This post was inspired by a recent twitter conversation between @woutgeo, @ddmeyer, @cheesemonkeysf (cool handle!), and myself. The 120-character at a time conversation revolved around a comment from @woutgeo:

“…am still mulling whether modeling Q’s can have correct, known answers”

Since this seems to be a common point of confusion, or even contention, I thought I’d talk about this idea a little today. That is, what do people mean when they make statements like “Modeling problems don’t have a single, unique, correct answer.”? If you read the introductory chapter to the new Annual Perspectives in Mathematics Education (APME) volume, Mathematical Modeling and Modeling Mathematics, you’ll find versions of this statement in several places. I’ll highlight two:

There are multiple paths open to the mathematical modeler, and no one, clear, unique approach or answer.

Mathematical modeling authentically connects to the real world, starting with ill-defined, often messy real-world problems, with no unique correct answer.

What do people, in particular, what do those of us who do mathematical modeling professionally, mean by such statements? What are we really trying to say with statements like “no unique correct answer”? There are actually multiple levels to this point, and I think it is worthwhile exploring a few of these levels here today.

At the simplest level, such statements express the point that the answer to a modeling problem is not like the answer to a typical textbook or classroom math problem. When we think of the idea of an “answer” to a math problem, due to many years of repetitive training, what we most often visualize is a number. “The” answer is 586, or 7, or 2+3i, or \sqrt{17}, or something like that. Perhaps, if we’re a bit more deeply immersed in algebra, or trigonometry, or calculus, our default vision of “answer” might be more like x^3 or \sin(3 \theta) or \frac{1}{x} or some such expression. Note that this default vision of an “answer” is some form of mathematical object and tied to that, perhaps so intimately that we don’t see it, is the idea that this answer is easily checked. It’s the result of “doing the math correctly,” and hence, of course, we should only get one such answer. But, when we talk about the answer to a modeling problem, these are not the types of objects we’re talking about. In one very important sense, the answer to a modeling problem is a model. And, here is the first place where this idea of “no unique answer” comes into play. Because models of a given real-world situation can be constructed using wildly different mathematical tools and are based on assumptions made by the modeler, it is often the case, likely even, that two modelers approaching the same problem produce different models, i.e., different “answers” to the same modeling problem. This is the point that the first of the two statements from the APME volume above is really making.

But, there is another level to this idea of “no unique answer” that’s worth exploring. The second statement from the APME volume mentioned above points to this next level. Let’s examine the statement again:

Mathematical modeling authentically connects to the real world, starting with ill-defined, often messy real-world problems, with no unique correct answer.

Here, note that “answer” is not referring to the answer to some modeling problem in the sense discussed above, but is referring to the “answer” to a real-world problem that the modeler is trying to address. Here, the notion of “no unique answer” points to the messiness and inherent uncertainty of the real-world. Because we can never hope to capture all of that messiness or tame all of that uncertainty, our models always remain provisional, approximate, and open to improvement. That is, we obtain “an answer” to our problem, and we evaluate whether or not it is good enough for our purposes, but we never get “the answer.”

Another way to see this is to always keep in mind that mathematical modeling is ultimately, a process. Hopefully, it’s a process that draws us closer and closer to the truth, but like an asymptote, never quite gets there. You can see this point of view and get a sense of the notion that there isn’t one right “answer” or model, but rather a never-ending array of possible “answers” or models, in the CCSS for mathematics:

In situations like these, the models devised depend on a number of factors: How precise an answer do we want or need? What aspects of the situation do we most need to understand, control, or optimize? What resources of time and tools do we have? The range of models that we can create and analyze is also constrained by the limitations of our mathematical, statistical, and technical skills, and our ability to recognize significant variables and relationships among them. 

There is a wonderful one-paragraph story by Jorge Luis Borges that is related to this second point. It’s called “On Exactitude in Science” and in this story, Borges explores the idea of modeling and uses absurdity to remind us that useful models are always incomplete. His story is short enough to reproduce here:

In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattered Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is no other Relic of the Disciplines of Geography.

The point, of course, that Borges is making is that a big part of science, a big part of understanding the real-world, is about making models and that the only perfectly complete model of reality is reality itself, but that such a model is also completely useless! This is true whether we’re talking about physical models like maps or more abstract mathematical models. The magic is that maps and models, approximate, imperfect, and ignoring vast parts of reality are incredibly useful and are our best tools for understanding, predicting, and controlling that reality.