Do mathematicians visualise what 4D would look like when they are working with abstract 4D concepts?

Question

I was wondering if mathematicians do truly have a sense of visualising what the fourth dimension would look like in general, such as the fourth dimension as found in hypercubes. I know that we, as 3D beings, cannot literally perceive 4D objects just as 2D objects can have no literal image of a 3D object. However, I do know that we can give the 2D objects a sense of whatvisualising 3D objects would look like if we introduce some abstract concepts, such as stereographical graphing, to them. So, my question is whether mathematicians have a rough concept of the visualisation of their work when they work with abstract concepts - in geometry - that involve dealing with the fourth dimension? And, I was also wondering if it would be useful to attempt to have an intuitive sense of 4D objects and planes before truly working with concepts involving the fourth dimension, because I fear I may end up like a machine lacking any fundamental intuition of their work and the concepts of their work which would truly irritate me as I always try proving concepts, intuitively understanding them, playing with them and even developing them further into more abstract concepts before I actually work the concept in problems.