Volume and area are two different measures. Consider a box 10 feet on a side. It’s volume is 10 x 10 x 10 = 1000 cubic feet, while its surface area is the sum of the area of each of the six sides, 6 x 10 x 10 = 60 square feet. The point is that volume is a measure of cubic feet while area is a measure of square feet. So we have to be a bit careful about how we compare them.
Location in a single dimension can be expressed with a single number (x); in two dimensions with a pair of numbers (x, y), and in three dimensions with a triple of numbers (x, y, z). If we look at our box that’s 10 feet on a side, we can think of the box corresponding to all coordinates (x, y, z) where x, y, and z take on all values between 0 and 10. The side on which the box rests is a square that corresponds to all coordinates (x, y, z) where x and y take on all values between 0 and 10 but z is limited to the single value 0. Analytically, the box can be thought of as made up of infinitely many squares stacked on top of one another, each square corresponding to a different value of z. Looked at in this way, we see that volume is very much a three-dimensional construct that appears infinitely larger than the two-dimensional surfaces. Is it reasonable to even make a comparison of the size of these two entities?
It all depends on what we mean by size. Size, in and of itself, isn’t a mathematically defined concept. That doesn’t mean the concept of size isn’t important. It just means that we have different definitions of size for different purposes. Two frequently used definitions are that of measure and that of cardinality.
Measure, in its various forms, is typically what we think of when dealing with areas, volumes and the like. It requires a deeper understanding of the foundations of math than can be covered here. However, I note that the definition of a measure is intimately related with the notion of dimension. The three-dimensional measure, or volume, of the side of a box, is zero. This agrees with our intuition that the surface of a box – or of Torricelli’s Trumpet – is totally inconsequential when compared with the volume of the same figure. From a measure-theoretic perspective, a mathematician might declare “yes, Torricelli’s trumpet has finite volume and infinite surface area, but the infinite surface is an infinite amount of nothing when compared with the trumpet’s volume.”
Even so, the fundamental conundrum remains. If we set out to measure the volume of the trumpet with tape measure and pencil (hypothetically, of course), we’d find it was finite, while we’d find the surface area was infinite using the same set of tools. We’re still left to ask “does this make intuitive sense”?
We can further complicate the issue by considering another approach to size, cardinality. Unlike measure, the concept of cardinality doesn’t depend on dimension. Instead, it focuses on the number of things in each set: how many points are in the surface when compared with the number of points in the box? While it may seem obvious that the number of points in the box is larger than the number of points in one side, Georg Cantor showed that the number was the same in that he could find a one-to-one correspondence between the two sets; that is, he could find a way to match one point in the one side of the box (or it’s entire surface) with exactly one point in the entire box. If this seems preposterous, it is. But the construction is reasonably straightforward (I cover it in detail in my book). So from the perspective of cardinality – the ability to show a one-to-one correspondence between the sets – the box and its surface are the same “size.” The same is true for Torricelli’s Trumpet and its surface.
Thus, at this more mathematical level, we might ask if it makes sense that two sets of the same cardinality –Torricelli’s Trumpet and its surface – have such different measures – one finite, the other infinite. The measure theorist might object that cardinality and measure are fundamentally different beasts. But the question raised by Torricelli’s Trumpet isn’t one of formalism. No matter how deeply we dive, we’re left with one, simple, basic question: does a figure with finite volume and infinite surface area make intuitive sense?