What would be the properties of that function? Well, for one thing, an utility of a region of space would not generally be equal to sum of utilities of parts, for the obvious reason that your head has bigger utility when it haven't been diced into perfect cubic blocks and then rearranged like a Rubik's cube.
I'm not entirely sure what the point of this comment was, but in that case, surely the problem occurs when said chunks die? I mean, if they magically kept working the same way, linking telepathically with the other chunks and processing information perfecty well, I don't see why they wouldn't be just as valuable, albeit rather grisly looking.
Finding out that the chunks will die (given the laws of physics as they are) is something that the function in question got to do. Likewise, finding out that they won't die with some magic, but would die if they weren't rearranged and the magic was applied (portal-ing the blood all over the place).
You just keep jumping to making an utility that is computed from the labels you already assign to the world.
edit: one could also subdivide it into very small regions of space, and note that you can't compute any kind of utility of the whole by going over every pi...
When someone complains that utilitarianism1 leads to the dust speck paradox or the trolley-car problem, I tell them that's a feature, not a bug. I'm not ready to say that respecting the utility monster is also a feature of utilitarianism, but it is what most people everywhere have always done. A model that doesn't allow for utility monsters can't model human behavior, and certainly shouldn't provoke indignant responses from philosophers who keep right on respecting their own utility monsters.
The utility monster is a creature that is somehow more capable of experiencing pleasure (or positive utility) than all others combined. Most people consider sacrificing everyone else's small utilities for the benefits of this monster to be repugnant.
Let's suppose the utility monster is a utility monster because it has a more highly-developed brain capable of making finer discriminations, higher-level abstractions, and more associations than all the lesser minds around it. Does that make it less repugnant? (If so, I lose you here. I invite you to post a comment explaining why utility-monster-by-smartness is an exception.) Suppose we have one utility monster and one million others. Everything we do, we do for the one utility monster. Repugnant?
Multiply by nine billion. We now have nine billion utility monsters and 9x1015 others. Still repugnant?
Yet these same enlightened, democratic societies whose philosophers decry the utility monster give approximately zero weight to the well-being of non-humans. We might try not to drive a species extinct, but when contemplating a new hydroelectric dam, nobody adds up the disutility to all the squirrels in the valley to be flooded.
If you believe the utility monster is a problem with utilitarianism, how do you take into account the well-being of squirrels? How about ants? Worms? Bacteria? You've gone to 1015 others just with ants.2 Maybe 1020 with nematodes.
"But humans are different!" our anti-utilitarian complains. "They're so much more intelligent and emotionally complex than nematodes that it would be repugnant to wipe out all humans to save any number of nematodes."
Well, that's what a real utility monster looks like.
The same people who believe this then turn around and say there's a problem with utilitarianism because (when unpacked into a plausible real-life example) it might kill all the nematodes to save one human. Given their beliefs, they should complain about the opposite "problem": For a sufficient number of nematodes, an instantiation of utilitarianism might say not to kill all the nematodes to save one human.
1. I use the term in a very general way, meaning any action selection system that uses a utility function—which in practice means any rational, deterministic action selection system in which action preferences are well-ordered.
2. This recent attempt to estimate the number of different living beings of different kinds gives some numbers. The web has many pages claiming there are 1015 ants, but I haven't found a citation of any original source.