I think that mathematicians view the world in a different perspective than some other people in the sense that they can find ways to fit Nature (such a complicated concept) into summaries using numbers and graphs. In that way, artists and mathematicians are similar. They try to express nature in their own ways. The former by painting and other medians and the latter by graphs and numbers.
I think that mathematicians make rules for the things that they see but they can only see what our world shows them, they don’t see beyond that. This relates to the story about the 3 blind men and the elephant. So in that sense they are both discoverers and inventors. They discover elements of nature and invent formulas and other theories to capture that small section of Nature.
The “Allegory of the Cave” was interesting to learn about. I think its true that we see only shadows of things and only the things that the world shows us. In the cave, they saw what the person showed them and held towards the light. It’s similar to real life because we only see what the world shows us and what our world is like. (A little bit like the Brain in the Vat and the Truman show.)