“What is the right dimensionality of this problem?” This question – deceptively simple – is a good way to avoid defaulting into the dimensionalities that are most available to us – 2 or 3 dimensions to a problem, and, frankly, mostly two. Often when we describe a problem we somehow come back to the simple graph where one variable is plotted against another. It is often whatever thing we are studying plotted against a time scale. This can be useful, but also deeply deceptive. The reality is that we probably rarely face problems of such low dimensionality – and when we ignore higher dimension versions of a problem we are ultimately at sea when discussing solutions.
Now, there is a reasonable counter argument here, and it goes something like this: we cannot effectively work with higher dimensions than, perhaps, 3. Understanding a problem requires that we can reduce its dimensionality down to something we can visualize, and when we cannot, well, then the problem is effectively intractable to us.
I like this argument, because it acknowledges that there are things we cannot understand. But I also think it underestimates our inventiveness and intelligence. I do think we can understand problems with a higher dimensionality than 3 – and that it is not just about visualization (although visualizing something always helps). But I also suspect that there are classes of problems such that their sheer dimensionality make them practically intractable.
Just asking the question, though, is a good start. The best example I can think of where we naturally assume high-dimensionality is health. Human health is a high-dimension problem, and the different metrics we may use gives us a complex picture of a person’s health – it is not a horrible idea to think about what the “health of X” would look like when exploring different mental models.