Its not really that physical, haven’t read the article but the way “dimensions” are used is best explained via algebra:

Say you have a formula x=6, you can visualise the value on a number line from 0 to 10, with a little mark at the 6. That’s one dimension, a line.

Bringing the next dimension in you can add a new variable/unknown like y. Given the formula x-y=6, often formatted as y=x-6, you can now visualise the value or the solution with a graph, the number line for x horizontally and y vertically (2D). Now instead of marking a dot on the number line you draw a line through the graph where y=x-6, e.g at coordinates (x=6,y=0).

In a similar way you can add a variable z, which you can draw as a new line perpendicular to the other two, turning the plane into a three dimensional space with 3 number lines, one for each variables.

This is where things get complicated, you add the fourth variable, bringing things into the fourth dimension. As you eluded to, it can sometimes be represented as time, so the contents of the 3D space from before can change with time, a bit harder to imagine but think of the new 4th dimension of time as being a slider like on a video player, dragging it around to “play the video” in the other three dimensions. Instead of the 4th dimension being time though, you can also think of it as another number line added to the graph with the other 3, the problem is that it’s impossible to visualise, we can’t draw a fourth line perpendicular to the other three. So we just say its there and keep solving the algebra as if it was there.

That’s the basics anywho, theres a bunch of material online, both text and videos if you wanna dive further, khan academy being a good place to start, or maybe brilliant.org.

In math, a dimension doesn’t have to be tied to our real physical dimensions. In two dimensions, you need two values to represent a point (a, b). In three dimensions, you need three values, (a, b, c). It simply keeps going, more dimensions means more values. Of course it gets harder to visualize, but the math keeps working.

Vectors are tuples of numbers, an ordered list. The length of the list is the dimension of the vector: (2,3) is a two-dimensional vector where the first component is 2 and the second one is 3. (2,6,5,9) would be a four-dimensional vector and of course they can be arbitrarily large.

You can string together any number of values you want to keep track of in a vector. Length, width and height typically being used for three-dimensional space, but you can add time to that to create a four-dimensional vector. You can add more if you want, add air humidity for a 5th dimension, wind speed for a 6th one etc. it all depends on how much you want to record/keep track of.

edit: just realised I didnt really answer the question. To “solve something in n dimensions” means that there is a problem that can be formulated in such a way that it makes sense when talking about lists of length n. So for instance spheres (having a length, width and height) can be described using 3 dimensional points. A common way to describe spheres is “all points which are length r away from the 0 point”. This naturally extends to other dimensions since we can measure length in any number of dimensions and thus talk about 4 dimensional, 32 dimensional or n dimensional spheres. Then we can try to solve the problem that was formulated in a certain dimension. The solution typically varies as more dimensions add more variety and using some quirks of certain numbers (prime/not prime, power of 2, divisible by 6 or whatever) its often possible to find a solution for certain dimensions, but not necessarily others. Having a problem solved for any dimension is typically the holy grail after which people will look into how make the problem more generic, or extend it in some other way.

I can only answer partially but I’ll give it my best shot. Dimensions don’t have to represent physical space, or anything related to it. Sometimes a fourth dimension is used to represent time, but really it could represent anything.

An example I saw frequently in school used different recipes for, say bread. They all have similar ingredients: flour, water, yeast. Each recipe can be represented as a vector in three dimensions. X might be flour, Y might be water, Z might be yeast. Easy to visualize. You can imagine a center at some arbitrary point in space, and we can say that any point in that space relative to it’s center is another recipe. If it helps to see what I mean, any point that lies in an axis would only have one ingredient. The very center represents the bread recipe with no amount of any ingredient. Those would be terrible recipes, since they wouldn’t actually make any bread, but that’s what those points would represent.

Since we’re imagining things as points in space, we can even use things like the distance formulas you learned in geometry to determine how far apart the recipes are in the imagined space. Well, obviously recipes can have more than three ingredients. Suppose we add salt and sugar to the recipes. Now the vectors representing those recipes have five dimensions. The cool thing is, those formulas we used still work in any number of dimensions. We just use a more general form of them.

I hope that helps sort of get you acquainted with the idea of higher dimensions in mathematics. I learned by taking two terms of linear algebra. The second term was pretty difficult.We were all very confused 😂

So you can relate dimensions to the physical world, but the way we tend to think of them in math and physics is more abstract. Generally I think describing them as “degrees of freedom” is accurate; they are extra ways of viewing the functions or concepts you’re working with. Higher dimensions allow us to discern information about the lower dimensions we exist in, much the same way higher order derivatives show us behavior of the lower level function. Dunno if this is helpful I might edit this in the morning when I’m not sleepy to make it more useful