I introduced the notion of a black box in an earlier post as some thing that has an internal organization that drives its behavior, and though we can see the behavior, we don’t have perfect knowledge of that internal organization. In short, a black box has an inside and an outside, and we only have access to the outside. What makes a black box interesting is when it has interesting behavior – but also because many different internal organizations could account for the behavior observed. The notion that the internal organization is under-determined by external behavior is part and parcel of our modern age.
We call two black boxes equivalent if they exhibit the very same behavior. This doesn’t require them to be identical. A simple example is the telephone. Let’s say I’ve been calling you every week for years, and I do the same today. I dial the number, and I expect you to pick it up and talk. Unbeknownst to me, you got a new phone yesterday, one with new-fangled bells and whistles: it plays music, takes pictures, and files your taxes. Yet those differences, so important to you, don’t affect my call to you at all. You still pick up and we still talk, and unless you tell me about your new phone, I would be none the wiser. From my end, your phone behaves exactly the same as it did before. My phone can still “talk” to your phone, and I can still talk to you. It wasn’t necessary for me to upgrade my phone at the same time you upgraded yours. For a phone to properly connect to the telephone network, it needs to behave in a particular way. But if it does, it can be yellow or green, play Bach or Beatles for its ring tone, take pictures, be wired or wireless, have a headset or a speaker phone, fit in your pocket or hang from the wall. The telephone network itself, the network to which telephones connect, has changed gradually: it used to only respond to the clicks of a rotary dial, then for a long time would accept either rotary dials or push button tones, and now many regions no longer support rotary dials. Rotary dials and push button tones are an example of phones that are not equivalent: they behave differently in ways that “count” as far as the telephone network is concerned. Land lines and cell phones (mobile phones) are likewise not equivalent, yet as long as the cell phone is within the reception area of the towers, the person on the other side of the telephone might not necessarily notice the difference.
Just like we can compare two black boxes and see if they have the same behavior, we can compare a black box and a model. Imagine we had a black box and tried to figure out what the organization inside is like. In doing so we might come up with a possible model for what the box is like. The model might explain the behavior of the black box or it might give us a way to replicate the box. If the model’s specification is too vague, none if this will work, but if it is precise enough, we can look at the functional match between the black box and the model.
For an example, let’s look at a particular black box with one input and one output. Whenever you put “1” on the input, you get “3” on the output, whenever you put “5” on the input, you get “11” on the output, and the following table records the results of the experiment:
On the left, I show the black box, and next to it a table of its behavior, to the extent we tested it in our experiment. To the right of the table is a model. In it, we figure that if we have a box that double the number fed in to it, and then feed its output to another box that adds one, we get a box that matches the behavior in the table. Is this model equivalent to the black box? Truth is, we don’t know. We haven’t done enough experiments to be confident that the pattern holds beyond the four values we established in the table. The model represents a generalization of what has been observed, it represents a hypothesis. It is a real question whether any amount of experimenting would ever establish the equivalence between the black box and the model – and this is a question coming back to at greater length. Right now, I’m more interested in comparing this model – the one with the “double” box inside of it – with the model shown on the right. This model, too, matches the behavior laid out in the table. I am confident that I can make a much stronger claim than that – that the two models will show identical behavior no matter what values are put on the input. I suggest that it is possible to show equivalence between the two models without relying on a large amount of experimentation. To do so (and I won’t do this in this post) requires a kind of thinking and a kind of making sense that’s of a very different nature. It is this kind of thinking that we usually call mathematical thinking.