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I'm really not good at math, and I'm trying to re-learn on Khan Academy. As I'm progressing, this question came up for me.

Learning math concepts that I can visualize in my mind or relate to some observable phenomenon in life are easy. For example, 2+2=4 is easy for me to understand. I can "see" in my mind two marbles in one cup, and then two more marbles being placed in that cup, and then seeing there are four marbles in the cup. I can even do this in real life, and not only imagine them, but literally see them, and I could even feel them with my fingers.

Using concrete, physical materials to teach abstract math concepts is the foundation of several early childhood instructional approaches, especially Montessori's. Also, if I understand this blog post, in the first paragraph Terence Tao states that in history, numbers were not always used in some forms of math, but rather shapes.

Yet, there always comes a point where I — and many other students I know — can no longer visualize, or give concrete representations of, the math we are learning. This is when math becomes difficult for me to understand. Complex statistical equations or engineering calculations baffle me. Without observable phenomenon, or even metaphors, to relate to the complex, abstract math, I feel lost. The only way forward that I have found is to exercise total faith in the theorems upon which my current math learning is being built. This ever-growing foundation upon which the more abstract math is built has to be a sure one, because I can no longer rely on my imagination or intuition to guide me. Only strong theorems can serve as the next layer of foundation to take me up further and further into the heights of more abstract math, more theorems. My questions are:

1. Is it actually possible to visualize, or physically represent, any mathematical concept, no matter how abstract, and I'm just weak-minded?

2. Are there mathematicians, perhaps certain geniuses, who can visualize in their minds extremely abstract mathematical concepts? Or, at some point, do we all have to simply trust the foundational theorems as we progress, leaving visualization behind?

JBR
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The questions are a bit too broad and general to give a very adequate answer in a reasonable amount of time, so I will just provide a general answer to give you a starting point to think about and maybe help you refine your questions.

Firstly, what do you really mean when you say "visualize an abstract concept"? What constitutes a visualization depends very much on your goals and where you're starting from. For instance, take the definition of the derivative as an example:

$$f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$

We can visualize this by drawing tangent lines against the function, to understand the connection between the derivative and slope. However, this doesn't answer all of the questions regarding its foundations. Clearly, the definition references limits, so the visualization can be extended using shrinking bounding boxes for the limit behavior. But all of this inherently implies that the limit is well-defined, and inherently implies the existence of real numbers, which inherently implies some axiomatic system. How far down the rabbit hole do you expect to be able to visualize this foundation? Hopefully, this example gives you context about two things in particular: where your expectations are for being able to visualize something, and that there do in fact exist visualizations for anything mathematical, depending on your definitions. Sometimes, there is a little bit of faith involved, and most of the time that's okay as long as the math works. There's too much math out there to be able to expect any individual to know all of the inner workings.

Secondly, just because you cannot visualize the concepts doesn't mean that you are necessarily weak at said visualizations. Being able to visualize a concept often times requires being able to take very complex structures and understanding how to map them onto some kind of simpler structure, and that process itself is sometimes a difficult math problem. We stand on the shoulders of giants, and so perhaps you just didn't find the right resource or the right teaching method that clicks for you. I think patience and perseverance, in addition to trying multiple methods, will eventually lead you to where you want to be going. John von Neumann once said "In mathematics you don't understand things. You just get used to them." While I have my own disagreements regarding this quote, there is a nugget of indispensable truth that promotes a healthy relationship with the level of math you describe when interpreted properly.

zhuli
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  • Zhuli, this is helpful, thank you! If you don't mind, what are your disagreements with the Neumann quote? – JBR Mar 24 '22 at 21:51
  • @JBR, you may want to read the extensive discussion at What are some interpretations of Von Neumann's quote? – Mark S. Mar 25 '22 at 09:18
  • @MarkS. Thank you! – JBR Mar 25 '22 at 17:31
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    @MarkS. Following that link lead me to a couple of other extremely helpful posts - I found them relevant to my current learning journey

    https://math.stackexchange.com/questions/617625/on-familiarity-or-how-to-avoid-going-down-the-math-rabbit-hole?noredirect=1&lq=1

    https://math.stackexchange.com/questions/913416/self-learning-mathematics-help-needed?noredirect=1&lq=1

     – JBR Mar 25 '22 at 20:14
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perhaps you may be interested in looking into Category theory. In there, we sometimes can make progressively more and more abstract statements out of diagrams. You will see for example vertices of 3d cubes as objects and arrows representing functions between them. It becomes more and more abstract, yet a lot of things are able to be presented with a visual argument. This is the field most likely to match your criteria.

Snared
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