Jiggly Pokery and Models

In the last piece I set out some arguments for why I’m sceptical of climate alarmism. They may not have been very good arguments, but I think I’m justified in being ultra sceptical, these days, of whatever the official Expert™ Approved narrative is. The absurd propaganda surrounding covid, the censorship and censure applied, and the collusion of Big Tech, media and government to present a very one-sided (and ultimately false) narrative has utterly destroyed any trust I might once have had in official institutions.

Much of the alarmism over both covid and the climate occurs because of a reliance on models. So, today I wanted to spend a bit of time thinking about models and what they are, and what they do for us.

At a basic (and somewhat abstract) level we all rely on models. We all use some kind of descriptive and/or interpretative internal framework with which to navigate the world and our lives in it. We might, justifiably, think of these as models.

If you’re one of those TERFy types then your internal frameworks will lead you into a view of the world in which, for example, men do not menstruate. This is a false model and nothing to do with biology because, as any good grievance study person will tell you, men have been socially-conditioned to be embarrassed when buying certain hygiene products and this is the principal reason why men do not menstruate. As our society loosens up a bit we’ll see more and more men reach for the chocolate and chainsaws once every month.

Here, I want to discuss more concrete and less speculative (and less sarcastic) models; those generated by physics using the tools of mathematics. I’m going to ignore the fact that physics and mathematics, as we know them, were largely created by white men as a tool of colonialist oppression. I hope you will forgive me for this egregious oversight.

I’m going to run with a definition of physics as a way of trying to understand why the stuff in the universe behaves in the way it does. As a result of Quantum Mechanics (QM), many modern physicists would bristle a bit at the use of ‘why’ here. They would maintain that physics is more descriptive and concerned with the ‘how’. This is something I want to talk about in a later article, because QM is really fascinating and has some interesting implications.

The first thing we’re going to need is some kind of language. Without consistent definitions and agreed conventions, discussions are going to degenerate into meaningless squabbles in which confusion reigns. There’s an important modern-day issue where the language is impossibly vague, subjective and ill-defined, but I can’t quite remember what it is now. Anyway . . .

Setting up the basics

If we’re going to describe the world, and how things move (etc) we’re going to need some kind of framework, some language, that allows us to tell us where stuff is. We will also need to figure out some way of describing where stuff is moving to and how fast it’s getting there.

We need to agree on this framework, otherwise it’s going to be chaos. We might think of an Amazon worker looking for some small ordered object that is to be stuffed into a huge box along with a small forest’s worth of packing paper. He might be given the instruction (3,4,5).

This would mean:
start here
walk 3m East
walk 4m North
look 5m up

and there is that desired knick-knack. This triple of numbers is enough to completely specify where an item is, provided you know where to start from.

This is what we call a coordinate system. So, a triple of numbers (x,y,z) are going to allow us to talk with one another and to unambiguously describe where stuff is, as long as we all have an agreed upon start point to work from.

Then we need to work out how to characterize motion. If something moves from a position x = 2 to position x = 6 and it takes 2s to do that, then we would say it’s moving at 2 metres per second (if the x numbers are in metres). So we get an idea that a speed can be worked out by dividing “change in position” by “time taken”.

So far, we’re not really doing ‘modelling’, as such, just setting up a suitable descriptive framework¹.

You may vaguely recall from school working with equations that look like x = vt, or an equation (in words) that reads x equals half a times t squared. These are part of the set of kinematic equations that describe motion when we have constant acceleration. These are not really ‘models’ - just a consequence of defining position, speed and acceleration in a certain natural way.

Once we’ve got our head round the notion of speed, we now need to think about what happens in very short time intervals. It’s easy to see how to work out a speed from something like “it moves 4m in 2s”, but what about if we consider 1s, or 0.1s or a nanosecond timescale? What if the speed itself changes? Maybe in a 2s time period something moves really fast to begin with and then slows down (think of slamming the brakes on in a car, here). This is where we have to use a derivative. It’s a way of going from this clunky picture of time in chunks to considering what happens in an ‘instant’ of time. It’s still the same idea, though. It’s still a change of distance divided by a change in time - just now we have very, very, very, very, small changes of distance and time.

So, we look at how position changes
To get the speed we use the (time) derivative of the position
To get the acceleration we use the (time) derivative of the speed

So, speed tells us about how fast the position is changing, and acceleration tells us how fast the speed is changing.

You can see why physics is not easy - and why it bores most people into a coma. We haven’t even started ‘modelling’ things yet and we’re already deep in maths with its strange-sounding words and squiggles.

But there’s no way round this - we need this common language, this common framework, in order to proceed and to make progress. It’s not as scary as it first appears to be, but it does take a bit of practice to get enough familiarity with it so that it stops looking scary.

