Posted by Jonathan Bartlett on Friday, 28 September, 2012 in Classical Christian Education
When we think about the world theologically, we are trying to understand the world as God sees it. We usually see the world through very finite lenses—our perspective does not extend beyond the end of our noses. Many people take what their immediate senses tell them, and use that information to define their existence and determine their course in life.
As believers, however, we are called to take a deeper view. God’s kingdom is not just for today, but for eternity. God’s original plan for creation was not for a broken creation, but for a creation that He could be with forever. As such, when we analyze claims, policies, or ideas, we should not just look at how they will affect tomorrow. We should look to see how they will affect the long-term future. We need to examine what happens to our ideas as they progress towards infinity.
Interestingly, mathematics already has many tools for examining the infinite. The ones we will utilize here have been available within mathematics for several centuries, but few people are aware of them. The primary tool for examining what will happen to a process if it continues forever is known as a limit.
Introduction to Limits in Mathematics
The limit of a function tells us what value a function tends to go to as the value of its unknown—its “x”—approaches some number. To take a simplistic notion, let us look at a very simple function: 2x + 3.
A limit asks: What is the value of the function as x approaches some number, irrespective of what it does when x reaches that number (though with this function they are the same thing)? Let us consider what the limit of the value of this function is as x approaches 3. To see this, we can calculate the values of the function when x is 2.8 (8.6), 2.9 (8.8), 2.95 (8.9), 2.99 (8.98), 2.999 (8.998), and so on. Without actually calculating the value of the function when x is 3, we can see that as x gets closer and closer to 3, the value of the function gets closer and closer to 9.
You might say that this is obvious, and then ask why we would not just calculate the answer directly? The answer is simple: There are many equations for which we cannot calculate certain values directly. For our purposes, the most important question is: What happens when x goes to infinity? Since infinity is not an actual number, we cannot just plug it into equations the same way that we plug in other numbers. However, we can speak of the limit of the function as x approaches infinity.
Analyzing Infinity with Limits
For this simple function, if x goes to infinity, 2x + 3 will also go to infinity. The fact that we are doubling the value of x simply does not matter. Two times infinity is still infinity, and infinity plus three is still infinity. Therefore, we would say that the limit of the value of 2x + 3 as x approaches infinity is positive infinity. Another slightly more concrete way of saying this is that as x increases without bound, the value of 2x + 3 also increases without bound.
Now let us look at another function: -2x^{2} + 5. In this equation, since we are multiplying the x value by a negative number, the limit of the function as x goes to infinity is a negative infinity. Or, to put it in a more concrete way, as x increases without bound, the value of the function will decrease without bound. For the purpose of this limit, the fact that x is being squared is largely irrelevant—it will simply increase without bound forever.
Now let us look at a function that is a little more irregular: 2x^{3} - 5x^{2} + 100. Here, you have a negative component and a positive component. In other words, x^3 is growing infinitely large, but -5x^2 is shrinking infinitely small. So which direction will this function take when it goes to infinity? You might consider graphing it. However, depending on where you look on the graph, this might not be helpful. The graph has multiple turns and therefore it is not always going in the same direction! So, how would we know which way this equation leads as it progresses to infinity?
The key to this is recognizing what controls the equation in different parts of the graph. When x is near 0, which component of the equation contributes the most to the value of the function? It is the constant, 100. When x is 0, the two other components (2x^3 and -5x^2) do not contribute anything. When x is 1, the first term contributes 2 and the second term contributes -5. Compared to the constant of 100, they still are not contributing much to the value. Likewise, for small values of x, the middle term contributes more to the value than the first term.
But what happens towards infinity? Well, since we are raising x to the third power in the first term, but only to the second power in the second term, then that means that as x gets larger and larger, the first term will grow faster and faster compared to the second term. To see why, look at the sequence of squares: 1^{2} = 1; 2^{2} = 4; 3^{2} = 9; 4^{2} = 16. Now, let us look at the sequence of cubes: 1^{3} = 1; 2^{3} = 8; 3^{3} = 27; 4^{3} = 64. As you can see, though they start on the same footing, the cubed numbers increase in value much faster than the squares. Therefore, as x progresses to infinity, the second term will count less and less towards the final value, and the first term will count more and more. So, as x progresses to infinity, only the largest exponent of x really matters. The fact that the second term is negative does not really matter in the long run.
More Complex Limits
Based on the examples offered above, it may be tempting to surmise that, as x goes to infinity, the limit will always be a positive or negative infinity. But this is not true. For instance, the limit of the value of 1/x, as x tends—in other words, as it moves—to infinity, is zero. Since x is dividing one, as you divide one into smaller and smaller pieces, the pieces themselves get closer and closer to zero. Therefore, the limit of 1/x as x approaches infinity is zero. There are other types of behaviors that happen with limits as well. For instance, the sin(x) function is an eternal cycle. Therefore, the limit of sin(x) as x approaches infinity is a range between -1 and 1.
Another interesting example of limits is the limit of the quotients of polynomials. For instance, what is the limit of (12x^{3} - 2x^{2} + 50x + 20) / (4x^{3} + 9x^{2} - 20x + 10)? To figure this out, let us deal with each side of the fraction individually. On the top, which term is contributing the most to the limit? It is 12x^{3}. Therefore, as x progresses to infinity, the top equation will look more and more like 12x^{3}; the other parts will become less and less relevant. On the bottom, which term is contributing the most to the limit? It is 4x^{3}. As x progresses to infinity, the other terms, since they have lower exponents, become less and less important. Therefore, for the limit, we are left with the simple fraction (12x^{3}) / (4x^{3}). When simplified, the x^{3}s cancel out, leaving 12/4, which is 3. So, as x approaches infinity, the value of the function will get closer and closer and closer to 3. At other points in the graph, the function has very complex behavior. But, at the far end of the graph, as x approaches infinity, the value of the function approaches 3.
