The Value of Calculus

I took three semesters of calculus in college. As a librarian, I don't have a strong need for advanced math education. Most of what I learned so many years ago has long since faded from memory. Still, calculus has some small enduring value to me. Or maybe it's “pre-calculus” where most people meet the concept of limits.

Perhaps the easiest way to understand limits is to look at what happens if one divides by zero. What is the result for dividing 1 by 0 on a standard calculator? The answer is that it gives an error. Dividing by zero is not an allowed math operation. So no one knows what 1/0 equals, right? Well, sort of. Here is where limits come in handy. What if I calculate the value of 1/x and don't actually make x equal to zero but make it very small? I see that 1/1 = 1; 1/0.1 = 10; 1/0.01 = 100; 1/0.001 = 1,000; 1/0.0001 = 10,000; etc. Although I can never make x equal to zero, I can figure out that the smaller x becomes, the larger result I will get. If I could divide by zero, the smallest possible number, I would get the largest possible number, which is infinitely large. Thus, although I can't divide by zero, I know what the result would be if I could because I know where my answer is headed for progressively smaller values of x: infinity.

At the other end of the scale, I could ask myself what 1/x would be if x were equal to infinity. Again, I can't actually put infinity into the calculator and get an answer. However, if I calculate the results for increasing values of x, I discover that the answer becomes progressively smaller. If I could take x all the way to infinity, I would end up with a result of zero. 1/0 = infinity. 1/infinity = 0. Both those answers are determined by letting x come as close as possible to either zero or infinity and watching where the result is heading. The result of dividing 1 by any number between 0 and infinity will fall somewhere between infinity and zero. Thus, 1/x+1 will approach 1 as x approaches infinity and will approach infinity as x approaches zero. For every positive value of x, 1/x+1 will lie somewhere between 1 and infinity.

I find this process of determining the boundaries of an equation's result useful for answering certain philosophical questions. For example, in my last post I pondered the question of whether it is possible for one person to make another person happy. To find my answer, I assigned the person responsible for creating happiness in the life of another person to provide the maximum service possible to that other person. I made that responsible person into a slave. Attentiveness from this person approaches the maximum humanly possible. I then mentally assessed the happiness of the person receiving the full attention of that slave. My conclusion was that humans have no lack of ability to be dissatisfied even in the face of total attentiveness by the person they have made responsible for their happiness. A slave can't read the mind of the master. A slave can only work so fast and so hard and can be only one place at a time. A slave requires food and sleep. My conclusion was that, because of these limitations and others, an unhappy person will continue to be unhappy even with a fully dedicated slave. Therefore, if even dedicated slave-level service doesn't satisfy an unhappy person, I can conclude that no lesser level of service and attentiveness will make an unhappy person happy. There needs to be another source for happiness. Can a person affect the happiness of another person? Yes. But the other person will always have the capability of being completely unhappy regardless of the level of service rendered.

This is the value of calculus to me. It helps me solve certain mental puzzles by giving me endpoints to the possible range of answers as the unknowns move across their full spectrum of possible values.

x approaches infinity questions:

• What if everybody did it?
• What if I did it perfectly?
• What is the worst that could happen (maximum unpleasantness)?

x approaches zero questions:

• What if I did nothing?
• What if nobody did it?
• What is the best that could happen (minimal unpleasantness)?

After I find the limits, it is then easier to figure out the range of answers for likely values of the unknowns in life's equations, e.g. the level of friendship and service I have to offer to an unhappy person in my life will never be sufficient to make that person happy.

Three semesters of calculus for that. Or at least I haven't noticed any other residual effect of those classes. Does anyone have any other practical uses for higher math?

I Could Be Happy If Only You Were a Better Person

Sometimes I get the impression that I'm the sole obstacle to another person's happiness. Unhappy people tell me how unhappy they are and either blame their unhappiness directly on me, citing something I did (or didn't do) that robbed them of their joy, or tell me what would make them happy without mentioning my own ability to provide that missing ingredient. In the latter case, I'm never quite sure whether or not I'm supposed to take the hint and provide them with what they need in order to be happy.

This is a lot of responsibility. I already take responsibility for my own happiness. And much of my happiness traces back to being available for service to other people, to many people. The happy people in my life accept what I give them with gratitude. Their happiness doesn't depend on me but I can add to it. Most of the unhappy people in my life are independently unhappy and don't particularly notice me. There are just a few who seem certain that they could be happier if only I focused more of my time and energy on pleasing them.

