This is a paper that I wrote a couple of years ago that resulted from some of my observations about mathematics and the human condition. It does not contain any deep theological content (as my posts rarely do) and is probably of little interest to the more theologically inclined folks in my readership. I have wanted feedback on it for some time, and thought that I would give it a chance here.

Approximating the Divine

Applying Numerical Methods to the Human Experience

As one studies elementary mathematics, he is faced with hundreds upon thousands of absolute statements about numbers, ratios, factors, and numerical operations. To prove his worth, he must apply these abstractions in simple word problems.

The train leaves the Chicago station at noon heading south at 50 miles per hour. At 1:30, a northbound train leaves Peoria at 45 miles per hour. If the two stations are 160 miles apart, where and at what time will the two trains collide?

We remember these conundrums all too well. The young student’s every challenge or exercise has a very particular end with very restrictive rules. There is a single right answer to which only the teacher has access. He must obtain this answer via a prescribed method – showing his work, of course. And he must express his answer using the proper cryptography and measurement, e.g. the answers shall not be given in Roman numerals and hectares when square inches are requested.

These rules stick with the student for most of his life as an attendant of primary and secondary school. Later on in his courses, he encounters vast quantities of formulae to memorise, such as the volume of a cylinder, the surface area of sphere, or the properties of a right triangle. This level of mathematical training, whilst being the most advanced that many attain, differs little from the rote memorization of times tables. As this remedial education approaches its terminus, the student decides either to continue with mathematics, or to abandon it entirely.

Thus, the picture of mathematics is, for many, quite dismal. Math people are seen as uncreative and boorish – how could anyone want to do all those silly problems for the rest of his life? People who ask such questions have likely never progressed passed the basic algebra. I do not say this to depict a lack of aptitude on their part – not at all. Rather, I would have the reader try to appreciate the vastly different paradigms used in the higher mathematics, of which the calculus stands firmly at the base.

Ideal or Real

In geometry, everything is ideal. Circles are perfect, right angles are exact, lines have zero thickness, planes extend to infinity, line segments are infinitely divisible, and points take up no space. Theorems are taught with absolute certitude (as well they ought) with the understanding that the fundamental axioms, whilst improvable, must be accepted if anything is to be proven.

Many people do not do well with the geometric proofs; they simply cannot see them mentally. They just see lines and curves on paper, and poorly labelled ones at that. They cannot identify with a world comprised of solid black lines against a stark white backdrop bounded by straight edges. To them, it is just not real. To others (like myself) it is as real as I make it. As an engineer, I see geometric patterns everywhere. When visiting an art gallery, the hidden construction lines of the building’s infrastructure and architectural design pop out at me revealing structure inside structure, whilst all the insignificant details – details that everyone else paid to admire – fade into oblivion.

As people, we must have purpose in what we do; we have no interest in solving problems without it. If mathematics were limited to calculating the performance of ridiculous trains that only travel at one constant velocity, I would not want any part of it myself. When any problem is excessively idealised to the point that it no longer reflects reality, the end result is the removal of useful purpose from the problem. Solving such a problem becomes a waste of time.

This is where mathematics is misunderstood. Geometric proofs are not employed by full time mathematicians attempting to document all extant congruencies of a right triangle inscribed in a semi-circle. The purpose of the rudimentary math problems is not to determine the previous arrival times of poorly engineered vehicles. Such problems are no more representative of mathematics than a kindergartner’s phonics primer is representative of 17^th century English literature. Elementary math problems are intended to teach the most fundamental mathematical concept – the art of problem solving.

The Closed Form Solution

There is some initial shock when at first a student begins his study of the calculus. He will see that so many of the rigid rules learnt in primary school are broken on an as-needed basis, if not discarded altogether – he is now in a completely different world. For instance, he discovers that sneaking in a calculator for a test is of no help. This realisation may happen slowly, but one day it occurs to him that he hasn’t read or written an actual Hindu-Arabic numeral for six months. He learns that he can now take the square roots of negative numbers and has no qualms about using the phrase “imaginary numbers” with a serious tone of voice. For problems involving integration, he finds that there are many ways to arrive at a single solution. Moreover, differential equations frequently have more than one legitimate solution. Some solutions require three pages of derivation when others may be obtained in a few lines. And slowly, the picture of true mathematics is illumined.

But the most important of all is the profound realization that most problems have no solution. The problems in algebra textbooks are contrived such that the same simple approach always results in the correct answer, which is usually a nice, round number. However, problems with real world applications are rarely so simple.