As you can see, using the aid of the mathematical concept of a limit allows us to better understand values of functions that we could never calculate directly.
Applications to Theology and Normative Behavior
So how do we apply this to theology? Remember, when we think about normative processes theologically, we are seeing how they will affect life not just in the short-term, but for the infinite limit. One way we can approach this is to make an analogy between functions and normative rules of society, with time being our free variable (i.e., x). Since, theologically, we are interested in how these norms affect the ability of human society to continue indefinitely, we want to understand the limits of normative rules as time progresses to infinity.
To use a simple example, let us take a look at the question of debt. Let us ask the question: Does debt increase prosperity? The short-term answer is, yes, it does. If I go into debt, I can get the car I always wanted, the house I always wanted, and all the electronic gadgets I can shake a stick at. What happens, however, if we take this as normative behavior, and extrapolate it to infinity? Will debt-living continue to increase prosperity as time continues forward? The answer is no.
To illustrate this, let us look at a hypothetical situation made up of simple elements: Let us say that I do not want to work, but I want to live like a king. Therefore, I decide to borrow $200,000 every year instead of working. Each loan is separately taken each year at a simple 10% interest rate, and each loan will come due after ten years. I am somewhat responsible with my money, however, so I save $100,000 a year from my loan. At the end of year one, I have $100,000 in cash and $100,000 in things that make me happy. Sounds good, right? After the second year, if I follow the same pattern, then I am doing even better. I have $200,000 in the bank, and I will have spent $200,000 doing things that make me happy. This seems like a good plan. But appearances are deceiving. Remember what happens at the tenth year? At the end of ten years, I have to begin paying back the first $200,000 loan. At that point, since I am saving $100,000 per year, I am not sweating it too much. I only have to pay $220,000 ($200,000 with interest of 10%, $20,000, added to it), but I have $1,000,000 in the bank because I have saved $100,000 every year for ten years. All still seems well. But take a look at the chart below to see that by the sixteenth year, even if I am taking out more loans, I will not have enough money to make loan payments, much less pay for my living expenses:
Year |
Annual Deposit* |
Bank Balance |
Loan Payment |
Final Balance |
10 |
+$ 100,000 |
$1,000,000 |
-$ 220,000 |
$ 780,000 |
11 |
+$ 100,000 |
$ 880,000 |
-$ 220,000 |
$ 660,000 |
12 |
+$ 100,000 |
$ 760,000 |
-$ 220,000 |
$ 540,000 |
13 |
+$ 100,000 |
$ 640,000 |
-$ 220,000 |
$ 420,000 |
14 |
+$ 100,000 |
$ 520,000 |
-$ 220,000 |
$ 300,000 |
15 |
+$ 100,000 |
$ 400,000 |
-$ 220,000 |
$ 180,000 |
16 |
+$ 100,000 |
$ 280,000 |
-$ 220,000 |
$ 60,000 |
17 |
+$ 100,000 |
$ 160,000 |
-$ 220,000 |
$ -60,000 |
18 |
+$ 100,000 |
$ 40,000 |
-$ 220,000 |
$-180,000 |
19 |
+$ 100,000 |
$ -80,000 |
-$ 220,000 |
$-300,000 |
20 |
+$ 100,000 |
$ -200,000 |
-$ 220,000 |
$-420,000 |
(*The $100,000 saved each year from the annual $200,000 loan.)
We can see from this illustration that, while I might be able to increase my borrowing in the short-term in order to solve the problem, ultimately this type of living does not work when extrapolated through time. It falls apart within the first two decades; when taken to infinity, the consequences would be even more drastic.
Interestingly, this is one of the problems with the five-year and ten-year budget projections that the government utilizes. The fact is, it does not matter how our actions today affect the budgets in five-year and ten-year terms. The above-proposed plan had us rolling in the dough at five and ten years, and was revenue-neutral over fifteen years! The question is, and always should be, how does this plan affect us as time continues unbounded?
Now, some questions are more difficult to answer using such simplistic mathematics. However, the nice thing about limit computation is that it often makes the mathematics much simpler. For instance, in the quotient of polynomials we discussed earlier, we were able to calculate what would happen in the limit to infinity even more simply than we could have calculated its value at any given point. Why is this? The long-term behavior of many functions is controlled by only a few terms, while short-term behavior is controlled by many more. Therefore, by identifying those controlling terms, we can greatly simplify finding the limit. In the debt example above, the controlling term was the difference between income and payback. Since this was negative, the long-range value of the function was negative, even if we were rolling in the dough for a few years.
One note of caution is in order. To take a limit out to infinity requires that we have accurately assessed the true nature of the process. We can take a presumed relationship out to infinity, and it will certainly tell us something. We can only be assured of correct results, however, if our knowledge of the relationship itself is accurate. We should always be ready to correct our views should other factors arise that we did not see clearly beforehand.
In summary, we need to remember that as theologically-minded people, when we make decisions about normative behavior—whether it is about policy, legislation, rituals, or social norms—we should be careful to think through the effects of these normative decisions as they progress continually through time, not just how they affect individuals and society today. The mathematical concept of a limit can aid us in this seemingly daunting task and provides hints on how to extrapolate ideas to the bounds of infinity.