Do We Need a New Math to Understand Physics?

Back before my instructional responsibilities forced me into turning my Fluid Mechanics Laboratory into an online adventure, I posted a link to an article entitled Does Time Really Flow? New Clues Come From a Century-Old Approach to Math, which discusses quantum physics and its relationship to mathematics. It’s an intriguing article, but many of the issues that it discusses have been around for some time.

As someone whose specialty (in part) is computational modeling, the problem can be posited as the way we model physical phenomena (mathematics) relates to the way it actually happens in the material world (physics, in this case quantum physics.) The difference between the two has generated the core problems the article discusses.

Modelling is something that has come front and centre in the current COVID-19 crisis. The weakness of the way modeling is done in medicine (and certainly the social sciences) is that it is an exercise in statistical extrapolation and not a predictive exercise based on an idea of the behaviour of the system, informed by correlations with the actual behaviour. Medicine is starting to move towards the latter type of modelling but it’s slow; if it could do so by, say, modelling the interactions of various drugs to various bacteria or viruses in the environment of the human body (with all the variations therein) we could select with more intelligence which clinical trials to pursue and understand their outcomes more completely.

But I digress…it wouldn’t hurt to start by understanding what modelling is. To do this I turn to Rutherford Aris’ Mathematical Modelling Techniques:

In these notes the term ‘mathematical model’–usually abbreviated to ‘model’–will be used for any complete and consistent set of mathematical equations which is thought to correspond to some other entity, its prototype. The prototype may be a physical, biological, social, psychological or conceptual entity, perhaps another mathematical model, though in detailed examples we shall be concerned with a few physico-chemical systems.

The nature of models, and their relationship with the prototype, is a complicated subject and one which Aris discusses at length. It is tied up with the type of modelling being used and the purpose for which the model is intended. The initial thought is always that the model most accurately reflect the behaviour of the prototype. The increase in computer power since Aris wrote this monograph has made achieving that objective simpler because the governing equations, the initial conditions and the boundary conditions (the three mathematical parameters which govern any system) can be made to reflect the behaviour of the prototype with greater precision. Whether it’s more accurate is another story.

Before we turn to the topic at hand, one more memorable quote from Aris (and the only time this author has ever been cited in a scientific work that I can recall) is the following:

It scarcely needs to be added that we shall not raise the old red herring about the model being less “real” than the prototype. Tolkien [178] has reminded us of the failure of the expression “real life” to live up to academic standards. “The notion,” he remarks, “that motor cares are more ‘alive” than, say, centaurs or dragons is curious; that they are more ‘real’ than, say, horses is pathetically absurd.”

Reference [178] is Leaf by Niggle. We will come back to this shortly.

With all that, the article goes off into the idea that we need a new mathematics (well, some people think we do) to describe the discrete phenomena that we have in quantum physics. From a modelling standpoint, I think there’s a simpler explanation for this, and one that would, if taken to heart, move things forward in a more efficient way.

The mathematics we have is “continuum” mathematics, and has been since the days of Dedekind and Cantor, at least (actually earlier with the calculus.) To oversimplify things, we have as many real number as we have places to put them; the more places we “create,” the more real numbers we have. This process goes on ad infinitum not only to actual infinity but between any two numbers we might specify.

Quantum mechanics is by definition a discrete phenomenon; the numbers we use may not run out of significant figures but the physics does. Physical phenomena being discrete, it makes sense that they should be modelled in a discrete way, recognising the “granularity” of the physics. That has, as the article suggests, some important implications for the way we view time, reversibility and determinism in physical events, and these should be incorporated into our models.

Those of us who deal with numerical methods have already seen this whether we recognise it or not. Most all numerical models are computed with digital computers (discrete by nature) and are limited to the fifteen significant figures of double precision (that includes our spreadsheets, by the way.) The things that happen in truncation can be significant up to and including crashing our model, depending upon how they play out in the calculations.

Also, even on a larger scale than quantum physics deals with, our use of continuum models is based on the assumption that the granularity of the medium we deal with is not significant relative to the behaviour of the system, or that same granularity is too hard to model either because of accuracy considerations or the limitations of our computer power. For example, in computational fluid dynamics a rule of thumb for fluids to be considered a continuum is that the Knudsen Number $K_{n}=\frac{\lambda}{L}\leq0.001$ , where $\lambda$ is the mean free length between molecules and $L$ is the length of the system. For fluids at the surface, this is easily met; in the vary highest regions of the atmosphere and space, it is not, and any interaction with the “fluid” must be done on a different basis.

The last quote from Aris above speaks to the issue the article raises about the existence of infinitely long numbers (the existence of $\pi$ should put paid to this kind of speculation in and of itself.) As mentioned at the start, the numbers exist; the physical reality they correspond to doesn’t, but that doesn’t impugn the existence of the former. I think this reflects a current trend towards discounting anything that does not have physical reality, but the long-term result of this will be a stunting in the ability of people to think abstractly, which is crucial for both mathematics and modelling. On the other hand, as quantum mechanics has done for a long time, the whole concept of determinism is challenged by the discrete nature of molecular and sub-molecular reality, but that is, as I am wont to say, another post.