Everything Jiggles

Feynman once stated that “to a first approximation, everything jiggles”. It’s a great way of underlining the fundamental importance of oscillations and it happens because the fundamental forces in the universe are conservative. This does not mean that they’re right-wing, but is a technical term to do with conservation of energy.

So, to illustrate a physics model, I want to look at jiggles.

We’re going to think about a small mass attached to a spring. We know that if we extend the spring by pulling on the mass and then let go, we’re going to see the mass ‘jiggle’. I think ‘jiggle’ is a nicer word than oscillate, which would be a more technical and professional way to describe the motion.

How are we going to ‘explain’ or ‘describe’ this motion? How are we going to model this? How are we going to use our descriptive framework of position, speed and acceleration to get some insight here?

I suspect that most of us remember sitting in that math class at school trying to solve quadratic equations and thinking something along the lines of “what in the ever-loving F is the point of this?”. I know I did. It got even worse when we were told about things like sines and cosines and enjoined to mumble math mantras like “adjacent over hypotenuse”.

Well, you might vaguely recall that sines and cosines are jiggle functions. They jiggle. Might it be that these objects of schoolwork futility are useful in describing what happens with our spring? It turns out that they are, and here’s why.

What I’ve shown here is a mass attached to a spring in 3 ‘states’.

State 1 : it’s at rest - no one is pulling or pushing on the spring
State 2 : we’ve pushed on the mass and compressed the spring
State 3 : we’ve pulled on the mass and extended the spring

We need to set up our descriptive framework and so we say that in state 1, where nothing is happening, we’ll say that the mass is at a position given by x = 0. We could have chosen any number here - it doesn’t actually matter - but it turns out (and experience shows us) that this choice of the zero makes life a lot easier for us.

Before we do anything, and before we dive in to any problem, it’s really important to ask ourselves what is it, exactly, that we’re trying to figure out here? What constitutes a ‘solution’ or an ‘answer’?

We compress (or extend) our spring and let go. What happens next?

What we want here is to be able to work out the position (the x value) of the mass at a point in time. Where is the mass after 1/4 of a second? Where is the mass after 4/5 of a second? We write this solution as x(t) which is a symbolic way of representing this idea. It’s called a function - and it tells us where (what x) we have at some time (the t). So if we can work out what this function is we can just plug in some value of t (say t = 1/4s) and it will tell us the value x has at this time.

So, you can see what our job is here - we need to try to figure out what this x(t) looks like. And we want to do it in a way that’s as general as possible. We want an answer that we can plug numbers into and not an answer that only works for a specific set of numbers (we don’t want to have to redo our calculation every time we have a different set of starting conditions). This generality is one of the benefits of algebra and an algebraic approach when modelling.

The Model

At this point it’s not clear how to proceed - because there’s a bit missing. We need to think.

State 2 : we’ve compressed the spring - we know, from experience of these things, that if we let go, the spring is going to ‘decompress’ and push the mass back towards the zero position - the x = 0 position. In other words, the spring is pushing on the mass in a certain direction - it’s exerting a force.

State 3 : we’ve extended the spring and we know that when we let go, the spring is going to ‘pull’ the mass back towards the x = 0 position. It exerts a force in the opposite direction to that of state 2.

At this point we recall another of our boring physics lessons from school and the equation F = ma, which relates force, mass and acceleration. This is Newton’s 2nd law of motion and we can think of this as a ‘model’ that represents, mathematically, how the world behaves.

We know from our setting up of our basic framework that if we take position (this is our x(t) function) and differentiate it, we get the speed. If we differentiate it again, we get the acceleration. So, we can write Newton’s 2nd law as

F = mx''(t)

The double dashes here are a shorthand way of writing differentiate twice. OK, now we might be getting somewhere - although it’s still all a bit murky at this point.

To proceed further, we need another model. This is the model of what force is exerted by a spring. This is known as Hooke’s Law and was an empirically derived model. Hooke found, by experiment, that if you didn’t overdo it and stretch or compress too much, the force exerted by a spring could be written as F = - kx

The k here is a constant which reflects the ‘stiffness’ of the spring. The negative part here is because the force acts in the opposite direction to the compression or extension. If you extend the spring (make x positive) the spring force wants to pull you back (the negative direction). If you extend or compress the spring more, the force wanting to ‘restore’ things gets bigger.

Now, as if by magic, we have what we need. Using both Hooke’s law and Newton’s 2nd law we can write

F = mx''(t) = - kx

So, we have something involving our desired solution x on the LHS, and something involving our desired solution x on the RHS.