I wonder. Has anyone ever found happiness in being served? Are people with devoted slaves to anticipate their every need happy people? I'm thinking not. Even the most devoted slave can't read the mind of his master and must sometimes fail to be fully pleasing. I can think of few more certain roads to frustration and anger than depending on someone else for one's contentment and joy.

So what do I do about the unhappy people who latch onto me as holding the key to their happiness? I can try to explain to them that joy comes through acts of service, but I don't think that is a concept that sells well.

In her book A Theology of Love, Mildred Bangs Wynkoop defined love as “impartial goodwill”. I think the best I can do for people looking to me to fix what's wrong with their world is to offer them the same level of service I offer to others. Sometimes it's easy to do less for those who show so little appreciation for small acts of service and consistently demand more. Other times, I find myself doing more -- service I don't want to provide and can't sustain.

Jesus Christ said that there will always be poor people among us. I suspect that's true not only in terms of financial poverty but in terms of emotional poverty. Serving those people with the same effort I invest in happy people might be the best I can do. If nothing else, doing so adds to my own sense of well-being. There is a certain satisfaction in neither walking away from a manipulator entirely nor giving into their demands but simply serving them as though they appreciated small acts of kindness.

Agree or Disagree: Prayer works

Just over a year ago, I wrote a post on the idea that "prayer works". You can read those thoughts here if you'd like, but it's not really necessary. Basically, I questioned what it means to say "prayer works". I have racked up a lot of unanswered prayers in my life and have watched people who are more optimistic than I am rack up even more.

One of my favorite people in the world is now on hospice care, dying of cancer at age 51. If prayer consistently worked for healing cancer patients, he would not be in this situation. I am far from the only one who values his friendship. The prayer that has bombarded heaven on his behalf is quite impressive by any measure. Hundreds of people spread across at least three continents are praying desperately for his healing. His mother has a history of seeing miraculous healings in response to her prayers. She is praying. Many of those praying have devoted their entire lives to God and the church, some serving in high levels of our denomination. Yet, he's dying. If "working" means always granting long life to those we love and for whom we pray, then, no, prayer does not work.

My experiments have revealed also that prayer doesn't work for getting signs from heaven. If you want God to prove His existence to you by giving you some sort of sign, you will likely be disappointed. Or at least it hasn't worked for me. In spite of my prayers, I have no proof of God's existence beyond what is available to all. My faith is at its core simply that -- faith.

So prayer doesn't work for keeping alive and healthy those we hold dear. And prayer doesn't work for giving us hard evidence concerning God. There's some disappointment there. What good is prayer if not for miracles of healing and signs of divine power at work among us?

This post was actually prompted not by disappointment with prayer but by a fresh realization of what a blessing prayer has been in my life. It takes perspective to see it, though.

Fifteen years ago, I prayed for people with whom to share my spiritual journey. After a while, one of the few people who filled that role in my life moved away. My prayers weren't working very well. However, from this perspective I can see that I was about to meet several people who have become dear friends. And the internet was about to open up a wealth of opportunities to communicate with like-minded people. Was it simply coincidence that my prayers back then were for the things that were about to happen?

Not quite so long ago, I started regularly practicing prayer and fasting for my church in response to a call by an internet friend to do so. After a year or two of praying that way, things were going so badly that I stopped the fasting part, suspecting that anyone involved would ask me to do so if they thought my prayers were behind the storm beating against the church. The results of those prayers were disastrous. However, from this perspective, I can see that we were about to enter a new era as a church. Some people aren't pleased with this new place where we've found ourselves, but I'm certainly not complaining.

One of my favorite verses from the Bible is Psalm 37:4 "Delight yourself in the Lord and he will give you the desires of your heart" (NIV).

Does prayer work? No, it doesn't. At least not like a working, well-stocked vending machine where you can expect a particular item to fall out if you put in the correct amount of change.

Does prayer work? I don't know, but sometimes when I pray terrible and unexpected things happen which precipitate delightful and unexpected blessings.

Does prayer work? It must! I pray and I am blessed at every turn.

On the other hand, a dear friend is dying too soon and all of the prayer in the world seems unable to prevent that from happening.

Does prayer work? On the survey I took that asked me that question, my response (blank) was tabulated as "not sure/no opinion". Maybe "I don't know" is the best answer I have. However, it bothers me a little that my response will show up in some set of ain't-it-awful statistics that will say, "___% of evangelical Christians responded that they don't know if prayer works."

Maybe what we need is a different question. Agree or disagree: Prayer is essential to my spiritual well-being. For that one I can confidently mark: Strongly agree.