(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! =

2601218943565795100204903227081043611191521875016945785727541837850835
6311569473822406785779581304570826199205758922472595366415651620520158
7379198458774083252910524469038881188412376434119195104550534665861624
3271940197113909845536727278537099345629855586719369774070003700430783
7589974206767840169672078462806292290321071616698672605489884455142571
9398549944893959449606404513236214026598619307324936977047760606768067
0176491669403034819961881455625195592566918830825514942947596537274845
6246288242345265977897377408964665539924359287862125159674832209760295
0569669992728467056374713753301924831358707612541268341586012944756601
1455420749589952563543068288634631084965650682771552996256790845235702
5521862223581300167008345234432368219357931847019565107297818043541738
9056072742804858399591972902172661229129842051606757903623233769945396
4191475175567557695392233803056825308599977441675784352815913461340394
6049012695420288383471013637338244845066600933484844407119312925376946
5735433737572477223018153403264717753198453734147867432704845798378661
8703257405938924215709695994630557521063203263493209220738320923356309
9232675044017017605720260108292880423356066430898887102973807975780130
5604957634283868305719066220529117482251053669775660302957404338798347
1518552602805333866357139101046336419769097397432285994219837046979109
9563033896046758898657957111765666700391567481531159439800436253993997
3120306649060132531130471902889849185620376666916446879112524919375442
5845895000311561682974304641142538074897281723375955380661719801404677
9356147936352662656833395097600000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000

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.

What is infinity and does it even exist? In our everyday experience, we find only finite things. A basket of eggs contains only a fixed number of eggs and no more. Our bodies are composed of particles (molecules, atoms, protons, quarks, etc.). But whatever particles describe our make up, we find only a finite number. It may be billions or trillions or more, but it still doesn’t get close to infinity. Even the known universe is finite – it’s only so many light-years in diameter and contains only so many elementary particles.

How, then, does one even get close to infinity? People have long realized that there’s no biggest number because it’s always possible to add 1 to any number and get still a bigger number. So numbers themselves, taken collectively, are infinite. Any given number is finite, but the mere fact that numbers go on forever – that’s infinite.

But what sort of infinite is this? The ancient Greek philosopher Aristotle (384– 322 BC) proposed that there are two types of infinity, a potential and an actual infinity. In a potential infinity, one can keep adding or subdividing without end, but one never actually reaches infinity. In a sense, a potential infinity is an endless process that at any point along the way is finite. By contrast, in an actual infinity, the infinite is viewed as a completed totality. Aristotle rejected actual infinity, claiming that only potential infinity exists.

So what, you say? For all practical purposes, we get on quite well with quite a bit less than even a potential infinity. Take the world’s most powerful supercomputer, Japan’s K Computer, which runs at 10 petaflops, using 705,024 SPARC64 processing cores. There are absolute limits to what this machine can do in terms of storage, retrieval, and processing. It’s safe to say that 10^100 (i.e., the number 1 followed by 100 zeros, aka “google”) sets an absolute limit on the amount of processing steps this machine will ever do, on the length of the longest number it can compute, and on the amount of bytes available to the machine’s memory.

And yet, the infinite is not so readily cast aside for practical reasons. Modern mathematics is done almost entirely in terms of sets (recall the “New Math”). Set theory treats just about anything as a set (the only things that are not sets are things too big to be sets – more on that in another post). Now numbers are sets. For instance, 0 is the empty set (it contains zero items). The number 1 is also a set (it is the set that contains zero, and thus is a set with one item).

But all the numbers taken collectively (0, 1, 2, etc.) also form a set, known to mathematicians as the natural numbers and represented as {0,1,2,3,…}. Ah, but what’s that ellipsis, those three dots (i.e., …), doing there? Doesn’t that tell us that the natural numbers are really just a potential infinity? Mathematicians don’t treat the natural numbers as a potential infinity but as an actual infinity – a completed totality that includes all numbers 0, 1, 2, etc.

But what do mathematicians know anyway? Perhaps treating the natural numbers as an actual infinity is just a convenient way to think about numbers and do calculations. If people’s concerns about infinity were left simply at the level of mathematics and its scientific applications, the debate over potential and actual infinities would be moot. But it turns out that this debate spills over into other areas, notably theology. If God is real, is he an actual infinite or is he just a potential infinite? Most religious believers see God also as unchanging, so if God is real and infinite, he must be an actual infinity.

Now it’s interesting that Georg Cantor, who invented set theory over 100 years ago, did so in part for theological reasons, seeing the infinite sets he came up with as a reflection of the infinity of God. Others, however, not believing that God exists or thinking that the very concept of an actual infinity is incoherent, reject the actual infinity and thus view Cantor’s so-called actual infinities as simply a device for describing much more mundane and finite processes. Yet it is a device that every working mathematician uses. As the great mathematician David Hilbert put it, “No one will drive us from the paradise which Cantor created for us.”

The debate over potential and actual infinities has been ongoing for centuries, and this short post won’t resolve it. Nonetheless, it’s worth noting that Cantor’s work on set theory has showed that the concept of an infinite set makes mathematical sense and avoids contradiction. Certain paradoxes, such as that infinite sets can be put in one-to-one correspondence with proper subsets (e.g., there are as many even numbers as natural numbers: 0à0, 1à2, 2à4, 3à6, etc.), may fly in the face of common intuitions, but science confronts us with lots of things that are counterintuitive.

In any case, modern mathematics, especially in its wholesale incorporation of set theory, has given the single biggest boost to the view that the actual infinite exists. Not that this proves the actual infinite exists – the nature of existence itself (a field philosophers refer to as “ontology” – the study of being) is itself up for grabs. But the mere fact that treating mathematical entities as actual infinities has yielded incredibly fruitful mathematical insights (Cantor’s paradise) gives the actual infinite breathing room that it never had in the past.

—–

References:

Joseph Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton: Princeton University Press, 1990).

Michael Hallett, Cantorian Set Theory and Limitation of Size (Oxford: Oxford University Press, 1984).

Are all numbers interesting?

Yes.

And we can prove it using a “proof by contradiction.”

1. Assume there are positive numbers that are not interesting.
2. Therefore, there is a smallest number that is not interesting.
3. Hey! That’s interesting!
4. Therefore, the assumption in (2) is violated. We have a contradiction and have proved that all positive numbers are interesting.

This proof is a bit whimsical. Here’s a similar, more serious question called Berry’s Paradox. [1] Define a positive number X as follows.

“The smallest positive integer not definable in under eleven words.”

This sentence has ten words. And it defines X. Therefore the sentence “The smallest positive integer not definable in under eleven words” defines X in 10 words. Think about it. The statement contradicts itself. We have defined the smallest positive integer that can’t be described in ten words using ten words. This is a contradiction. A paradox.

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