However, although we may use many abstract dimensions in our thought processes, most people cannot form a mental picture of anything with more than three physical dimensions. In other words, we may use many dimensions more than four to think, but we still cannot visualize higher dimensions, in the sense of mutually perpendicular directions.
You have limited yourself to understanding a higher dimension with only one of the five senses. You are assuming that a higher dimension involves a physical direction. An idea is not a physical direction.
I will have to disagree with you on this point. Math is a very useful tool, especially in dealing with dimensions beyond human comprehension. Certainly it's more interesting to try to visualize higher dimensions than to represent them with mathematical formulas, as true visualization would give us a more intuitive grasp of higher dimensions. But if we use math, we can at least put ourselves on the right track to understanding them, especially since we need math in order to simulate them.
I did not say that math was not a usefull tool. Just not in this instance. Again you go on the assumption that a higher dimension has physical direction and that it can be visualized.
I am saying we experience a higher dimension already. We just do not see it for what it is.