"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?
Robin, of course it's not obvious. It's only an obvious conclusion if the global utility function from the dust specks is an additive function of the individual utilities, and since we know that utility functions must be bounded to avoid Dutch books, we know that the global utility function cannot possibly be additive -- otherwise you could break the bound by choosing a large enough number of people (say, 3^^^3).
From a more metamathematical perspective, you can also question whether 3^^3 is a number at all. It's perfectly straightforward to construct a perfectly consistent mathematics that rejects the axiom of infinity. Besides the philosophical justification for ultrafinitism (ie, infinite sets don't really exist), these theories corresponds to various notions of bounded computation (such as logspace or polytime). This is a natural requirement, if we want to require moral judgements to be made quickly enough to be relevant to decision making -- and that rules out seriously computing with numbers like 3^^^3.
I once read the following story about a Russian mathematician. I can't find the source right now.
Cast: Russian mathematician RM, other guy OG
RM: "Truly large numbers don't really exist in the same sense that small ones do."
OG: "That's ridiculous. Consider the powers of two. Does 2ˆ1 exist?""
RM: "Yes."
OG: "OK, does 2ˆ2 exist?"
RM: ".Yes."
OG: "So you'd agree that 2ˆ3 exists?"
RM: "...Yes."
OG: "How about 2ˆ4?"
RM: ".......Yes."
OG: "So this is silly. Where would yo... (read more)