Models as maps and maps as interfaces

One of my favorite conceptual metaphors from David Basanta is Mathematical Models as Maps. From this perspective, we as scientists are exploring an unknown realm of our particular domain of study. And we want to share with others what we’ve learned, maybe so that they can follow us. So we build a model — we draw a map. At first, we might not know how to identify prominent landmarks, or orient ourselves in our fields. The initial maps are vague sketches that are not useful to anybody but ourselves. Eventually, though, we identify landmarks — key experiments and procedures — and create more useful maps that others can start to use. We publish good, re-usable models.

In this post, I want to discuss the Models as Map metaphors. In particular, I want to trace through how it can take us from a naive realist, to critical realist, to interface theory view of models.

I’ve used the Models as Maps metaphor a number of times on the blog: most notable when asking: are all models wrong? It is on this question that people usually turn to Borges’ On Exactitude in Science. Borges opens with:

… 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.

A reminder that a map can only be perfect if it occupies the whole territory of what it represents. Or in modeling terms: if it becomes the thing we’re modelling. This is the naive realist view of models: for a good model, the world is exactly as the model represents it.

But this is silly. So Borges concludes:

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.

And this introduces us to the importance of a model being useful. It also suggests a method for that usefulness: omission. Ignore the irrelevant details and represent only the key factors needed for our purposes. This is the critical realist view of models: a good model resembles reality, but it doesn’t capture all of it. To borrow an image from Kevin Song: if naive realism is a perfect picture then critical realism is a blurry one.

But omissions aren’t the only — or the most important — way that maps don’t match reality. For this, it is useful to look at the difference between globes and flat maps. Vox offers a fun 6 minute video:

In the above video, Johnny Harris reminds us that it is mathematically impossible to turn a globe into a flat map without at least some distortion. In other words, if we want to represent a globe as a map, we not only have to omit things but by omitting things, we have to misrepresent the world. I think that this is the default for modelling.

No model is right. Similar to how no map represents the world. Rather, we have to choose which model to use based on what we are using it for. In particular, Harris discusses why the Mercator projection was so popular: if we want to navigate, and our only tool is setting angles then we want a map that preserves angles. On such a map, if we draw a line between two points then it gives us an angle that we can follow to navigate. This makes the Mercator projection good for navigation but bad for judging the relative size of countries. If, instead, we want to use our map to compare the area of countries then we should switch to the Gall-Peters projection. You can find more projections from Mike Bostock.

Something similar can happen with models. This is the interface theory view of models: our model does not resemble reality, rather it allows us to act in a useful way.

This means that models can have both omissions and distortions, as long as the resulting models are tractable and allow us to make useful decisions. This also means that the model as a tool is paired with the problem that it aims to solve. And any given model can only solve certain problems. Thus, if we have many problems then we might need many different models. Further, these models might not be mutually consistent in their ontological commitments, and none of those commitments might represent reality.

I subscribe to this view of models as interfaces.

And I think it is the only view that we can embrace while respecting history.

Here I want to turn from Borges to Asimov’s Relativity of Wrong. There, Asimov talks of a interlocutor that wrote to him:

in every century people have thought they understood the universe at last, and in every century they were proved to be wrong. It follows that the one thing we can say about our modern “knowledge” is that it is wrong.

This is surely focused primarily on the ontological grounding of theories. But to it Asimov rightly replied:

John, when people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together.

From the perspective of interface theory, this means that our interfaces have improved. We have gotten better at doing things with our models. But our interfaces do not differ in just what they omit, but they can differ foundationally. Thus, we can talk about them being better at solving problems: less wrong. But just because they’re better at solving problems, we cannot conclude that their grounding is ‘correct’. It is just useful. For now.

As we use the models, we will generate new field-endogenous problems or realize that the model isn’t useful for solving some field-exogenous problem. These new problems will then need new models (or refinements) for us to act on them. These new models might require completely different ontological grounding.

And so the cycle of generating new models will continue.