Eschatology, Celibacy, and the Exponential Distribution

Mathematics explains to us how Christians today have no more reason to reject Paul's recommendations to live celibacy than the first Christians did.

It is a tradition of The Loyal Lion to dedicate our November issues to the last things, and especially their connection to the doctrines of celibacy and virginity which we work to restore to prominence in the modern Catholic Church. This is fitting because as Advent looms, the Lectionary centers on Christian beliefs about the end times, or as theologians say, our eschatology. This science, from the Greek word eschaton meaning ‘end,’ deals with Christ’s return at the end of days, the judgment, and the possibilities of heaven or hell.

In this article we correct an erroneous view about Jesus’ return that we despise for the way it interferes with the true Catholic teachings about celibacy. A big part of Christianity is our anxious wait for the return of the Lord. But the nature of this wait turns out to be a theological bone of contention. To develop our point we must embark on a mathematical discussion involving probability theory. We need to make use of math’s ability to characterize three types of waiting. Indulge us.

First, imagine a bus that comes every twenty minutes. A man walks up to the bus stop unaware of when it last came. He can expect that at most he has to wait twenty minutes. But he might get lucky and show up just as the bus arrives. On average though, he knows he can expect a ten-minute wait. In mathematical language, we say the wait time for the bus is uniformly distributed between zero and twenty minutes. No wait time is more likely than any other. Over the long haul, he’ll have to wait two minutes as often as twelve and nine minutes as often as nineteen.

But notice with uniform distributions like this, the expected waiting time changes based on extra information. So let’s suppose that someone else was waiting at the bus stop when our rider shows up and she tells the newbie that she’s been waiting ten minutes. With this extra information, the man now expects to wait ten minutes more at most, expecting average wait of five minutes, not ten. That is, the longer one has waited, the more likely it is that the bus will come soon.

The man waiting for a bus will have a wait time drawn from the uniform distribution (blue). The mother waiting to pick up her kid has a wait time described by the normal distribution (pink). The person waiting on the next phone call will have a wait time specified by the exponential distribution (yellow). In each of the three situations, a person can expect to wait ten minutes on average, but the exponential distribution has a peculiar memoryless property that the uniform and normal distributions don't. Yet, this property makes the exponential distribution best for describing Jesus’ promised return.

For our second scenario consider a mom waiting to pick up a kid from school. The bell rings at 3 p.m., so she arrives at this time every day. We can imagine how her wait might vary from day to day. Let’s say that it takes twenty minutes for every last kid to get out. Now her kid probably won’t be the first one out the door, nor the last, but will come somewhere in between. Like the man at the bus stop, on average she has to wait ten minutes. But unlike the bus every wait time is not equally likely. It will probably happen that a kid or two trickles out right after the bell, followed by a few more, followed by a big flock of kids which then starts to thin out ultimately to a trickle of the last few kids who were tying their shoes when the bell rang. In this case, the time that the mother must wait is well-modeled by the normal or Gaussian distribution, with its familiar bell-shaped curve shown in pink in the figure. Unlike the uniform wait of the bus rider, the curve shows that she’ll have to wait ten minutes a lot more than five minutes, for example.

Now here like with the uniform distribution, the longer the mom waits, the less she expects to wait longer. So suppose she wants to talk to one of the other moms in line. After waiting five minutes, she knows that it’s not too likely her kid will come out in the next minute, so she runs over and say ‘hi’ thinking she probably won’t miss her kid. But after waiting ten minutes, it’s more likely the kid will come out in the next few minutes and she should stay put.

Now most people have lots of experience with the uniform and normal distributions. Therefore, it is easy to get an intuitive feel for the math. But there is another type of distribution that mathematicians use to describe wait times that is not so intuitive. It’s called the exponential distribution and it has a lot of real-life applications. For example, call centers like a 911 dispatch use the exponential distribution to model how much time operators have between calls. Suppose for our purposes that the wait times between 911 calls are exponentially distributed, and on average there are six calls per hour. This means that the average wait between calls is ten minutes. The wait-time percentages for such a distribution are shown in the figure by the yellow curve. Notice that it’s very likely the next call will come in less than ten minutes, but unlike the uniform and normal cases, it’s also fairly probable that it may be a while until the next call. This feature makes the exponential distribution is a good fit when waiting for events that might happen soon, but that could also be a long time in coming. Scientists use it to predict when electronics like calculators will fail or how long marriages will last.

But the quintessential feature of the exponential distribution is called the memoryless property. This means that unlike the uniform and the normal case, the expected wait doesn’t change based on the amount time already spent waiting. To illustrate this counter-intuitive property, suppose at the 911 call center there hasn’t been a call in thirty minutes. This means it’s very likely that a call will come soon, right? Wrong. With the exponentially-distributed wait times, the time spent waiting doesn’t change the expected wait. The time already spent waiting gives no additional information as to the likelihood that the event will occur soon. They must always expect to wait the average time between calls, or ten more minutes. Even after an hour of no calls, they will still have to wait on average ten more minutes for a call. Nobody at the 911 dispatch should ever say, “There hasn’t been a 911 call all day, so there’s bound to be one shortly.” But nor can they say, “There hasn’t been a call all day, so let’s go play volleyball.” It doesn’t work that way. Rather, the dispatch has to always be ready, prepared to handle a call every ten minutes.

The memoryless property means that no matter how long you’ve been waiting, you know nothing about how much longer you have to wait. Now this property of the exponential distribution is useful for our theological purposes, as it is a perfect model of the situation we Christians face who await Jesus’ return. The wait time on Jesus’ return is exponentially distributed! Since nobody knows the hour of His coming, we must be ever-prepared, regardless of how long we have been waiting.

Now what does this have to do with celibacy and virginity, the champion causes of our apostolate? A lot. Many ASJ doctrines have their Scriptural root in the writings of St. Paul to the early Church. For example, Paul tells us in 1 Cor. 7:29, “I tell you, brothers, the time is running out. From now on, let those having wives act as not having them….” But many modern scholars brazenly assert that such words don’t apply to modern Christians. They say Paul only said that out of an “incorrect” belief that Christ’s return was imminent, something we now know to be “false.” These scholars act like since Jesus hasn’t come yet in 2000 years that it is somehow unlikely that He’s going to come tomorrow. But this is false and distorts the received faith! ASJ exposes the flaw in this reasoning with the mathematical example of the exponential distribution. Because we can draw no inference whatsoever about Jesus’ return from the time we’ve already spent waiting, St. Paul’s advice on celibacy is just as valid today as on the day he first gave it.

Math doesn’t lie: the chance that Jesus will come in 2007 is no less than it was in the year 107. We have no justification for discounting Paul’s authoritative advice like that which recommends celibacy for the unmarried and continence for married Christians.

 This article appeared in the November 15, 2006 issue of The Loyal Lion.