Mathematical Modeling

The purpose of models is not to describe reality but to clarify the conditions under which a statement is true.
- Robert Skidelsky, Emeritus Professor of Political Economy at the University of Warwick

Models for population growth, too, are brushed aside by some ecologists as being far too simple to be real. But the models are intended as thought experiments, and not as pictures of real life. A model must be properly reduced, to the point of caricature, for the decisive elements to stand out…. Abstraction is always caricature: an exaggeration of essential traits, or at least an omission of irrelevant details. [emphasis mine]
Karl Sigmund (from Games of Life: Explorations in Ecology, Evolution and Behavior)

Between the idea
And the reality

Falls the Shadow
TS Elliot

Introduction — the basic concept.

It’s 1975. My college roommate, one James Hoff, is a math genius. I’d always been able to do math and had steadily taken math courses until, in an existential crisis midway through my freshman year in college, I stopped. From there on it was only the occasionally required stats course for me.

In all that math, I’d never achieved any real understanding of what it was all about. Even stats which theoretically describes the world, while eminently doable cookbook style, gave me no insight into what was really up.

Now that was changing. We’d smoked a doobie and Mr James Hoff, math genius, is running the voodoo down.

There are two components of a model, he explains:

First, there are the formal properties of the mathematical model.

Take your standard Bell curve. There are a variety of measures that can be used to describe the shape’s relative height, spread, and so on giving us common stats such as standard deviation or median.

Second, and this is always the tricky part, there is the likelihood that this formal model applies to the matter under consideration.

Sometimes this is trivial. (Even the trivial is often non-trivial, but let’s not muddy the water, yet.) Clump a bunch of folks together sorted out on a field and clustered by height from shortest to tallest. Look at it from above. Chances are you have a Bell curve. The clusters of medium height folks will be thickest tapering in both directions until you find only one or two people on either end in the ‘very short’ or ‘very tall’ clusters.

What it all means is: if there’s a reasonably high chance that your model applies to a situation; then you can meaningfully use the formal properties of your model to talk about the situation…hopefully even generate testable hypotheses!

It scales up from there through increasing complex models but that’s the gist.

Thank you, Mr. Hoff.

Mathematical Modeling —Even Simple Examples Aren’t Simple

Let’s take Hoff’s model of the model and use it with an example that’s simple…way simple…and examine the model: 1 + 1 = 2

Part 1: what are the formal properties of the model?

We should begin by noting that 1+1=2 is actually only a fragment of a model.

I’m using a current standard version of addition with a model involving a number line and points along it. Addition is movement along that line. We have moved 1 unit up from a starting point at positive 1 and arrived at positive 2. The model has precise formal properties and addition can be defined clearly using it. (There are, of course, historical predecessors to this model, current variants, and there could quite possibly be a successor to the standard.) We could get into details of the current model, I guess, but suffice to say the model will confidently show you that 1 + 1 does in fact equal 2 and that 1 + 1 will never ever,ever equal 3. That’s all we need for this discussion.
 
Part 2: is there a meaningful connection between this model and whatever it is we’re modeling.

Ok, now for the fun.

1 + 1 = 2 meaningfully informs 1 apple + 1 apple = 2 apples. Also, 1 apple + 1 orange = 2 pieces of fruit. But the solar system and my Aunt Joan are also 2. And 1 explosion + 1 cat = 2.

Now the usefulness of this model is starting to slip.

1+1=2 is not ever less ‘true’ but it is certainly less meaningfully connected to what we are modeling in some case than in others.

Here’s a few more I like: 1 quark + 1 quark = 2 quarks. Right?

And the one statement “1–1=0” and a second statement “Never was there a time when you, I, nor these kings did not exist; nor in the future shall we ever cease” are two statements.

The conclusion?

Adding an apple and an apple to get two is different than adding Mount Everest and Daffy Duck or a meson and your momma. Not everything that human thought or language calls out is necessarily an entity that can be added to all other entities in any meaningful manner beyond the obvious sense that they’re both roughly parallel components of a sentence. As we move from the concrete to the goofy, our venerable model loses coherence.

(Note that I do not believe this is a spurious set of examples. The history of philosophy and theology is littered with arguments based on enumerated lists grouped into false categories.)

Here’s another simple example with a commonly used and often misleading descriptive statistic: the simple average. Each of the following groups have the same average, i.e. 10: 
{4,12,18,12,4}
{10,10,10,10,10}
{25,0,0,0,25}
{-10000, 3, 1942,8497,-392}.

Clearly without an indication that the data fits into some pattern or model that makes the average a meaningful statistic, it could easily be dangerous nonsense. Without knowing that the data looks like it is hard to say what average means.

Ok, so?

I’m going to be doing a little mentoring on Lean Startup methods. A key component of that is hardening a verbal description of the business’s value proposition into meaningful metrics that allow hypothesis testing…basically employing our most effective learning engine, science, to more quickly evolve a business. I thought it might be helpful to step back and take a close look at metrics themselves.

Working with metrics might be science but figuring the proper metrics is an art.

I’m hoping this helps. Next up: adding complexity.