Quote Originally Posted by Maat Mons View Post
The summation that underlies the integral to find a volume, as you noted, assigns a thickness. Area times arbitrarily small thickness dx is an arbitrarily small volume. So, you’re adding an infinite number of infinitely small volumes to get a finite volume. That works. The example I was responding to was adding an infinite number of masses that aren’t arbitrarily small. Hence the problem.
The mass of an infinitessimal 4d volume element will also be an infinitessimal. The math, at least, works, even if the poster was not explicit about the dx (but in talking about adding up cross-sections rather than going and writing down an integral or sum explicitly, I find that to be implied taking the message of that post in good faith). It's like complaining that someone wrote down an indefinite integral when really what they mean is 'pick the limits appropriately to what you're trying to model'.

As for the physical intuition justifying the math, next paragraph...

I’m well aware that a 4D density is not the same as a 3D density. That is, in fact, the whole point of the post I made. I order for the post I was responding to to make any sense, the each 3D mass would need to be multiplied by a one dimensional density and an arbitrarily small length. This would have made each summand an arbitrarily small 4D hypermass, and the result would have been a finite 4D hypermass.
To connect this to a physical picture, we have to identify what's the physically real thing when we change the dimensionality. Is the 3d density the physically real thing, so now all 4d objects have an infinite mass? That doesn't make sense even in 3d, because ultimately the density is a summary statistic over a finite volume and infinitessimal mass densities aren't physically real in a literal sense (infinitessimal probability amplitudes on the other hand, maybe).

The natural assumption is to extend the rotational invariance of physics in 3d to 4d, meaning that the distance between particles comprising a material (which is summarized as a density when considering a sufficiently large volume) is the invariant. So that justifies saying that the 1d, 2d, 3d, and 4d densities of materials would have a particular relationship due to the forces between their component pieces only caring about distance at some small enough level, rather than explicitly caring about direction.

Or in terms of dimensional analysis, the thing that gets you from a 3d density to a 4d density is a characteristic length-scale associated with spacing between particles in a material. That is the thinnest a '3d' volume of that material in a 4d space can be in the 4th direction.