The answers to calculus problems are rarely numbers; they are functions of variables. If a function can be obtained and expressed purely symbolically, then the problem is said to have a closed-form solution. The closed-form solution is a highly sought-after ideal.

And it is quite an ideal. Closed-form solutions reveal more than simply numerical answers; they tell the whole story for the generic problem. They frequently uncover previously unseen relationships and allow for accurate predictions of the behaviour of some physical system. When understood and used properly, they can serve as abstract models in lieu of empirical data. Unlike the extremely limited solutions that are so typical of elementary maths, closed-form solutions are applicable over a range of parametres and conceivable cases, including every possible permutation within the constraints defined.

To illustrate the point better, let us look at an example of a closed form solution to a simple engineering problem:

In the figure above, a cantilever beam is shown with the outline of its deflected state after a load has been applied.

In the above equation, u(x) is a function representing the deflection of a cantilever beam of length l with respect to the x‑axis. The x term is a variable representing the entire set of locations along the length of the beam; an x value of zero means the root of the beam, i.e. where the beam is clamped, whilst an x value equal to l means the free end of the beam. The EI term in the denominator of the right-hand side of the equation represents the bending stiffness of the beam; the higher the stiffness, the lower the deflection. Finally, P represents the load applied to the beam. Since it is in the numerator on the right-hand side of the equation, the higher the load is, the higher the deflection will be.

Notice now that given any cantilever beam of constant cross section, with any length and any load, this equation can be used to determine the deflection at any point on the beam.

But closed-form solutions are limited – and the one above is certainly no exception. Closed-form solutions for simple one and two dimensional problems are useful and readily obtainable. But as the mathematician improves his model to account for more “real” phenomena (i.e., as he considers a less idealised problem) the closed-form solutions become all the more elusive. The more dimensions included and boundary conditions considered, the more the closed-form solutions that form the basis of his model increase in complexity and require more time to derive. And as they become more complex, they become all the more unwieldy.

In the case of the deflection equation above, it is limited to modelling the behaviour of a beam solely in its elastic state. Once the load reaches a certain point, the stresses in the beam cause the material to yield into a plastic state. If enough load is applied, the beam fractures. Outside of purely elastic behaviour, the above equation does not apply at all.

It is worth noting that the deflection of a simply-supported flat rectangular plate under a uniform load has also been solved in closed form. But truthfully nobody ever uses the solution except in teaching. This is primarily because it is so idealised compared to the problems that are being solved currently (rarely are the edges of structural plates simply-supported in actuality). Accurately modelled problems in modern engineering applications simply cannot be solved in closed form.

Numerical Approximation

What do you do about all these problems with no solutions? You do the only human thing: approximate it. And there are as many ways to approximate solutions as there are humans. Some approximations are good and others are not so good. What makes an approximation good depends largely on what the immediate goal is, what the time and space constraints are, what the budget can tolerate, and just about everything else in the world. In short, approximations, like all of the fine arts, require a keen sense of resources and a healthy dose of creativity.

Approximations of complex problems can usually be evaluated with much fewer resources and a lot less time than actually working out the full closed-form solution. One of the common approaches is iteration/recursion. Starting with an initial value, successive iterations are performed, each iteration yielding a more accurate value than the previous one. Thus, regardless of the initial value chosen, if enough successive iterations are executed, you will eventually be so close to the answer that it is good enough.

Consider a man who is mulling a problem over on the way home from work and he wants a quick answer. One approximation method may converge quickly with relatively few iterations, but is too complicated to implement on a piece scrap paper while riding the subway. Another takes at least three dozen iterations, but is simple enough that he can do the calculations in his sleep. Given such a straight-forward scenario, the decision seems obvious.

The predominant methods of real analysis in to-day’s scientific and engineering fields are numerical methods. They’re faster and more generic, they get us what we need, and they do it well. But the real beauty of finding (or deriving) an approximation can be appreciated best by a numerical analyst who understands the fine art that it is. Parameter constraints must be carefully chosen; low-level computer programmes must be written efficiently; initial values should be carefully defined. Such aspects of approximation require finesse, patience, and experience.

And there is no “one right answer” to the solution. The more assumptions shared in common, the more the solutions of various approaches will be in agreement. They all may end up within ten per cent of each other after all the solutions have converged and everyone goes home happy.

