Comment by dsacco | Hacker News Reader

Comment by dsacco | Hacker News Reader

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You want to be careful with that. Pictures can be helpful in mathematical education (especially in something like a first course in real analysis), but if you get accustomed to relying on learning with pictures your mathematical intuition can break down once you arrive at a place where pictures don’t make sense.

Once you’re working in k-dimensions, where k > 3 (or maybe 4), you can’t rely on pictures anymore because there is no satisfying depiction of 4 or greater dimensions on a 2 dimensional medium. Complex numbers can be learned this way because they’re (simplified) the set R^2. Number systems in general are sort of okay to learn this way because they typically just extend other number systems, so you can successively build intuition on top of previous abstractions.

But you won’t really “learn” a lot of much more complicated topological or geometric (and therein analytic) concepts if you can’t build intuition without “seeing” it. As a very simple example, contrast these two approaches to defining a trivial concept in topology, which you’d probably come across before complex numbers in an analysis course:

1. A neighborhood of a point p in a Euclidean space is the set of all q such that d(p, q) < r, where r is some radius greater than 0.

2. A neighborhood is a circle (or sphere) with a radius of r surrounding a point p.

One of those feels more immediately intuitive, but that’s only because we’re dealing with the simple case of k = 2 or 3. You can’t “dodge” the complexity by visualizing it once you get into higher k, you have to just formalize it to develop intuition rigorously. Definition 1 is useful because it abstracts to conceptual domains we cannot practically visualize. Definition 2 would be useful if you could build a visual intuition of a 3-sphere, or 4-sphere, and so on. But building a visual intuition of k > 3 dimensions just shifts the problem away from what you’re trying to learn to something that is just as difficult: eventually you have to just get comfortable staring at a page of frustrating numbers, unfortunately.

> 1. A neighborhood of a point p in a Euclidean space is the set of all q such that d(p, q) < r, where r is some radius greater than 0.

Isn't that what mathematicians call a ball rather than a neighbourhood? A name which rather suggests that they are visualising it.

Not really, no. It’s a neighborhood. A ball is one description of a neighborhood in a specific context. I hesitate to link to Wikipedia, but: https://en.m.wikipedia.org/wiki/Neighbourhood_(mathematics)

It would not be correct to use terminology like “ball” to describe a “neighborhood”, even if, in specific instances, they are the same representation in different contexts (i.e. geometric vs analytic).

I am not using 'ball' to describe a neighbourhood, you are using 'neighbourhood' to describe a ball! From your own link:

In a metric space M=(X,d), a set V is a neighbourhood of a point p if there exists an open ball with centre p and radius r>0, such that B_r (p)=B(p;r)={x∈X | d(x,p)<r} is contained in V. (My emphasis.)

Balls are definable in metric spaces, neighborhoods in general topological ones, so "neighborhood" is indeed the more general term.

That said, once we have a metric space, the metric balls give a natural basis for the topology, which means that every theorem about neighborhoods has a corresponding one that uses balls instead. So in practice, when in a metric space we just use balls everywhere because they let us exploit their extra structure if it's needed. This is why you see the abuse of language calling neighborhoods balls, despite that being technically incorrect.

A sphere is the surface of a ball. Your criticism applies to (2) but not to (1).

However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to emphasize that one should not visualize it.

> However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to emphasize that one should not visualize it.

You’re right, but in fairness I’ve seen multiple analysis textbook authors use terms like “circle” and “ball” to describe a neighborhood in topology. This is usually in the fashion, “...(in other words, E is a ball.”

This is precisely the sort of thing I dislike, because aside from being technically incorrect like you say, it can (for that section) encourage a student to think, “Oh, why didn’t he just say that?” and skip the heavier definition -> theorem -> proof sequence preceding it, to the detriment of understanding later material.

That's actually untrue. 3 Blue 1 Brown has an excellent example of how you can use visuals to aid intuition in higher dimensional thinking. It's just not the naive representation.

Yes, I think this advice isn't quite right.

A pictorial formalism is possible for most if not all mathematical concepts as much as a symbolic.

It just won't be naive and may not have been systematically developed and communicated / taught anywhere.

The issue here is mostly just an awareness that, naively, the pictures change fast than the formula. (r^1, r^2, r^3... look more a like than lines, circles and spheres).

I agree with what you and the parent are saying here, and I suppose I didn’t well enough communicate my meaning. To clarify, I am focusing on the naive representations, which I find people rely on too much. It is easy to learn lower dimensions naively and feel like they make sense, but you won’t bridge that understanding to higher dimensions as seamlessly as going from a point to a line to a circle to a sphere, etc.

And for the pictorial representations which do exist, implicit in understanding them is the symbolic information. Conversely, you can understand common naive representations without an implicit understanding of the formalism. For example, we all know what a circle is before we know about irrational numbers, but it’s silly to try and understand a 5-cell or a tesseract without a broader understanding of k-dimensional space. It’s precisely that naive intuition that doesn’t generalize well.

In other words, my point more simply was caution: use pictures, but don’t replace symbolic formalism with them, just use them augment your understanding once you’ve at least sort of got it.

Yeah, good point. I like 3Blue1Brown’s videos but in fairness I would say that video is quite a bit closer to the formal side of things than the visual side. I was talking about naive representations - if you build up to the compound visualization 3B1B walks through you’ve covered a lot of the rigor implicitly.

It's okay if you also develop the skill of constructing a duplicate mental picture from a sufficiently detailed description of it.

Build up your picture-mirroring skill on 2-dimensional and 3-dimensional pictures, and you can eventually imagine pictures with more dimensions. Maybe you could even manage an animation in 4-dimensions. About the only thing I can reliably imagine in 6 or more dimensions is a hypersphere, and even that might be incorrect if one of the dimensions is not mutually perpendicular to all the others or if its basis vector doesn't square to itself.

Would you say there is a use for imagery, or should it never be applied because there is a risk of someone, somewhere, becoming reliant on it?

I would say there is absolutely a use for imagery - I wouldn’t advise learning real or complex analysis of without visual aids :). I just think they are easily abused and used to mask gaps in formal knowledge.

You have to start somewhere. Once idea is visualized and internalized on lower dimensions, you move to the next level, where images not needed; however to get to that step 0, where you understand why complex numbers used in AC: lots of folks never bother to explain that it's just the best way to describe rotation.

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