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.

I am sure that we may come up with new ideas in mathematics to deal with this situation.  But a new mathematics?  I doubt it.

Quantifying the risk of random contact with infectious individuals — carnotcycle

For citizens like me, old enough and perhaps wise enough to take a cautious view of mingling again with one’s fellow man after a period of lockdown in a time of pandemic, it’s natural to wonder about the risk of contact with infectious individuals as one re-establishes one’s daily routines. I thought this over and […]

via Quantifying the risk of random contact with infectious individuals — carnotcycle

Advice to Graduates: The Two Promises I Made to Myself

In this time of no graduation ceremonies, I think it appropriate to repost this from August 2016, when I faced getting my PhD without a graduation ceremony because my institution didn’t do an August graduation.  This year the institution is forgoing May graduation for–wait for it–an August graduation.  (I eventually did get to walk, in December.)  Life and careers don’t move in straight lines; I think the advice I gave then deserves to be said again, especially now.

It may seem an odd time to do a pseudo-graduation piece. Obviously the University of Tennessee thinks so: this weekend I am supposed to officially receive my PhD degree, but the university, having spent a great deal of money on a new, traditional looking quad, doesn’t do an August graduation ceremony, with a graduation speech of any kind. So this will have to suffice.

In accreditation standards, this degree is referred to as the “terminal degree.” I agree: by the time you’re done with it, you’re just about dead. But I have other things to commemorate this year. One of those is the twentieth anniversary of our family divesting itself of our business. Accompanied by the loss of my father and brother, it was one of those times when everything was different at the end than it was at the beginning. In the wake of those events I took stock of things, sought God and made myself two interrelated promises that I have pretty much kept in the score that followed. I think they’re worth passing on because, in the midst of swelling words, it’s easy to lose sight of practicalities.

The first was that I would never again allow myself to be dependent upon one source of income. Up until that point the family business—a company with one product to boot—had been my main livelihood for eighteen years. In those years it was impressed upon me that, from a professional standpoint, the business should be like segregation to George Wallace: first, last and always. Although I had the usual consulting contracts, they wouldn’t last that long, and there were the equally usual non-compete agreements in them. With the unhappy memory of every day being a “hero or zero” event, I decided to diversify my income. It’s been very helpful. We’re supposed to sleep a third of the time; that decision made that third (and the other two-thirds) a lot happier.

One of those diversifications has been my online activity, which started the year after the business went away. It hasn’t been the most lucrative thing, but in the process of putting stuff up I’ve delved back into our family history. We’ve been successful since we’ve been here, and for my father’s family that’s about a century and a half. Much of that success has been due to the diverse nature of the income: my great-grandfather’s yachts, my grandfather’s cars and airplanes, etc. Even the “one product” family business, at the turn of the last century, had a diverse offering which included bridges, dredges, and other products. There was a historical lesson that had been forgotten, and this is a country which habitually forgets historical lessons.

To make that really work involves another family habit: living below your means and staying out of debt to the greatest extent possible. That flies in the face of a credit-driven society driven by instant gratification, and it isn’t always easy in a country where wages are compressed the way they are. That being so, without it, the advantage in your life will always shift towards those who make the payments.

The second was that I would never let my professional (or other) identity be taken over by another institution or individual. This will take a little more explaining.

When your family has been in our business as long as ours was, the public image of the two tend to run together. But which came first? My great-great-grandfather started the company in 1852, sold it eleven years later, his sons bought it back in 1881, we got out of it in 1996. It should be obvious that the company was ours as long as we had it. But that wasn’t the message I heard, especially from the family and those in the company. The message I heard all too often was that the business made us what we were and that we owed the business in perpetuity because of that. That justified the aforementioned idea that it should be the sole source of income.

Getting out of the business didn’t solve that problem. I worked for people who wanted my professional identity completely contained in the work and institution which they ran. That wasn’t any better at what was strictly a job than it was at my own business. But there are others who saw it to their advantage to let “me be me” and they reap the benefits from that. In those cases it’s been a “win-win” situation for everyone. (Remember that, in job hunting, they’re not only choosing you; you’re choosing them.)

There are two parts to this issue: the practical and the “theoretical.” From a practical standpoint, in a world where companies, institutions and even lines of work are in a perpetual state of upheaval, it doesn’t make sense to have one’s reputation in the marketplace dependent upon one institution. Sometimes one can end up the “last man (or woman) standing” in a profession, where the skill set has gone out of currency and you’re the “go-to” person. But even then the reputation needs to be yours, not your employer’s.

The “theoretical” part is a little trickier but just as important, because it goes to how you look at life in general, which in turn will determine where that life goes.

Christianity teaches that we derive our worth and value from God who created us and made our salvation possible. That being the case, it’s always amazing that, in what has been up until now a predominantly Christian country, that so many in church every Sunday pursue personal validation in this society with such gusto. We insist on driving the proper car, living in the proper house, and raising the proper children to communicate the message of success, when the Gospel tells us that none of these things is necessary for happiness.

Secularizing the country will only make this problem worse, because it takes away the alternative to worldly success without obviating the need for perpetual validation in the society. The enforced online groupthink, where we are forced to go along with the herd’s course or else, is only the most distasteful manifestation of this problem. Consider the matter of same-sex civil marriage; in a society as polarized as ours is and where cohabitation is as common as it is, it’s really strange that neither or both sides could bring themselves to pitch the institution of civil marriage altogether. Everyone argued under the assumption that the state had to validate a marriage in order for it to be one. The same thing goes for our elite institutions. Whether they provide a better education is open to question; whether they confer on those who endure their degree programs a glow of respectability is not.

I used to think that my family I was born into didn’t like my Christianity because it put God in charge of things, not them. That’s true as far as it goes, but the more I think about it the more I realize that they didn’t like the fact that God defined who I was and not them. The person who defines who you are controls you, which is why identity is such a big deal in this society. My God loves and forgives, and that’s more than I can say about many people and institutions in this world.

These, then, are the two promises I made to myself past the mid-point. I am glad I did. I think you will be glad if you do too. May God richly bless you.