I taught myself perspective drawing in 5th grade, which I recall because my friends asked me to explain how to do it, upon which I went to the library to learn it proper. I was surprised to read how late in the history of painting it was that artists got the perspective right. Upon closer introspection I guess though I didn’t actually learn it from watching the three dimensional world carefully, but by watching carefully images and photos that were already two dimensional.

Example of perspective drawing Pietro Perugino, about 1481 Source: Wikipedia Commons |

A painting is, in a very simple sense, a model of the world, and understanding perspective drawing must have had people realize that there is a mathematical basis of the world that’s waiting to be recognized. If you make an accurate drawing of, say, what you see out of your window, if you have sufficient details about how the mountains look like and where the river is, you might be able to “predict” from your drawing that there must be a tree standing over there.

I used to think of our theories as being maps, essentially, from mathematics to reality. I wrote about this earlier and will just reproduce here the accompanying diagram.

There is the world of mathematics, the eternal platonic ideal, and we take a part of it and identify it with the real world. The mathematical part we can call “the model” and “the theory” is the mechanism of identification with the real world, essentially how you compute observables and connect them to data. (I am aware that’s not how other people might be using these words, but arguing about words is pointless.)

This picture of the way we describe the world however raises the question if there is a distinction between these two areas, the question whether mathematics is equally real as that computer screen you are looking at, a question that is some thousand years old, minus the computer. Note that to ask that question, I don’t have to tell you what “real” means. I am just asking if there is a difference between a mathematical object and something that you can throw at me. Max Tegmark famously does not believe there is a distinction.

Most people I know believe there is.

However, it occurred to me, the mapping that the image suggests is actually not what we do if we build a model or apply a theory. What we always do, instead, is that we map one system of the real world to some other system, where the idea is that the one system is better to understand or to use.

Think of the painting: the painting is not a mathematical object. It’s an abstraction, all right, one that can make use of mathematical tools, but it’s not in and by itself a platonic idea. The same is true for all other models that we use. A computer simulation is not a mathematical object, it is a re-building, usually also a simplification, of another part of nature that we want to compare it too. And a calculation that you do in your head is not platonic either, it’s some firing of neurons and a lot of chemical reactions going on, and so on. And it is again, essentially some simulation, approximation, extrapolation, of another part of nature that you want to compare it to, to the end of making a prediction because you want to know if you got it right.

So where does that leave mathematics then? Mathematics is a tool that we use to improve on our models, it’s a technique that we force our thoughts through because it has proven to be incredibly useful. Nevertheless, the point I am trying to make is that this usefulness doesn’t mean a model actually extracts some mathematical “substance” from reality.

You will wonder now what does it matter. The reason it matters to me is that for reasons I elaborated on in this earlier post, I think that the occurrence of the multiverse in its various forms is unavoidable and a consequence of relying exclusively on mathematical consistency. The multiverse tells us that mathematics is not sufficient. What is, I don’t know.

The question is of course if we can conceive of any type of model and a theory to map it that is not mathematical. One thing that came to my mind here is analog gravity, basically the idea to study some types of gravitational phenomena with condensed matter or fluid analogies (thus the name), an idea that has caught on during the last years. I am not terribly excited about this because I don’t really see what we learn from this about quantum gravity. But the point is that it’s an example where you have a model (the “analogue”) that is mapped to the system you want to describe (spacetime) and the model in this case is not a mathematical structure.

Or in other words, if it should be the case that nature cannot be described by mathematics alone, this type of models could still be used.

So much about my latest thoughts on the question whether, at some point in the history of science, we will have to find a way to go beyond mathematics to make progress, and what that could possibly mean.