"What's the worst that can happen?" goes the optimistic saying. It's probably a bad question to ask anyone with a creative imagination. Let's consider the problem on an individual level: it's not really the worst that can happen, but would nonetheless be fairly bad, if you were horribly tortured for a number of years. This is one of the worse things that can realistically happen to one person in today's world.
What's the least bad, bad thing that can happen? Well, suppose a dust speck floated into your eye and irritated it just a little, for a fraction of a second, barely enough to make you notice before you blink and wipe away the dust speck.
For our next ingredient, we need a large number. Let's use 3^^^3, written in Knuth's up-arrow notation:
- 3^3 = 27.
- 3^^3 = (3^(3^3)) = 3^27 = 7625597484987.
- 3^^^3 = (3^^(3^^3)) = 3^^7625597484987 = (3^(3^(3^(... 7625597484987 times ...)))).
3^^^3 is an exponential tower of 3s which is 7,625,597,484,987 layers tall. You start with 1; raise 3 to the power of 1 to get 3; raise 3 to the power of 3 to get 27; raise 3 to the power of 27 to get 7625597484987; raise 3 to the power of 7625597484987 to get a number much larger than the number of atoms in the universe, but which could still be written down in base 10, on 100 square kilometers of paper; then raise 3 to that power; and continue until you've exponentiated 7625597484987 times. That's 3^^^3. It's the smallest simple inconceivably huge number I know.
Now here's the moral dilemma. If neither event is going to happen to you personally, but you still had to choose one or the other:
Would you prefer that one person be horribly tortured for fifty years without hope or rest, or that 3^^^3 people get dust specks in their eyes?
I think the answer is obvious. How about you?
Bravo, Eliezer. Anyone who says the answer to this is obvious is either WAY smarter than I am, or isn't thinking through the implications.
Suppose we want to define Utility as a function of pain/discomfort on the continuum of [dust speck, torture] and including the number of people afflicted. We can choose whatever desiderata we want (e.g. positive real valued, monotonic, commutative under addition).
But what if we choose as one desideratum, "There is no number n large enough such that Utility(n dust specks) > Utility(50 yrs torture)." What does that imply about the function? It can't be analytic in n (even if n were continuous). That rules out multaplicative functions trivially.
Would it have singularities? If so, how would we combine utility functions at singular values? Take limits? How, exactly?
Or must dust specks and torture live in different spaces, and is there no basis that can be used to map one to the other?
The bottom line: is it possible to consistently define utility using the above desideratum? It seems like it must be so, since the answer is obvious. It seems like it must not be so, because of the implications for the utility function as the arguments change.
Edit: After discussing with my local meetup, this is somewhat resolved. The above desiderata require the utility to be bounded in the number of people, n. For example, it could be a staurating exponential function. This is self-consistent, but inconsistent with the notion that because experience is independent, utilities should add.
Interestingly, it puts strict mathematical rules on how utility can scale with n.