We have the equation mx''(t) = - kx and we need to solve it to get our desired answer. This is an example of something called a differential equation and to set it up we’ve had to (a) apply a framework using coordinates and (b) apply models of motion and a force.

The differential equation is asking, in math terms, the following question :

what function, x(t), when I differentiate it twice, gives me the function again, multiplied by a negative constant?

I’m not going to go into how this equation is actually solved, but it should come as no surprise that the answer we get is one of those jiggle functions - a sine or a cosine.

Now, after having quite thoroughly bored the arse off most of you (assuming you’ve even read this far), let’s make a few observations.

What, in the ever-loving F, is the point of all this?

The first thing to notice is the amount of work and explanation that I’ve needed to even get things into a form that could be ‘solved’ - and this is a really, really simple physical system. It is, perhaps, the simplest non-trivial physical model we can think of. It also happens to be of critical importance for much of physics².

It’s a good starting point to think about what a ‘model’ looks like in physics.

Is it realistic? Well, not really. We know that if we set a spring jiggling it isn’t going to do that forever. Extending and compressing a spring is going to generate some heat (energy will be lost to the environment), there are going to be other resistive forces - and so the motion is going to stop unless we keep pushing and pulling. We need to add a damping force into our model if we want to get a closer representation of reality. This is adding in yet another ‘model’ to account for damping (it’s usually modelled by adding in a term proportional to the speed as a first attempt).

So, already we know that we’re going to have to add in some more complexity.

What if we had two masses and two springs connected together? In this case we’d do essentially the same set-up, but now we have the motion of one mass affecting the motion of the other. We end up with 2 separate differential equations (one for the position of each mass) - but these equations are coupled. We have to solve them as a system.

We very quickly are in a whole new layer of complexity - and we’re not even looking at a particularly complicated system.

This physics model can be tested. It makes quantitative predictions. The underlying framework and models (Newton’s laws, Hooke’s law) have been tested to kingdom come and back again. We have a pretty good handle on their domains of application and where they work. Hooke’s Law, for example, is only an approximation and if we over-extend the spring it fails and we need another model to describe the behaviour in this instance.

What about covid models, or climate models?

The first thing to notice is that they are hugely more complex than the simple spring/mass system. Just as our spring/mass system model required some input from another model (Hooke’s Law) the covid and climate models require this kind of input too.

In the spring/mass system we had a single thing to work out - our x function. When we had things coupled together (connecting two springs) we had to work out a system of equations where each individual equation depended on the other.

With covid or climate models you end up with a horrible system of many coupled differential equations to solve - and these are probably also going to be stochastic differential equations (yet another layer of complexity requiring assumptions and models about the precise nature of the stochasticity).

These models can’t be solved ‘by hand’ - they’re going to need some serious computer time and some serious coding by people who (a) know how to code and (b) how to handle the accumulation of errors introduced by necessary truncations.

The other aspect I haven’t even touched on is the issue of nonlinearity and feedback. With most simple models we have an intuitive feeling that if we’re out by say 0.1% with our conditions (we might not measure the extension of the spring with better than this accuracy, for example) then it’s not going to affect our final solution that much. When you add nonlinearity into the mix we can sometimes lose this illusory safety - and most of you will probably have heard of something called the ‘butterfly effect’ where even a small difference in initial conditions can lead to vastly different behaviours as a result.

None of what I’m saying here should be taken to imply an automatic rejection of complex models. But it very much means that very significant levels of care and attention to detail are required to generate a model for a complex system that lays claim to be any representation of reality.

The acid test for any model that purports to be a representation of reality is that its predictions match what actually happens. Notice that even getting this right does not imply that the model is ‘correct’. It’s a necessary, but not sufficient, condition that the predictions match observation. We do know, however, that if our model does not match reality then it’s not correct (it may be incorrect, or incomplete, or both).

The climate modelling is, at least, based somewhat on tested physical models (Newton’s laws, fluid dynamics, etc). Covid models rely on largely unproven assumptions about things like infection rate, virulence, population movements, etc - they’re not even starting from a well-grounded set of physical laws.

But the main problem with both the covid and climate models that have been used to generate fear is that they’re just crap at making predictions.

It’s not good enough. If we’re going to shut down the world, or make everyone eat cockroaches, then I think we deserve better than the crappy models that have been delivered so far.

1. It could be argued that the descriptive framework is, itself, a kind of ‘model’ which makes implicit assumptions about the way the world ‘is’. This would be true, but I don’t want to get too far down any recursive Riemannian rabbit hole here.

2. Photons, for example, can be described as occupations of quantised field modes - and these ‘modes’ arise from an oscillator model that is fundamentally no different from this simple spring system.