Mathematics & the interpretation of science

In his essay Form, Substance & Difference, the ecologist and anthropologist Gregory Bateson wrote:

It all starts with the Pythagoreans. The argument took the shape “Do you ask what it is made of – earth, fire, water, etc?” Or do you ask, “What is its pattern?”

We can reasonably say that this argument has now been resolved decisively in favour of pattern. The basic ontology of nature that we have is an ontology of events rather than entities. The models of physics are mathematical models that describe transformations in space and time of measurable quantities: momentum, energy, charge and so on. This isn’t always apparent though, because the compositional way of thinking persists and is embedded in the way we use language.

The mathematical formulation of physics raises a number of questions for the understanding and interpretation of natural science. This description from Roger Penrose illustrates the issues quite well, I think.

Questioner: Roger, how accurately does math describe the physical world?

Roger Penrose: Well it is extraordinarily precise, but I think people often find it puzzling that something abstract as mathematics could really describe reality as we understand it…

I mean reality… you think of something like a chair or something you know something made of solid stuff, and then you say… well, what’s our best scientific understanding of what that is? Well, you say, it’s made of fibers and cells, and so on… and these are made of molecules, and those molecules are made of atoms, those atoms are made out of nuclei and electrons going around, and you say well… what’s a nucleus? Then you say… well it’s protons and neutrons, and they’re held together by things called gluons… and neutrons and protons are made of things called quarks, and so on…

And then you say, well what is an electron? And what’s a quark? And at that stage, the best you can do is to describe some mathematical structure… you say, they’re things that satisfy the Dirac equation, or something like that… which you can’t understand what that means, without mathematics.”

Roger Penrose | Is Mathematics Invented or Discovered? | Closer to the Truth, YouTube | Note: Shortened quote

The question of precision is an interesting one. One issue, which is perhaps more for practitioners, is what level of precision counts as adequate. From a philosophical standpoint, the question is whether we think this degree of precision, whatever it is, is a function of the nature of mathematics or a function of the nature of the world. This is an aspect of the issue of philosophical realism.

However, in this post, I want to ask about the significance of the fact that we can’t understand physics without mathematics.

I don’t think Penrose means that the mathematics is too complicated. Mathematics is used in many fields but we don’t usually say that we can’t understand the idea of, say, future value in finance or standard deviation in statistics if we can’t set up and complete the calculations. In these cases, the concepts can be grasped independently of the calculations.

I also don’t think that he means that only someone who has the necessary mathematical ability can interpret what the equations mean. If it is possible to translate the mathematical formulation into a conceptual interpretation, then that interpretation can be communicated. In a similar way, if you can speak both German and English, anything you can translate from German into English you can explain to the same level to another English speaker.

The reason why we can’t understand what is meant without the mathematics is that there is no other description of the events. The structure of language misleads here. Electrons are not entities in the same way that cells are entities. We use the same linguistic structure, attaching predicates to a subject, to describe both, but when we talk about the attributes and behaviour of a cell there is a prior subject to which we are applying the predicates. When we talk about electrons, there is no prior subject independent of the predicate.

Thinking about the designs for a building can illustrate this point. Architects must calculate the effect of different loads on the structural elements of a building. These loads are forces that create stresses on the structure’s components. The primary force is gravitational, the weight of the structure and its occupants, but there are also the lateral forces of the wind and consequent changes in pressure, the effects of temperature on expansion and contraction, the adjustments that shrinkage and settlement will impose, and the impact of dynamic events, of things shifting, falling, and oscillating in or near the structure. In general, a building must be functional and at the same time strong enough and stable enough to keep its shape when subject to all of these forces.

Although we wouldn’t normally go to the trouble of understanding the calculations unless we were actually trying to build a house, it is not difficult to interpret the fundamental ideas of stress, strain, elasticity, plasticity, shearing, deformation, compression and tension that the calculations are quantifying. This is because we can visualize the load bearing entities, beams, posts and walls and so on, that these forces are applied to and we have an intuitive sense of how steel, timber, masonry and concrete will behave under such pressures. In an architect’s drawing the forces in play are not usually represented. The drawings show only the structure put in place to resist those forces. Gravity is not represented by symbolic arrows showing the magnitude and direction of the force but coded into the thickness of the beams and the stoutness of the posts.

In physics, on the other hand, only the forces can be represented. What we have to imagine is something like an architect’s drawing but one where, instead of the dimensions, material composition and configuration of the structural components, the drawings show only the forces being exerted at each point in the space occupied by the structure.

We can say that at the macro-scale of an architect’s drawing we have a choice, we can either represent the pattern or we can represent the components. At the micro-scale of physics, we have no choice, we can only represent the pattern.

We have no intuitive sense of what these micro-scale systems actually are, only of how they behave. Although we talk about particles, these are idealizations, zero-dimensional points in space and time, not composite structures like posts and beams only on a much smaller scale. We mustn’t imagine that a particle in this sense is a grain of something. A photon is not a grain of light. At the scale of molecules and smaller, what we have are systems in which electromagnetic and nuclear forces are in play at every point in the volume occupied by the system.