Ten per cent? Yes, sometimes more. The real irony of it is that even if somebody had derived a closed form solution to a perfectly defined real problem, he wouldn’t be able to duplicate the results of the analysis in a real world experiment without an appreciable amount of human error. The instrumentation and measurements alone cost most laboratories ten percentage points of error. When the dust had settled, whether a scientist perfectly duplicated the experiment or not is something he would never know. It is as if we were destined to always be approaching and never arriving like Achilles in pursuit of the Tortoise in Zeno’s famous paradox about motion – there is always half the distance to go.

Semantic Approximation

Approximation, then, is the human condition. We are always approximating no matter what we may be doing. Words cannot suffice to explain how we feel much of the time, but we use them anyway, often to excess when our thoughts are the most disjointed. We keep talking and talking, trying to find that word on the tips of our tongues, yet failing adequately to express (approximate) our thoughts. In the end we may confess that “words are not enough” to describe how we feel – and yet that very statement may be the greatest approximation possible. The fact that one’s thoughts and feelings are oftentimes ineffable seems obvious – what may not be as obvious is the striking correlation between numerical methods in mathematics and human conversation.

The concept of interrogation or interview is the most obvious example. One party is seeking to elicit a certain amount of information from the other. The first question is like the initial value. When the first question is answered, one iteration is completed. Based on the first answer, question two is formulated and so on, until one of two things happens:

1) Convergence. The questions and answers slowly hone in on a solution of sorts, which converges when the questioner is satisfied. The one being questioned may be lying the whole time, and a solution may still be found, provided the questioner’s approach is sound.

2) Divergence. The questions and answers may initially appear to be moving forward, but begin to meander around many irrelevant topics. Both parties begin talking past each other and frustration builds. Rashly spoken words are interjected and misunderstood, resulting in further confusion and an eventual breakdown of communication. The two may be well meaning and love each other and still this happens, as anyone who is married will agree.

Obviously, other outcomes are possible, but everyone knows how good conversations can bring resolution, and bad ones can leave everyone discouraged. Sometimes amidst such frustration, we wish that there were some simple and easy approach to communicate with people – if only we could get our hands on a semantic closed-form solution. And in spite of such shortcomings, we pursue until we find the resolution we are longing for.

Consider the proverb of Solomon: “To speak a word in due time, is like apples of gold on beds of silver.” (Prov. XXV:xi) Notice what high esteem he has for a man who can speak accurately and in season. He is not referring to empty talk or idle chatter, lofty verbiage or mere flowery language. He is specifically praising the man who can, with a single, well-chosen word or phrase, bring resolution out of conflict, provide comfort in sorrow, or give instruction amidst ignorance. To carry out the analogy, this man can bring about convergence when all other methods have failed to do so. This man is praised so highly because he can transcend the human weaknesses that impair communication and offer, in a single iteration, an approximation sufficiently crafted for the situation.

Still Human

Everything we do on this earth involves this relentless pursuit for perfection. In a sense, it is our conscious self-awareness that exposes our plight, and it is the very knowledge of ourselves as imperfect beings that seems to fuel our desire to continue the search for closed-form solutions. But at the end of the day, if we ever do decide to apply the closed-form solution to a real life application, we still end up plugging values into a formula, and generating an approximate numerical answer using a calculator or the back of an envelope.

But there is something inherently transcendent about closed-form solutions. They are, in short, a picture of the Divine. They are a metaphor of God’s certainties over all time and space, a reminder of constancy in what sometimes appears to be an inconsistent universe. If a closed-form solution did exist for all the physical forces of the universe, we would not know it if we saw it – it would be utter nonsense to our imperfect minds. We, as humans, are always striving for the Divine, ceaselessly running after and working for a goal that we know will not and cannot be attained in this life. And yet if we abandoned our perpetual quest of striving for that which we know is impossible, we would be less than human, no better than beasts.

For centuries, saints have searched the great volumes of Holy Writ, seeking to describe God and all His greatness, and while they have by no means failed, their best results are only approximations – approximations that get closer and closer with time and energy – but still approximations nonetheless. The beauty of the process can be seen clearly throughout the history of the church, from the ante-Nicene days all the way through II Nicæa, the Lateran Councils, and beyond. And after millennia, the Church has not grown weary of her pursuit of the Divine, the ineffably sublime.

So we continue to fill up scratch paper with pencil markings and cover our trousers with eraser shavings, fearlessly in search for the closed-form solution, knowing that when we get there, we will have but a small inkling of an image of a picture of a piece of the mind of God. That is, provided the axioms used to arrive at the closed-form solution are true to what the closed-form solution is purported to model, which is what, unfortunately, cannot be known.

So then, what is a closed-form solution really? It is mankind’s most brilliant approximation ever.