A Mathematical Model of Spiritual Development

I was inspired after a recent conversation (on humans’ psychological relationship with our own spiritual development) to do something I have been meaning to do for a long time, namely to put into practice some of the methodologies that I considered very powerful after reading Alister McGrath’s A Scientific Theology trilogy. Essentially, I want to bring the well-known concept of mathematical modeling to bear on a theological issue (spiritual development). In doing so, I am hoping to achieve what all models are meant to achieve–an analogy that captures as much of the data as possible while providing a simple, coherent, abstract picture that can be investigated in its own right. I am emphatically not trying to define or otherwise delineate all aspects of spiritual development via such a model; spiritual development is the reality, and any model is simply a way of attempting to capture that reality and present it in such a way as to spark further exploration. That is all! Indeed, just in virtue of the fact that the model I am about to propose is mathematical, we should not expect it to speak to all (or even most) of the phenomena surrounding spiritual development, which exists in a stratum of reality distinct from that of mathematics. Still, some surprising correlations can be made, and I hope these may be found useful.

These, then, are the data I am taking as given, and trying to model (notice that they are not physical data, but theological and psychological, even moral data):

  1. Good and evil are infinitely incommensurable.
  2. If we presume that there are beings which exhibit infinite goodness or infinite evil, these correspond to God and the devil, and “infinite goodness” would describe the character of God, as would “infinite evil” describe the character of the devil.
  3. The state of spiritual development of a human being can exist on a spectrum anywhere between infinite goodness and infinite evil (though it cannot exist at either endpoint). This datum reflects the observation that humans are never found to be perfectly good or perfectly evil (excepting, in Christian theology, the special case of Jesus).
  4. It is easier to be morally mediocre than morally perfect
  5. Incremental advances in spiritual development (morality) are easier at a “less moral” stage than a “more moral” stage.
  6. Paradoxically, one’s awareness of one’s own depravity (immorality) becomes greater the less depraved one actually becomes.

Now we may take a look at the model. I propose that all these data may be plotted in various ways around the function y = -a/bx and various ways of graphing it: (Please note that all of the graphs here may appear too small to read, in which case you have simply to click on them to load a larger version)

We have chosen for our values of a and b the number 1, since, as we will see, the actual function slope will not matter as much as its relation to certain factors. If it were possible to quantify absolutely any of what follows (which appears to be doubtful), we could experimentally determine a and b, but that is beside the point here.

Here are the initial correspondences (INF := infinity):

  • The values of the function y = -1/x correspond to possible states in the spiritual development of a human being.
  • Perfect goodness corresponds to the y axis–specifically at the point x=0,y=INF.
  • Perfect evil corresponds to the x axis–specifically at the point x=-INF,y=0.

And we can already begin to see how the model agrees with some of our data:

  • Perfect good and perfect evil are infinitely incommensurable (data 1 and 2 above): First, they are located at points infinitely far from each other in 2-dimensional space. Second, they are not even in a continuous spectrum of points–no line which connects them can be drawn, for such a line would correspond to the impossible function y = x + INF. Moreover, they exist in two completely orthogonal dimensions.
  • Our observation that human spiritual development can be on a spectrum anywhere between, but not including, perfect good and perfect evil (datum 3 above) is captured perfectly by saying that it is plotted on the graph y = -1/x. The curve never meets the x and y axes, but passes through every other x and y value (though of course not every x/y combination).

Now let us extend the model. Let us define the notion of “spiritual work” to mean the effort (whether internal or external, and relative to persons) required to move along the spiritual development curve towards good. We can quantify this notion, determining the specific amount of work required to move between two points on the spiritual development curve, using the integral of the curve. The integral of the curve between two x-values is the appropriate mathematical construct for our purposes because it tells us the area underneath the curve between those x-values. This area, importantly, reflects our observation that incremental advances in spiritual development become more difficult further along the curve. Let’s look at some examples.

First, take a graphical depiction of the area underneath our curve between the (arbitrarily-chosen) x-values -8.0 and -5.6 (for a span of length 2.4–corresponding to the amount of spiritual “progress” made):

The area depicted happens to be about 0.36, so we can say that for this arbitrarily-chosen progress of spiritual development (a progress corresponding to the number 2.4), the spiritual work required is 0.36. Now, it is important to note that neither the x-values chosen, the “progress” value of 2.4, nor the “work” value of 0.36 have been given any real-world correspondents. The main thing here is that they are related to each other in a certain way–mapping such quantities to the reality of spiritual development is outside the scope of this model (an important limitation of it), and possibly outside the scope of possibility in general.

Now, let us look at another progress-graph, this time progressing from the x-value of -3.2 to -0.8. Notice that again we have a total progress of 2.4.

