hope I could make it clear crystal.
You may also like to read these: