Closed-Form Solutions & the Case for God

I was recently watching a series of classroom videos on Finite Element Analysis (FEA), and the professor mentioned that FEA is not a classical closed-form solution, but rather an iterative, open-form solution. What on earth does that mean, and how could it possibly relate to looking at the case for God? Let’s work through that this week.

First, let me give some background so you can maybe see why a nerd like me would make that connection. A closed-form solution, in this context, is where you can simply solve an equation  to find the unknown variable. For instance, in my practice of structural engineering, the deflection of a cantilever beam may be something I need to know as I’m sizing the beam. If the beam conforms to certain assumptions like a constant cross section, constant material stiffness, a uniform load, and so forth, I have a simple equation: $\Delta = \frac{wL^4} {8EI}$. If it’s a concentrated load at the end, there’s a slightly different equation. These equations are each derived from beam deflection theory for a specific boundary condition, like a cantilever, or a simple span beam, and they provide exact answers. We engineers like exact answers. It’s nice to be able to say “this beam will only deflect 1.21 inches under that load, which is still acceptable.” I like closed form solutions because they are directly solvable for the variable I’m looking for, but sometimes, even with tables of equations for dozens of different conditions, there are no closed-form solutions, or they are too complex to use, or it would take a while to derive the equation from scratch. An open-form solution like the approximation methods used in FEA is iterative and relies on the results of previous attempts. FEA models a component like a beam with lots of little pieces that can each respond differently, so I’m not quite as limited by simplifying assumptions. Think of a beam made out of lots of LEGO® bricks.  Each brick (a “finite element”) is connected to multiple other bricks, and the total strength of the beam depends on the behavior of all of those individual connections. In general, the smaller the bricks, the more accurately you can represent the beam. But as the number of LEGO® bricks increases, the time to calculate all of those interconnections increases exponentially. That type of solution gets complex pretty quickly, and requires a computer for any problem worth solving. But it also doesn’t produce an exact solution. It iterates, or repeats the calculations with different input values until the successive estimates begins to converge. In other words, it runs through the thousands of equations over and over until the results aren’t changing much with each pass, and are within a tolerance the user sets for what is “close enough”. And what is “close enough”? That’s going to vary with the user and the type of problem being solved. Also, another engineer could try solving the exact same problem with a different mesh size (i.e. bigger or smaller LEGOs) and arrive at different results since it’s not just the beam properties that determine the answer now, but the modeling choices like mesh sizes, convergence tolerances, and iteration method.  So why would I want to use a complicated, inexact, and sometimes difficult to verify process like that? FEA lets me solve things I couldn’t otherwise. Some problems get far outside the simplifying assumptions of our various formulas, and FEA (done correctly), is the best option for finding a solution, even if it isn’t exact.

Now, what if our search for “proof” of the existence of God is like that open-form calculation? I often read forum comments from skeptics wanting “proof” of God’s existence: an end-all silver bullet that would provide a 100% certain, undeniable answer. And until they get that 100% certainty, they refuse to believe. But that’s like wanting that exact, closed-form solution to some complex engineering problem for which there is no formula. Also, even if I had a nice formula for something, those “exact” answers are often based on simplifying assumptions, such that the “approximate” solution from my finite element model of a complex design may actually match up better with reality. If I held out for an exact solution, I might never get my answer even though maybe a minute of my computer working through Newton’s Method will get me close enough to finish designing that component and move on to the next task. Moreover, we don’t normally expect anywhere near 100% certainty about anything else in life. Most decisions in life are made with far more uncertainty because, whether we know it or not, we use a process called “abductive reasoning”, or “reasoning to the best inference,” to arrive at a reasonable answer in the face of uncertainty or missing information. In the case of the existence of God, there isn’t one knockdown argument that yields certainty for everyone, but there are a host of different arguments that all converge on the same answer: the world we observe is the result of intentional, purposeful, goal-oriented interaction best explained by the God revealed in the Bible. It’s a cumulative case that becomes more and more certain as we see more lines of evidence and reasoning trending toward the same result. And at some point, which varies from person to person, the convergence of the objective evidence gets within our subjective tolerance where we finally have to either accept where the evidence is leading and bow the knee to our Lord, or deny reason itself to maintain our rejection of Him.

If you’ve been a skeptic toward God, I encourage you to look at your expectations. Have you set an unreasonable standard? Let me ask you, if Christianity were true, would you believe it? If not, that should concern you. If you’d answer “Of course, but it’s not true,” think about what evidence you would accept. Don’t be content with “I just need more”, but seriously contemplate what your “convergence tolerance” is. What kind of evidence will you accept? How much is enough? Why do you discount some types of evidence, and are those really good reasons to do so? Will you gamble with your eternal soul over a 10% uncertainty? How about 1%? Or 0.01%?  Interestingly, in FEA, a sufficiently accurate result for most structural problems can be had in seconds to minutes, while asking for more accurate approximations can make the solution time increase exponentially, requiring hours to days, and often with only marginal differences that don’t change the final design and don’t justify the extra time. Tomorrow’s not guaranteed for any of us, so don’t waste your life demanding the nth degree of proof when the evidence you’ve already seen is sufficient to know the truth and decide accordingly. As the Bible warns, “Today if you hear His voice, do not harden your hearts.” [Heb 4:7]