(Or Jonathan Edwards Meets Leonhard Euler)
Visions of hell abound, not just in Christian literature but also in mythology. The Greeks, for instance, had Sisyphus, who was condemned to an eternity of futility, constantly rolling a rock up a hill, only to have it come tumbling back once he was almost at the crest. Or consider Tantalus, who was condemned to viewing all sort of sensuous delights, only to have them fall out of reach as soon as he tried to grasp them.
While religious and mythological imaginations are quite respectable at depicting the horrors of hell, one may ask what a mathematician might do if asked to design hell. Here’s one possibility. In the theory of infinite sequences, some sequences converge to a fixed number and others go off to infinity. Thus, for instance, the sequence 1, 2, 4, 8, 16, etc., where the general term is 2^n, goes off to infinity whereas 1, 1/2, 1/3, 1/4, 1/5, etc., where the general term is 1/n, converges to 0.
Now, some numbers go off to infinity much faster than others. The sequence 1, 2, 4, 8, 16, etc. with general term 2^n has an exponential rate of growth and thus goes off to infinity pretty fast. But mathematicians can think up much faster rates of growth. Consider, for instance, the number m! (i.e., “m factorial”), which equals 1 x 2 x 3 x … x m, and now define the superfactorial m!!n recursively as m! for n = 1, (m!)! for n = 2, and in general (m!!(n-1))!. Basically, m!!n means take m and apply the factorial operator to it n times. Hence SUPERfactorial.
If we now compare 2^n with n!!n, we find that n!!n goes to infinity much much faster than 2^n, in fact so much so that if we divide out n!!n by 2^n, the sequence n!!n/2^n still goes off to infinity. Granted 1!!1 is just 1 and 2!!2 is just 2, but 3!!3 is a 1,747-digit number: 3!!! = 6!! = 720! =
With this bit of background, we can now describe how a mathematician might design hell. At even intervals n = 0, 2, 4, 6, etc., provide a brief respite from torment. In fact, provide sheer delight during those intervals, but do so for only 1/n seconds. At odd intervals n = 1, 3, 5, 7, etc., provide n!!n seconds of torment, and let the intensity of torment double at each second of the interval. So, for the 1,747-digit number of seconds making up interval 3, if the intensity of torment is some value during a given second (let’s set the initial level of torment at passing a large kidney stone), then it doubles each subsequent second thereafter for all the seconds comprising this 1,747-digit number.
Interestingly, the total number of moments of torment and delight in this design of hell will be the same because the harmonic series 1 + 1/2 + 1/3 + 1/4 + etc. goes to infinity (check your basic calculus text if you doubt this) and, of course, the superfactorial series 1!!1 + 2!!2 + 3!!3 + etc. also goes to infinity. So, stretched to eternity, this scenario provides as much torment as delight. Only because of the timing in the way the torment and delight are experienced will the torment seem to totally dominate and outweigh the delight.
You might wonder why put in the delight at all. It turns out, as a matter of human psychology, it actually makes a painful task all the more painful if it is interspersed with a little pleasure rather than just getting it all out of the way at once. The ever briefer moments of delight, interspersed among oceans of pain, will highlight the pain and make the torments of hell all the more poignant. Indeed, the moment of delight call to mind all the pleasures that were missed by being consigned to (the mathematician’s) hell.
The mathematics here is completely elementary and could easily have been conveyed by mathematician Leonhard Euler (1707-1783) to theologian Jonathan Edward (1703-1758), who is best remembered for his sermon “Sinners in the Hands of an Angry God.” (Go here for the sermon and here for a rationalization of why one should not be utterly appalled at it.) In that sermon, Edwards writes:
The God that holds you over the pit of hell, much as one holds a spider, or some loathsome insect, over the fire, abhors you, and is dreadfully provoked; his wrath towards you burns like fire; he looks upon you as worthy of nothing else, but to be cast into the fire; he is of purer eyes than to bear to have you in his sight; you are ten thousand times so abominable in his eyes as the most hateful venomous serpent is in ours. You have offended him infinitely more than ever a stubborn rebel did his prince.
With the help of some basic mathematics, Edwards may have fleshed out his picture of hell along the lines sketched here.