Most of us, myself included, don’t have either the time or the expertise to get involved in the nitty-gritty of the mathematical details of things like the climate models or the covid epidemiological models. But a smattering of basic mathematical understanding can help us frame our ideas on these subjects.
Most of us actually use the things I’m going to be talking about without realizing it. Even those people who, understandably, have a flash of PTSD from those days in school when some weird person at the front tried to tell you what a function was, along with various other things written in squigglish.
For example, if you’ve ever thought that exposure to social media results in an increase in mental health conditions (such as depression or anxiety) in adolescents, you’re effectively thinking in mathematical function terms. You’re thinking in terms of a ‘model’ in which you have some input variable (amount of exposure) and some output variable (amount of mental health problems) and postulating a relationship that says more exposure = more problem.
Many people have a kind of instinctive ‘shut down’ reaction whenever they see squigglish (i.e. math formulae). I’m no different, except that I have learned not to shut down for more simple squigglish. But a page full of advanced research math has the same effect on me; cold sweats, crossed eyes, that sort of thing.
The two great propagandistic plagues of our time, covid and climate, are also beset by these kinds of simple math ‘functions’ when it comes to how things are phrased. Examples of this would be
More people wearing masks, fewer infections
More vaccinated people, fewer deaths
More Carbon Dioxide, more global warming
Once you’re aware of this kind of thing, you’ll see that an awful lot of policy is made on the basis of these kinds of arguments. It’s not just restricted to covid and climate.
So, in my view, it’s very useful to have at least a basic idea of functions in a maths sense, and how to think about them.
Mathematicians love to take a simple idea and turn it into something impossibly complicated. Actually, there are very good reasons for doing this, but if I admitted that I’d lose my opportunity to have a friendly dig at mathematicians. So, the proper notion of what a function is, mathematically, can be quite complicated and sophisticated. I’m going to take a simple approach that is good enough for our purposes.
A function is, basically, a theoretical input-output machine. You stick your input in one end and out the other end comes the output. A bit like a sausage machine in which the input is the scrapings from the abattoir floor and the output is a lovely tube of flavoursome beauty.
A different machine (a different function) might take the same scrapings and turn it into a burger patty, for example.
We write this theoretical thing as
where y is the output, x is the input, and the f, the function, represents what kind of ‘machine’ we have.
We can model all of these arguments that inform policy in this way.
We could have
y (the output) : number of adolescents reporting anxiety
x (the input) : number of hours spent on social media, on average
The function, then, would tell us how ‘bad’ the problem was. If we double the number of hours do we see double the number of reported cases of anxiety? Or do we have a different relationship?
It’s the function part that is often either glossed over or ignored entirely when politicians warble their warnings of dire outcomes.
I’ll come back to this, but there’s also another ‘trick’ that is often used, and that’s to reduce everything to a single variable. We might, for example, have a function that looks like this
Here, the z is now the output and the x and y are the inputs. The output might still be a sausage, but the inputs here might be the scrapings (x) and some herbs (y). Focusing on only one input variable is going to give you an incomplete picture.
For the moment, let’s just focus on one input, one variable of interest.
Entirely at random, let’s think of some variable of interest. Oh, I don’t know, should we, for some inexplicable reason, pick the concentration of Carbon Dioxide in our atmosphere? Should we? OK, let’s do that.
So, the claim by the climate panic peeps can be reduced to the following; more CO2 means a bigger temperature (usually the parameter is a global mean offset from some chosen baseline).
Mathematicians call these kinds of functions monotonically increasing, which means that as we keep inputting higher ‘x’ we get a higher output ‘y’.
Let’s accept this claim as true for the climate.
Quick, let’s stop burning stuff and eat cockroaches. It’s the only thing we can do to prevent catastrophe.
Not quite, because what’s missing in that claim is HOW the function is increasing.
Since the covid clownery we’ve all become experts in what an exponential function is, because the Experts™ and media blathered on about them so much and tried to scare the crap out of us, so is the temperature offset rising exponentially, or in some other way?
Let’s take a look at 3 basic functions, all of which are examples of monotonically increasing functions.
