Torture vs. Dust Specks

“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?