As is visually obvious, the area (corresponding to work) under the curve between these points is much greater. As it happens, it is 1.39, almost 4 times greater than our previous work value of 0.36! We could, of course, continue to belabor the point with more and more examples, culminating with the discovery that, at any point where we wish to integrate from one x-value to 0, the area under the curve becomes infinite (in terms of work: impossible). Also needless to say, even the particular way in which 1.39 is related to 0.36 is not important–what is important here is that there is an increasing difficulty (in terms of work) as we move closer to pure goodness.

It appears, then, that the model accurately handles our observations of how spiritual progress is made, and its corresponding difficulty (data 4 and 5 above). Already some theological fruit is ripe for the plucking as a result of the model, particularly in regards to work: statements in the New Testament such as “With man it is impossible, but with God all things are possible” take on new meaning as we see via the model that (if it is accurate), becoming purely good is indeed impossible for us. Perhaps, then, these words of Jesus are not referring to such actions as we would normally consider “supernatural” (throwing mountains into the sea, having power over the weather, etc..), but rather to the spiritual development we normally find impossible. Alas, in order to stay on task, we should save such ruminations for a later essay.

Now we come to the last datum under discussion (6 above), namely the psychological observation that, the closer we get to God’s nature (perfect goodness), the farther we often feel from it. A cursory reading of the Apostle Paul’s epistles is enough to prove that this feeling has been with Christians since the beginning. How can this psychological fact possibly be modeled using the same graphs with which we have been working? The key here is in the realization that the datum refers to our own perspective, our own awareness, rather than a definite aspect of the reality of spiritual development. That is, in reality, the closer we get to God’s nature, the closer we are to pure goodness, however we might feel about it. All this amounts to saying is that, if we have made spiritual progress, we have made spiritual progress! The tautology arises because we have simply defined “spiritual progress” as “growth in character which brings that character closer to God’s”. So, given that the datum under discussion is not about our actual progress on the path of spiritual development (the actual location on the curve in the model), it stands to reason that there should be no “real” change to the model necessary to accommodate the datum. And, in fact, this is what we find.

It turns out that, just as the datum concerns perception in the reality of spiritual development, so it concerns perception (better called “perspective”) in our model. Specifically, our feelings about our spiritual development (i.e., our closeness to God) can be mapped to a relation of (a) our position on the curve, and (b) the specific way the curve (always mathematically the same) is represented to our visual senses, via properties of whatever “viewport” (screen, paper) we use to look at the curve.

Take, for example, this graph (the same as our first):

The dot at -0.4, 2.5 corresponds, let’s say, to our current position on the spiritual development curve. (If only we could be so lucky in real life!) Now there are two values we need to fill out the relation between (a) and (b) above. First, there is the x-distance between where we are and pure goodness (call it the “actual remaining progress”). We get this by subtracting our x-location from the x-location of pure goodness; in this case it’s the value 0.4. So close, yet so far away! Next, we need what I will call the “width of the viewport”. That is, the maximum x-value shown on the graph minus the minimum x-value shown on the graph. In this case it is 10. Finally, the value we want is the actual remaining progress divided by the width of the viewport (in our case, 0.4 / 10.0 = 0.04). We can call this value the “perceived remaining progress”. So for this graph, we have a perceived remaining progress of 0.04.

Now, let’s look at another graph:

The dot is now at -0.38, 2.63, meaning that the actual remaining progress is now less than in the previous graph: 0.38. We have become better! However, the dimensions of the viewport have changed. Even though the curve looks much the same, the numbers on the axes give the lie: the width of the viewport is now 5, not 10. (To further deceive, I made it so that the x and y axes are no longer equivalently scaled). In other words, we have “zoomed in”. The perceived remaining progress is then 0.38 / 5.0 = 0.076. It is almost twice the perceived remaining progress of the previous graph, which reflects our observed intuitions that spiritual progress, coupled presumably with an increased awareness of the goodness of God, can actually result in a greater perceived remaining progress (a greater distance from God). This “increased awareness”, then, is modeled here by the act of “zooming in”, which we have already more rigorously defined. The net result is that, even though in the second of the two recent graphs we have made progress, we appear to be further from our goal.

In order to determine what laws, if any, govern the moments at and degrees to which this “zoom” takes place, we would need to engage in empirical observation. We could start with the basic assumption, for example, that every decrease in actual remaining progress brings about a corresponding decrease in viewport width, so that perceived remaining progress is always more or less the same. Of course, this assumption is probably erroneous, if only because it ignores the spring-like oscillations that relate changes in viewport width to changes in actual remaining progress. At any rate, the point here is just to show that the model can handle the data under consideration, regardless of whether we can somehow shoehorn it into another model which can make accurate predictions.

Now, since we have seen that the mathematical model I proposed does indeed handle data 1-6 above, we are left with a few questions. First, is there any feature of the realities of spiritual development as we experience them that cannot be countenanced under this model? We already mentioned the theological claim that Jesus was both a human being and perfectly good. Does this wreck the model as a whole, or can we make special exception for him given his somewhat transpositional nature (much as a physicist does not treat very small quantum particles with the same rules as macroparticles)? In other words, what are the limits of this model?

