Ashby, Measure, and Fractality (February 2021)

This is a techincal addendum to the talk I gave at Emily Harvey in 2018, which can be found on my vimeo. At the time I did not have the mathematical knowledge to clearly and rigorously articulate certain points. Still, this is written in plain English and not extremely technical.

You can't simply talk about Asbhy's Law in terms of the size of a set of choices, because some of those choices might be different versions of some more general choice; for example, "hit it with a hammer hard" is a more specific version of "hit it with a hammer", if you do the former you've also done the latter but not vice versa, so you cannot just place them side by side in a list and treat them as totally separate things.

So you need what's called a "sigma algebra" in measure theory, which instead of treating each choice as an item on a list treats each one as (for all intents and purposes) a spatial area. In this case, "hit it with a hammer hard" is an area inside "hit it with a hammer".

But it's not exactly spatial, yet: you need to give this collection of sets geometric rules, specifically what's called a "measure", which assigns sizes in a logically consistent fashion. For example, if I assign a measure of X to one thing, and a measure of Y to another thing, then the measure of those two things put together has to be exactly X+Y (this rule is known as additivity.)

So now you don't necessarily get more area by adding another choice (one might be contained in the other). But this doesn't quite add up either: in life we learn how to do things in more specific ways and this gives us more options. When you play a key on the piano a certain way, you can make new music: even if "play the key like this" is a subset of "play the key", there's clearly something new in being able to do a more specific version of the same thing.

So if we just picture these sets as basic shapes, where we can arbitrarily subdivide them without a change in the total area covered, we can't capture this idea.

But now consider a different shape for describing these areas: fractals. What makes fractals differ from other shapes is that they're infinitely jagged: no matter how much you zoom in, there are new shapes protruding from segments of the perimeter. This matters because the size of your ruler is no longer arbitrary: if you measure a fractal shape centimeter by centimeter, then that centimeter length of your ruler will cut across some protrusion, and in doing so it will underestimate the total perimeter of your shape. This means that the smaller you make your units of measurement, that is, the more you subdivide the perimeter, the larger a perimeter you get.

So now, a more specific action actually does add to the total "amount" of choice you have, insofar as it changes the measure.

If you now imagine an organism or some other kind of systme, you can think of this perimeter as all of its possible interactions with the world. The greater the perimeter, the more choices, and the more choices, the more ways to adapt to a situation (Asbhy's Law).

Add to this that if you can think of the volume of such an organism, it would be extremely inefficient to be a Euclidean shape, as it would increase in size faster than surface area as its perimeter grows (imagine a square, where increasing perimeter from 40 to 44 inches would increase area from 100 to 121 inches). Physically, this would correspond to completely untenable demands on metabolism.