[tsimionescu]

> I see this a lot with math concepts as they begin to get more abstract: strange visualizations to try to build intuition. I think this is ultimately a dead-end approach which misleads rather than enlightens.

Isn't this how people arrived at most of these concepts historically, how the intuition arose that these are meaningful concepts at all?

For example, the notion of a continuous function arose from a desire to explicitly classify functions whose graph "looks smooth and unbroken". People started with the visual representation, and then started to build a formalism that explains it. Once they found a formalism that was satisfying for regular cases, they could now apply it to cases where the visual intuition fails, such as functions on infinite-dimensional spaces. But the concept of a continuous function remains tied to the visual idea, fundamentally that's where it comes from.

Similalrly with vectors, you have to first develop an intuition of the visual representation of what vector operations mean in a simple to understand vector space like Newtonian two-dimensional or three-dimensional space. Only after you build this clean and visual intuition can you really start understanding the formalization of vectors, and then start extending the same concepts to spaces that are much harder or impossible to visualize. But that doesn't mean that vector addition is an arbitrary operation labeled + - vector addition is a meaningful concept for spatial vectors, one that you can formally extend to other operations if they follow certain rules while retaining many properties of the two-dimensional case.

[chongli]

the notion of a continuous function arose from a desire to explicitly classify functions whose graph "looks smooth and unbroken"

This notion falls down when you get to topology where you have continuous functions on topological spaces (which need not have any concept of distance nor even "smoothness"), since a topology can be defined on a finite (or infinite) set (of objects which may not even be numbers).

But that doesn't mean that vector addition is an arbitrary operation labeled + - vector addition is a meaningful concept for spatial vectors, one that you can formally extend to other operations if they follow certain rules while retaining many properties of the two-dimensional case

Vector addition need not even look like addition. For example, the positive real numbers can be defined as a vector space over the real numbers, with:

* (+) vector addition: u + v = uv (adding two vectors by multiplying two positive real numbers)

* (.) scalar multiplication: av = v^a

* (0) zero vector: 1 (the identity for multiplication)

Now we can verify some of the vector space axioms:

* let a be a scalar and u, v be vectors, then: a(u + v) = a(uv) = (uv)^a = (u^a)(v^a) = au + av, thus distributivity of scalar multiplication over vector addition holds

* let a, b be scalars and v be a vector, then: (a + b)v = v^(a + b) = (v^a)(v^b) = av + bv, thus distributivity of scalar multiplication over scalar addition holds

The rest can also be similarly verified but you get the picture.

Another weird vector space is the set of spanning subgraphs of a finite, simple, undirected graph over the finite field F[2] (which yields only 0 and 1 as scalars). In this one the idea of vector addition between subgraphs G + H is about determining whether two vertices are adjacent in one or the other of G or H, or adjacent in neither or both. This isn't really like addition at all, so none of the intuitions you might develop for vector addition in a Euclidean two-dimensional space would apply at all.

[tsimionescu]

None of these examples contradict what I've said. The historical fact is that continuous functions and vector addition were discovered as interesting mathematical properties based on certain visual intuitions. Only later were did they come to be extended to more general notions applicable to wildly different conditions.

What I claim in addition is that it's still useful for your intuition to understand this history and the leaps that were made by extremely talented mathematicians of the past who came up with these formalizations of intuitive properties.

I'd also claim that they couldn't have ever arrived at the current formal systems if they hadn't started with certain intuitions for simple systems.

[Tainnor]

You're arguing for including historical motivation in order to make a definition seem more immediately useful and intuitive. I agree with that.

What the person you replied to is suggesting, however, is that this can only get you so far. At some you have to sit down and study the formal definition and understand how it relates to the examples you've seen so far. Otherwise it's very easy to have your intuition lead you astray.

[tsimionescu]

I'd that's what they're arguing, then I fully agree. My fear was that they're arguing that the formal definition is the only thing that you should spend time on, that the original motivation is irrelevant - this is what I was trying to argue against.