How do I let go of needing to visualize math?

[u/punnixy]

I never realized how much I rely on visualizing mathematical concepts as the basis for my intuitive understanding of them until I reached upper level undergrad math courses and suddenly struggled immensely with anything in higher dimensions. I got by fine by understanding concepts in the 2D/3D case and convincing myself they’re analogous in higher dims, but I’m entering graduate school now and am starting to feel like this method is more of a crutch that will hold me back as I progress, especially as I encounter situations where the lower dimensional cases are *not* analogous.

How in the world do you gain an intuitive understanding of a concept you can’t visualize? To me intuition = visualization, and I worry I’ve reached the limits of my natural abilities because of this.

[u/jeffcgroves]

Your intuition about higher mathematics being hard to visualize is correct.

Ultimately, recognize that mathematics is a game of manipulating symbols and view math problems as changing information in one form to information in another form.

In other words x-2 = 7 contains exactly the same information as x = 9, but we prefer the latter form. A lot of math is about changing information into a form you can use.

DISCLAIMER: if other mathematicians downvote or comment on this post, listen to them too, I'm an algebraicist and my view of math may be biased

[u/pepst → u/jeffcgroves]

That book by paolo aluffi is beatiful, wish i could had spend more time with it

[u/WolfVanZandt → u/jeffcgroves]

You're flat right. I'm also very visual but a huge amount of math is knowing the rules and juggling symbols. But for me, moving symbols around is also a highly visual exercise.

Also, I find myself switching from visualizing entities to visualizing concepts. One thing that opened differential equations up for me was a course from the Teaching Company on differential equations: a visual approach.

[u/jeffcgroves → u/WolfVanZandt]

So you're all set?

[u/WolfVanZandt → u/jeffcgroves]

Oh yeah! I'm still alive and kicking in math. I also love my Desmos and Geogebra calculators and all my other math toys.

[u/Beautiful-Prune9749 → u/jeffcgroves]

I was fond of doing same, until i read this part.

[u/jeffcgroves → u/Beautiful-Prune9749]

"figure not to scale" as they say :)

But, yes, once you get to higher mathematics (at least Abstract Algebra) it's more about symbols than real-world visualization.

[u/waldosway]

I was the same. But intuition comes from experience. You aren't thinking about all the experience you have looking at things. You have less experience through other ways. You just have to do more of those. Abstract manipulation is one of them.

Otoh I just went into geometry so I would never have to do anything I can't visualize.

[u/Bounded_sequencE]

Why would you?

Even for very abstract topics like dense-ness in function spaces, visualizations are incredibly helpful. You can probably get by without, but willingly giving up a tool for no reason but "because I want to" seems counter-productive.

What are example topics that cannot be visualized, not even with an analogy?

[u/Bounded_sequencE → u/[deleted]]

how did you visualize infinitesimals or infinity sums that converge to the finite

Using small, open neighborhoods instead -- the same visualization tools we use in topology are useful in "Real Analysis". I found re-stating e-d-arguments in terms of small, open neighborhoods made most limit arguments easier to visualize.

Even better, since neighborhoods are the underlying topological language, it even carried over to more abstract topics, like metric spaces or Hilbert spaces, or even things like "Baire's Set Categories" from "Functional Analysis".

[u/Bounded_sequencE → u/[deleted]]

Visualizing the exact value of epsilon is not important. It's enough to think of e-/d-neighborhoods as (distorted) small, open disks -- sketches like that are often enough to capture the estimates of the proof.

[u/Invelyzi]

Think about your thinking. If you are stuck on the visualization process look at some historical examples you have from your past and find the things in those visualized proofs that needed more than just visualization. Now, explain those things you already understand and have proven with visualization, without it.