In the early stages of one’s mathematics education, the human capacity for geometric and spatial reasoning was paramount to the understanding mathematical concepts. Numbers were symbolized using colored blocks, fractions as slices of a circle, even in introductory calculus courses, the concepts of integration and differentiation are introduced with a geometric interpretation. However, as I have moved up in my mathematics education, I have noticed that the more advanced topics appear to benefit less from a keen sense of space and figure, which has caused great difficulty in some of my more advanced classes. Even in courses like applied numerical analysis, where the geometric application would be assumed to be fairly straightforwards, I find myself confounded when attempting to picture the mathematical activity in my mind’s eye.
Perhaps not so surprisingly to some, I have found that Locke has some interesting things to say on this topic even before the invention of infinitesimal calculus. See, I posit that this difficulty of geometric interpretation in higher math arises from the fact that they often involved quantities that are beyond those that the human mind can properly understand. That is not to say that we can’t reason our way about infinite sums and the like, but when we do, we aren’t considering the quantity itself, but rather the summation process.
As Locke himself says, “…both in addition and division, either of space or duration, when the idea under consideration become very big or very small,its precise bulk becomes very obscure and confused; and it is the number of repeated additions or divisions that alone remains clear and distinct” (Es. Conc. Hum. Undr. 143). This is quite the hot take, especially considering the modern mathematical landscape inhabited by both computation heavy super-computation and theoreticians that endure almost no calculation at all. This reveals quite a surprising rift between the methods of action of numerical methods (i.e. simulation, computation, etc) and more theoretical types of work that human mathematicians are apt to do.
But perhaps the biggest takeaway from this is that when humans work with mathematics, they do not simulate the system in question in the same way that a computer does, rather the use of heuristic devices and the like are used to generate information about the system in a general and more abstract form. With this in mind, it makes much more sense that higher level mathematics programs tend to put one’s mind off attempting to visualize the systems in question, as this is the type of thinking in which mathematicians most typically engage. However, I suspect that there is quite a great difference between arduously simulating an entire topology and the mere mental visualization of such a thing that is conducive to visual problem solving. I’m sure you will agree. Discarding the simple explanation of the ideas being two difficult to visualize in one’s head, such as when working in n-dimensional spaces and the like, what could cause this spatial disconnect in problems that otherwise be rendered in one’s mind’s eye?
Perhaps some of the difficulties with mathematical visual thinking can be illuminated by the associated Stanford Plate entry (https://plato.stanford.edu/archives/spr2020/entries/epistemology-visual-thinking), which mentions an interesting fact about visual proof, which is that they can often over-generalize or under-generalize a proof when used. Since here we aren’t talking strictly about the power that visual thinking has in proof, I will mainly disregard the portions that directly correlate to that topic, but I find that many of the reasons why visual thinking might lead to error in proof may lead explain its difficulty of use in some situations. The reasons why visual proofs often over or under-generalize is that there has to be a perfect balance between minimalist elegance and comprehensive details so that the image formed is perfectly representative of the mathematical concept. If there is excess detail, the picture under-generalizes, if not enough, over. I conjecture that there are similar reasons why visual thinking can be such a task in higher mathematics, where extremely precise definitions, axioms and theorems are the substratum from which our visual notions emerge. These mathematical entities can become quite complicated as one advances up to more complicated mathematics, making it a much more daunting task for one’s mind to determine what is immediately relevant to the task at hand from all of the other extraneous information included in the network of definitions and theorems. This is quite different from the type of geometric interpretation of lower mathematics, which is mainly concerned with the geometric relations of quantity, such as length, volume or area.
With these considerations in mind, perhaps it is not that geometric interpretation is to be wholly disregarded from higher mathematics, when appropriate, but rather than the mind is not as naturally conversant in the language of theorems and definitions as it is with the modes of quantity. This is a message I have learned the hard way in my undergraduate education. Given that visual thinking can occur with any mathematical entity as its content, not just numbers, I find that there should be no inherent property of higher mathematics that should bar it from the use of visual techniques. This is, of course, excluding those ideas that are not suited to the three or two dimension visualizations that the human mind can produce.