I was asked how I understand four dimensions. Or, for that matter, 196,833 dimensions. The question wasn't about how the novice can understand many dimensions; the question is how I, a experienced mathematician, do it.
Well, the first answer is that I don't. Not in the sense that I can imagine, and actually see, one and two and three dimensions. I can just look at a line or a surface or a box and see it. I can certainly imagine seeing it. In fact, I was imagining it as I typed that, and further, was looking at the two-dimensional computer screen just now, while typing, when I noticed a typo and corrected it. No. I don't see four dimensions, let alone 196,883 dimensions, like that.
But I can still make definite statements about four- and many-dimensional geometry and know they are right. And I can judge other mathematicians' claims about it, if they provide enough evidence for thier claims. How? By analogy. reasoning, axioms, and algebraic formulations.
I learned about four-dimensional geometry from a book by Manning called "The geometry of four dimensions". He starts by meticulously developing ordinary Euclidean geometry, using a set of axioms much more detailed than Euclid's. Euclid's were a monumental discovery, but they still leaned too much on intuition. They just weren't precise enough. It wan't until the late 1800's that a precise formulation of geometry was achieved.
Manning took a version of these axioms, and derived elementary geometry using them.
He started with the geometry of the line, postulating that there were two distinct points and (using his axioms) startes working with the unique line joining them.
Then he went on to geometry of the plane. He postulated that there were three points not all on one line, and derived the geometry of the pland joininng them. Connecting these with lines, and connecting all the points on these lines with more lines gave him a plane.
Then on to three-dimensional space. He postulated that there were a four points not all lying in the same plane. Connecting these with lines, and the points on those lines, giving planes, and the points on those places gave him a three-dimensional space, which (if I remember correctly) he called a hyperplane.
The step to four dimensions was now clear -- postulate five points not all within one same hyperplane. In principle, he could have gone on to five-dimensional geometry, and so on, but the subject of the book was four dimensions, so he stopped there.
The reasoning he perpetrates in each dimension was of the same style -- meticulously detailed text and symbols, with statements following from one another by the ordinary mechanisms of mathematical reasoning.
But there's another way to approach higher dimensions. Consider analytic geometry. This is a way of doing geometry with algebra. Points are represented by coordinates, which are numbers. In a plane, points are represented by a pair of numbers. A line is represented by an equation of a particular form in two variables. In three dimensinos, points are represented by a triple of numbers. A plane is represented by an equation of the same form, but in three variables, Again, the step to four dimensions is obvious. And taking it yields essentially the same geometry as Manning's.
Analogy sometimes helps understand these things. While purely formal manipulations of algebraic or logical symbols may suffice to prove results, I find analogy helps me to get a feeling for what's going on. For example, in two dimensions a wheel rotates about its centre, a point. In three, a ball rotates about an axis, a line. In four, a hyperball (let's call it that) rotates about an axis too, but now the axis is a plane. I can slice an ordinary three-dimensional ball by parallel planes perpendicular to its axis, and find that the rotating ball has become a stack of disks, each rotating around its centre, which is a point on the axis. All the centres of those disks centres form the axis that the ball rotates around. So I can further imagine slicing the hyperball into spheres, each rotating around its axis, and all these axes form the plane which is the hyperball's axis. Such intuitions provide the motivation for the more formal calculations and reasonings, which are necessary to validate the mental pictures.
Once we have made the step to coordinate geometry, we've made the step into a abstract thinking, where reasoning about n dimensions is just reasoning about an algebra with n variables. Algebra we can do. And so thinking about 196,883 dimensions is just like doing algebra in n variables, with n set to 196,833. And no, I don't even try to draw a picture of this.