Mathematics and God Mathematics offers


Mathematics and God

Mathematics offers a lot of wonderful and weird corners to kick around in thinking about God. I'll share one of my favorites.

It has been shown that in iterative systems of non-linear equations certain sets of things can happen. Systems can immediately stabilize, or have an "orbit" of 1. That is if f(x)=x^3 and you start with a value of 1 and use the solution for the next iteration of the function, you will discover that the solution is always 1.

Another thing that can happen is that the value can waver between 2 numbers and you can have an orbit of period 2 in which the two are always feeding back and forth, or you can achieve a "hopf bifurcation" in which as the value approaches the bifurcating point it eventually "lands" on one of two options and then follows a trajectory from that point, no longer wavering between two values. Up to the value of bifurcation, there is a tendency of the value to "wobble" back and forth. Many other things can happen as well. There are stable orbits of periods 2, 4, and higher numbers, where the values in the system will always be one of two or four numbers.

But the number three is particularly important. If a system achieves an orbit of period 3, it is considered the gateway to chaos. Technically chaos looks a lot like randomness, but it is nothing of the sort. Chaos might be better termed weak determinism. That is, if you knew every factor affecting the system, and factored them all in, you would be able to predict what would happen for the next two or three cycles of the system, but then your predictions would become less and less accurate. This is one reason why long-term weather forecasting is unfeasible. We don't know all of the factors affecting the system AND even if we did, our ability to model weather mathematically has a finite limit defined by how many decimal places we use to calculate the next cycle, and it is in a place beyond the last decimal used that there is enough variation to create wildly different scenarios.

I once modeled the logistical difference equation using double precision (so about 16 decimal places) and starting the model with an initial difference of point 15 zeros 1. What this means is that I ran a "mathematical experiment" twice. You plug a value into the equation, then you take the results and put them back into the equation. You do this for as many rounds as you like. I did this for one hundred rounds on the first trial. I started the second trial with a value infinitesimally different. After only 5 repetitions, small differences had emerged. At ten repetitions the graphs of the two experiments looked like I had started with diametrically opposed numbers. From that point on, there was no similarity in the graphs.

Now, what I find most interesting in this is the number of different systems that can be modeled using non-linear dynamics. In fact, it seems, almost anything can be modeled in this way. Computer "random-number generators" are in fact, weakly deterministic in this way.

For the theologically minded this whole complicated mess appears to suggest that behind the apparent chaos at the surface of things, in fact, most systems are "weakly deterministic" that is weakly predictable, given all possible information, and ultimately guided very strictly along their paths, although these paths are never seen.

Okay, so I've guided you through the morass to this conclusion--it appears that many systems are both governed and guided by some overarching considerations and if one could know all possible parameters affecting the system at all times, one could predict what would happen in the system.

To me, that speaks profoundly of the action of God. He is, after all, omniscient and omnipotent. So He indeed can know every factor that affects a system, and He can know the outcomes that are hidden from all of us. Now, admittedly, we are talking merely metaphorical references here. But the study of Chaotic Dynamics was only one of the things that profoundly convinced me of the presence of God's guiding hand in all that is.

This is NOT a proof of the existence of God. Or, if it is, it is flawed in the way many such proofs are by presupposing elements that would tend toward that existence anyway. In other words, if one starts by believing, this simply is an element of reinforcing belief, but if one is an unbeliever, this argument by analogy and metaphor is not a convincing proof, or even a very good support for one's contention.

But I am stunned by these "hidden jewels" that open up worlds of thought, speculation, and appreciation for the Divine Grace that penetrates and permeates all that is. In thinking through these complex questions, one catches a glimpse of Divine Action as one allows the Divine to enter into one's way of perceiving the world. As I said, in the course of my studies, this was only one among many of the marvelous "proofs" or evidences of God's action in the world. I hope in the future to share others.

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This page contains a single entry by Steven Riddle published on October 22, 2002 7:40 AM.

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