Second, are there any more correspondences that we could make under this model? Perhaps, for example, we could do more with the y-dimension, treating it analogously to distance from a certain massive object. Then we would employ the physical rules of gravitation to explain why there appears to be a constant “downward” (evil-ward) force with regard to morality (explained in Christian terms as a propensity to do evil as a result of the Fall). For another example, we could attach meaning to the derivative of the spiritual development curve at a given point (the slope of the line tangent to that point on the curve), saying it represents somehow “spiritual momentum” or “commitment”, on a scale of 0 to infinity.

Lastly, in order to make the model useful in application as well as reflection, is there any way to empirically determine values to replace any of the arbitrarily-chosen I have used here? In other words, is there any way to make these values correspond to actual states of spiritual development? No doubt there are some features of spiritual development that are general and could be explored in the same way by everyone, but some are no doubt relative to individual persons. Could the model be extended to include, say, “spiritual constants”, much as the physical laws of motion are extended to handle constants of friction for certain surfaces? Or on the other hand does the very nature of the phenomena under investigation preclude any kind of testable or repeatable introspection (or examination of others)? I am forced to admit that I am not optimistic about finding positive answers to these questions.

Still, I don’t think the model wholly useless, and believe that (as good models should), it can provide clear abstraction of some points of the spiritual development process which help us to better understand what is going on, even if these abstractions necessarily have their limits. This clarity is, of course, what all mathematical models grant to investigators who use them to model reality. At different times and for different strata of reality, they cohere more or less exactly with their subject phenomena, and I hope in this essay to have illuminated at least one possible path of researching and modeling spiritual development that uses the resource of mathematics to its advantage.

By Jonathan Lipps

Jonathan worked as a programmer in tech startups for several decades, but is also passionate about all kinds of creative pursuits and academic discussion. Jonathan has master’s degrees in philosophy and linguistics, from Stanford and Oxford respectively, and is working on another in theology. An American-Canadian, he lives in Vancouver, BC and has way too many hobbies.

6 replies on “A Mathematical Model of Spiritual Development”

A minor detail is that I think you meant ‘first derivative’ when you said ‘second.’ The first gives the curve’s slope while the second gives the concavity, which itself can have a spiritual interpretation according to your model (the inflection point indicating the reversal of an accelearting or decelerating trend in spiritual progress).
Second, while you adequately explain how moving ‘forward’ on the curve results in the appearance of being in fact further behind on it, it’s still hand wavy. It requires skewing the aspect ratio of the graph’s axes, and the explanation that spiritual progress corresponds to a scrunching of the x-axis seems a bit contrived. I might as well plot the y-axis on a logarithmic scale and say I’m making linear spiritual progress.
It’s fun, but honestly I think your powers would best be used elsewhere 🙂

Actually, I think you might be misunderstanding the model.

As to the first point about derivatives, I do mean second derivative, though I agree that it is unclear why that should be specially important. The first and second derivatives of a x^-n based formula are not appreciatively different (the way they are for, say, a x^n based formula) between -INF and 0. I was just thinking of more mathematical properties that might have meaning in the model. Not wed to second derivative at all!

As to the second point about the perceived remaining progress, you are right that it’s “hand wavy”. That’s the whole point! If it weren’t so, my model wouldn’t be a good one. It’s the same way in real life–as we progress in character towards God, we don’t actually become farther away from his level of goodness; it merely seems like it. This “seeming” is adequately captured by a skewing of the lines of the graph. Such is what is essentially happening in reality–our perspective of the same data is being skewed (perhaps rightly). In fact, maybe the only fully “true” perspective is that where the point (0, INF) of goodness takes up the entire viewport (in other words, where we are infinitely focused on God). From the perspective of this point, since each point is (mathematically speaking) infinitely far from the one next to it, no matter how close we may approach it on the spiritual development curve, we are infinitely far away.

As to third point about remaking the model on a logarithmic scale, you could indeed plot all of this on a logarithmic scale and get a completely different picture. However, I claim that it would not be as natural a model, and to ensure the correspondences with the data, you would need to essentially translate through the model I am proposing anyway. (To drive home this point I should probably modify datum #5 to include the surely-admissible psychological component that the difference in difficulty feels exponential, not linear).

One assumption you seem to make is that humans start out at (-INF, 0) and spend their whole lives on the curve you drew, trying to get as far right as they can. But do humans really start out at (-INF, 0)? Why don’t we start out at (0,0)?

Also, why are good and evil orthogonal? Also, if they’re not different directions on a line, why does it make more sense to represent them in 2 dimensions, instead of more?

Some classic worldview questions.

This is an interesting approach. From an empirical study standpoint, is it possible to design a mixed-model research project that covers the quantitative and qualitative aspects? Is there a validated survey instrument one could use? Pavi and I worked a bit on a proposal for spiritual assistive technology and having a good, assessment instrument was key to verifying results. Thoughts and advice are welcomed.

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