Maps, models and meaning

In the book Notes from a Small Island, Bill Bryson describes his fascination with the London Underground map: “What a piece of perfection it is, created in 1931 by a forgotten hero named Harry Beck, an out-of-work draughtsman who realized that when you are underground it doesn’t matter where you are. Beck saw – and what an intuitive stroke this was – that as long as the stations were presented in their right sequence with their interchanges clearly delineated, he could freely distort scale, indeed abandon it altogether. He gave his map the orderly precision of an electrical wiring system, and in so doing created an entirely new, imaginary London that has very little to do with the disorderly geography of the city above.”

When I take the Tube, I sometimes think about where I am in London, i.e., what buildings and streets are above me. And I give up straight away. It is so much easier to put your blind trust in the Tube map. I am not the only one doing this. Guo (2011), for example, showed that passengers in London often trust the Tube map more than their own travel experience when deciding the best travel path. However, interestingly, when there was a strike on the Underground, commuters started experimenting with new routes that they kept to once the strike was over, especially for commuters in areas where the Tube map was more distorted (cf. Larcom et al. 2017).

Geographic accuracy is overrated when you are in the Tube. You might even call the geographic inaccuracies for ‘lies‘, but even if we wanted, we would not be able to make a ‘perfect’ map of the Tube – or London, or the world. When you are on the Tube, you are primarily interested in what stations you are connected to and where you can switch to another line (if relevant). Everything else is noise. We make a map because we have a specific model of the world in mind.

Maps are simplifications. This is best described by Jorge Luis Borges in On Exactitude in Science. If you want a perfect map of the world, you need to reproduce the world, and then it is not a great map. Similarly, in Alice in Wonderland, the character Mein Herr makes a perfect map of the country that is the same size as the country. However, while (good) maps will always be simplifications, they simplify reality with human perception in mind, i.e., the meaning we can derive from them. For example, how sunlight is shown on maps matter, because humans tend to believe that light comes from above. In a study of Paris Metro maps, Roberts et al. (2013) found that traditional design rules are not always useful, and even breaking them can improve usability. In other words, maps do not speak for themselves, even if we have a specific model in mind.

Gunitsky (2019) describes how maps lie in the same way theories ‘lie’, i.e., to emphasise relevant features of the world. Specifically, he writes: “Key to the notion of cartographic-epistemological parsimony is the idea that explanations require generalization, and generalization requires abstraction. A theory abstracts from the world in order to make sense of it, not out of a conviction that abstraction is more elegant or more descriptively accurate. And like theories, different maps may focus on different elements of the landscape depending on their purpose. A map of soil types used by agronomists will look different than a map of hiking trails used by tourists even when covering the same small patch of countryside. Thus, a theory of democracy may focus on different factors than a theory of trade policy even when looking at the same country during the same time period.”

Similarly, Nagaraj and Stern 2020 write: “Maps are not made at random but by mapmakers who exercise significant discretion and agency, whose choices are shaped by the economic, strategic, and institutional environment in which a particular map is produced. Two key elements of mapmaking are worthwhile to distinguish: the gathering and organizing geospatial information (data) and, conditional on that data, the use of geospatial tools and visualizations to create a particular map (design).” While this is literally about the construction of maps, it can also serve as a metaphor about how we aim to derive meaning from maps.

Maps are not neutral, but rely upon implicit and explicit assumptions about the world. Kate Crawford describes in The Atlas of AI how maps are related to power: “Maps, at their best, offer us a compendium of open pathways—shared ways of knowing—that can be mixed and combined to make new interconnections. But there are also maps of domination, those national maps where territory is carved along the fault lines of power: from the direct interventions of drawing borders across contested spaces to revealing the colonial paths of empires.”

When we create a map, we select the features of the world we care about, exlcuding what we do not care about. We do this because we want a map to be used as a model. In the chapter by Smaldino (2017), “Models Are Stupid, and We Need More of Them”, it is described how simple (or, stupid) models are useful because we cannot as humans derive meaning directly from complex systems: “Stupid models are extremely useful. They are useful because humans are boundedly rational and because language is imprecise. It is often only by formalizing a complex system that we can make progress in understanding it. Formal models should be a necessary component of the behavioral scientist’s toolkit. Models are stupid, and we need more of them.”

We do not need more complex models as adding details to a model is not an improvement in and by itself. On the contrary, it might just add more noise to our model. As Healy (2017) argues in his article Fuck Nuance: “This sort of nuance is fundamentally anti-theoretical. It blocks the process of abstraction that theory depends on.” The problem is related to the bias-variance trade-off. More model complexity can lead to higher sensitivity to changes in our training data and, accordingly, worse generalisations (i.e., overfitting).

When we work with a model, we are often interested in predicting a response $Y$. To predict values of $Y$, we can introduce $p$ different variables/predictors, $X_{1}, X_{2}, …,X_{p}$. $Y$ is related with $X_{1},…,X_{p}$, and the relationship can be written in a general form as:

$$ Y = f(X_{1}, X_{2}, …, X_{p}) + \epsilon $$

The function, $f()$, represents the systematic component we can use to explain $Y$ (what we find important in our model). Key to this is the idea that we cannot explain everything with our model. We can predict something (and everything else is ‘error’). We can identify a signal (and everything else is ‘noise’). We have an explained part (and an unexplained part). We have a deterministic part (and a ‘random’ part). $\epsilon$ represents the error term, i.e., everything we miss with our model. However, adding an extra predictor our model ($X_{p+1}$) will not be an improvement on its own (emphasising the importance of including a penalty term when we are not sure about the correct model specification).

A linear regression model is one kind of model, but it does not provide meaning on its own. Or, it does not provide any causal inferences, only correlations/predictions. We can easily obtain coefficients, but we cannot get meaning out of them without a (causal) model.

A good simple model is parsimonious. Gunitsky (2019) identities three different conceptions of parsimony. First, the aesthetic conception (a theory’s elegance and clarity). Second, the ontological conception (the world is governed by simple fundamental laws). Third, the epistemological conception (abstracting from reality to highlight recurring patterns and build testable propositions). When we talk about maps as simplifications, we are often talking about the third conception of parsimony. In the table in the paper, there is a great overview of the different types of parsimony and how they relate to social science theory.

Last, let us return to the London Underground map. Benoit and Laver (2012) describe how we can think about the dimensionality of the European policy space with examples from the Tube map. First, they demonstrate how a physical description using one dimension will serve Central Line passengers well. However, in one dimension, it would be difficult to serve Circle Line passengers well. In other words, a the complexity of a model, in relation to a map, is all conditional upon how we are going to use it and what meaning we will need to get out of it. Specifically, here is what Benoit and Laver (2012, p. 216) conclude:

“Just as there is not one true map, neither is there ‘one true dimensionality’ for any given political setting. Sometimes, as with London’s Central Line, a one-dimensional map will tell us everything we want to know. Sometimes, as with London’s Circle Line, it will not. Nearly always, we ignore potentially salient dimensions in the name of a parsimonious description of the world, just as we typically do not seek three-dimensional maps of London Tube stations – despite the fact that these stations do in fact have substantively meaningful geographical coordinates in three dimensions.”

When we create a map, we select the features that are important and thereby remove features that are unimportant. However, we also have a specific model of the world in mind that can help the users of such maps (commuters, scientists, citizens, etc.) when interpreting the connections between the selected and non-selected features.