On the Importance of Postulates

Today Kate (my wife) and I had a conversation about statistics eventually leading to the concepts of Fractional Factorial Design and the Sparsity-of-Effects Principle. In Fractional Factorial Design (FFD from here on), one seeks to limit the number of experiments required to evaluate the relative importance of the factors to be studied.

For example, if you wanted investigate what factors produce a paper airplane that flies as far as possible you might look at various possibilities for the wing shape, wing width, and paper type. But if there are 5 options for each of these combinations you would have to make 125 paper airplanes (5 wing shapes x 5 wing widths x 5 paper types) in order to find the best combination. You may not have that much time or paper. Or patience.

So instead of testing all of the options to find the best paper airplane design, you may instead decide to look for whether one of the factors is more important than the others and go from there. Using FFD you could find the, for example, 8 most optimal paper airplanes to make in order to determine which factor is most important. If multiple factors work together you may miss out on that relationship but if not this could be a powerful way of saving yourself time (and paper).

If you’re interested, here are the 8 combinations to try (X1=Wing shape #, X2=Wing width #, X3=Paper type #)*:

 X1 X2 X3
----------
  1  1  1
  1  1  5
  1  5  1
  1  5  5
  5  1  1
  5  1  5
  5  5  1
  5  5  5

* table generated using the AlgDesign library in R

Regardless, all of this got me thinking about the fact that many algorithmic tools (such as FFD), and even entire fields of study, often emerge from a single assumption or definition. Usually one with far more importance than is immediately apparent. In the case above, it’s not immediately apparent that you can safely prioritize or limit the number of combinations you test and still obtain a meaningful result. However, by making a single assumption (i.e. the sparsity-of-effects principle) you can turn a problem too large to feasibly solve into one that’s much more manageable (albeit with some caveats).

Similarly, special relativity is derived from two postulates (from Wikipedia):

• The laws of physics are invariant (i.e. identical) in all inertial systems (non-accelerating frames of reference).
• The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

The first postulate is straightforward and, while I will admit the second postulate is a somewhat non-obvious, there was reason for Einstein to suspect this. However, it’s not clear upon seeing these assumptions that time dilation should occur for objects moving at high speeds. We now know that this occurs, but it was the predictions of this theory that brought us to that understanding.

I think recognizing this pattern, of limiting oneself in order to find a more concrete logical space, is extremely important in terms of appreciating mathematics or really any formalized field of study. Only by defining the structure of the domain on which one wishes to operate, and the assumptions therein, does it become possible to derive truly meaningful results. Unfortunately, I think the desire for this kind of formalism separates academia and business far more often than it should. Even among college graduates who studied the same field, ideas are far more often communicated and discussed in something much closer to layman’s terms than the carefully chosen, well-defined language it was originally conveyed to them in. The only apparent reverence is for the concepts and tools we don’t understand, labeling them simply as having been figured out by people smarter than us.

The fact of the matter, though, is that this works more often than not. If the non-trivial results of academia were common and easily understood, they would necessarily be less impactful. So instead of disgracing their fields, these workers often do well. After all, they aren’t wasting time with pedantic language or preparing for unlikely scenarios. They instead solve the common case and move on to the next problem. There are often checks and balances for the cases where these kinds of mistakes were made in past with severe consequences, of course, which pushes the probability of failure down even further. But not to zero.

When the next rare case occurs will the time we saved by not worrying about it be enough to justify it? Will we even know what happened? Are the giants upon whose shoulders’ we stand so tall that we can no longer see their shoes?

I, for one, am pulling out the binoculars.