However, I believe it’s completely possible to encounter shadows of things that are beyond comprehension. I’ve touched upon some examples in my book, but there are many more. I maintain an ever-growing list, and I’d be happy to hear from readers with their thoughts, ideas and insights.
So the more appropriate question is "when are we peering at shadows of the incomprehensible?" To a large extent the answer is personal. That which is beyond comprehension is that which strikes you as perplexing to the point of incomprehensibility. I’m not suggesting the answer is entirely subjective, as there exist phenomena that have no reasonable explanation – physics is a treasure trove of such phenomena. But many examples require internal examination.
Let me take up one example today and come back to others in future posts.
It’s possible to construct any number of containers that have finite volume but stretch to infinity and have no bottom. Torricelli’s trumpet, covered in my book, is but one example. The question that presents itself is the following. How do you fill a container that has a hole in it? It would seem that any liquid you filled it with would simply continue forever downward, so that you could continue filling it without stop. Thus, how could the container have finite volume?
It’s true that if we attempted to fill any such container with a real liquid such as water, the water molecules would eventually get jammed in the neck of the container when it became too thin. But the bottomless containers we’re talking about are mathematical constructions, so we should feel free to fill them with another mathematical construction: dimensionless water drops.
It’s at this point that you must ponder the alternatives for yourself. What’s going on? For the mathematically inclined it’s possible to get into all sorts of detailed arguments about this and that. None are fully satisfying. Many miss the point. My personal perspective is that the problem stems from the way in which different mathematical dimensions relate to one another. There’s no "problem" about dimensions that one can point to, but the type of situation encountered here pops up in various scenarios. And even in all their mathematical glory, these scenarios remain head scratching.
But the simple point we can all agree upon, whether mathematically inclined or not, is the following: there exist infinite, bottomless containers that have finite volume. "This bottomless container contains six gallons" is a perfectly valid statement. But does it make any sense?