Will there be a time when drawings stop being useful in mathematics?

[Eclair_de_XII]

TL;DR

When I am reviewing my calculus problems, especially those that require visualization, I find it immensely helpful to draw a rough sketch of what the problem is asking for. However, I wonder if I should not rely on sketches too often, because I fear that they will lose any usefulness in the higher math subjects. Is this a na\"ive fear or should I learn to stop relying on drawings as much as I do now?

I mean, as a simple example, vector analysis involves the study of higher-dimensional objects in Euclidean space, objects that are downright impossible to visualize using simply pencil and paper. When my professor taught it, he often explained the topics using analogies that tied into vector spaces that would be familiar to undergraduate students, such as 2-space or 3-space. Anyway, will there ever be a point in time when drawings and sketches, not necessarily hand-drawn, stop being useful in mathematics?

 

[FactChecker]

I think that visualization of the problem will often be useful for mathematics with a geometric application or interpretation.

 

[fresh_42]

Anyway, will there ever be a point in time when drawings and sketches, not necessarily hand-drawn, stop being useful in mathematics?

No.

And this answer has nothing to do with mathematics. It has merely to do with humans as a species. I have forgotten the correct figures, but I think to remember that at least 75% of us prefer visuality over any other sense like hearing, tasting, or smelling. Prefer means that it is their favorite way of recognition, and the first sense they analyze. This is also the reason why the by far major part of communication takes part in body language. A fact we shouldn't forget while communicating without this input.

 

[RPinPA]

However, I wonder if I should not rely on sketches too often, because I fear that they will lose any usefulness in the higher math subjects.

As a counterexample, when I took Real Analysis, I found drawings to usually be the key to working out a proof. Even though they were proofs of fairly abstract things generalized to n-dimensional spaces, a little doodling in 2-D was often enough to show me why the theorem was true. In the case of a theorem like "If A and B and C then D", using the sketch to come up with situations where D was not true, would generally show me why you need all of A, B and C as pre-conditions.

A drawing is not a proof, but a drawing is potentially a powerful compact shorthand for structuring your thinking.

 

[romsofia]

At the end of the day, it's YOUR life. When I took PDEs back in high school, my prof said something at the start of the course that stuck with me: "Some people think geometrically, and some think algebraically." And since she thought geometrically, she would teach the course that way.

I don't find graphs or sketches useful, and I do research in general relativity. A lot of my colleagues, however, like to sketch ideas when explaining them. Whatever leads you to greater understanding should be your goal! Don't be pressured by others to drop something that helps you.*

*This is advice for outside of a course, if you're currently in a course and your professor asks you to solve a problem a certain way, you better know that way.

 

[FactChecker]

Different subjects lend themselves to be most easily understood in different ways. And different people think best in different ways -- some geometrically, some symbolically, and others in other ways. And geniuses think in -- who am I fooling? -- I have no clue -- they seem to intuitively understand things that I will never understand.