The first just takes the input and squares it; double the input and you will quadruple the output. The second is a straight linear function; double the input and you’ll double the output. The third is a logarithmic function, which is the ‘inverse’ of the exponential function.
They’re not overly informative in squigglish unless you’ve studied squigglish, so let’s see what they look like using a graphical representation.
Now, all of these functions are on their way to ‘infinity’ - they just keep getting bigger and bigger - but they’re each heading there at a different rate.
The blue line, the square function, looks like a kind of runaway scenario. The green line, the linear (straight line) function, looks a bit worrying. The red line, the logarithmic function, looks like things are sort of tapering off and maybe not quite as panic-inducing.
So, which of these 3 ‘types’ of function do we have when it comes to the climate? Dunno. It’s hard to say either from the models or the data.
The reason why I’ve been prompted to look at this is the recent article in The Daily Sceptic which reported on a paper that argues that there is a “saturation” effect. The more CO2 you add, the less effective at warming it gets.
They’re arguing that what we really have is the logarithm-type scenario when it comes to CO2 concentration in the atmosphere.
We all intuitively understand this kind of thing. If you’re putting some protective wood stain on that faded wooden chair in the garden the first coat makes a huge difference. The second coat looks a bit better, but you’ve not doubled the impact by spending twice as long (adding the 2nd coat). Add more coats and you’re in “law of diminishing return” territory.
We can use differentiation to get an idea of how this works for these idealized scenario ‘types’ represented by our 3 basic functions. We use the Greek letter capital delta, Δ, to represent a change and we can get an approximation for how the output changes when we make a small change to the input. The formula looks like this :
which reads as the change in y is equal to the derivative of the function times the change in the input x. For our 3 function types above we have that
Let’s suppose we’re currently at a value of x = 4 (which might stand for 400ppm for example) and we increase x by 0.1. We would then have
Change in y (square function) : Δy = 0.8
Change in y (linear function) : Δy = 0.1
Change in y (logarithmic function) : Δy = 0.02
That’s quite a big difference between the 3 example function ‘types’.
If we differentiate again, we’re looking at the ‘acceleration’ of these function types.
The square function has a constant ‘acceleration’ and so the ‘speed’ is just getting faster. The linear function has a zero ‘acceleration’ and so we’re just tootling along at the same ‘speed’. The logarithmic function has a negative ‘acceleration’ and so the ‘speed’ is slowing down.
Which of these 3 different regimes of ‘acceleration’ are we in when it comes to the climate? The paper reported in The Daily Sceptic would suggest we’re more in the ‘logarithmic’ kind of regime with a negative ‘acceleration’.
But look how different each of these regimes are. They all have the same basic property of increasing as x increases - which is the basic claimed property of the climate anxious - but they’re doing that in very different ways.
The mere claim, even if it’s true, that some thing y is getting worse as you increase x, is nowhere near enough information upon which to base any policy decision; you also need to know HOW it’s getting worse.
But that’s not all. You see what we’ve done here is ‘reduced’ the entirety of the climate problem to a single variable; the concentration of CO2 in the atmosphere.
What we have, in practice, is a function like y = f(a,b,c,d,e, . . .) where there’s a whole bunch of inputs that go into affecting the output. We also don’t know what things would have been like if man hadn’t been invented. What would the ‘natural’ climate regime be right now if we hadn’t seen the emergence of humanity? There’s good reason to suppose that we’re in the middle of a ‘natural’ warming cycle anyway, and so the question of how much CO2 is perturbing that natural evolution is a very relevant one.
Analysing a simple function of one variable is one thing, but when you’ve got two (or more) inputs, or variables, then it gets significantly more complicated. You can’t even properly ‘picture’ these functions, in general. We can represent graphically a function of two variables like z = f(x,y) using a 3D plot that might look something like this
Here the ‘height’ of the graph is the output, the z. You can see that focusing on one variable (the x say) is like taking a ‘slice’ at some value of y. You get some information from a slice, but you need the whole thing to properly see what’s going on. If you have more than 2 inputs you can’t even draw a graph like this; you run out of spatial dimensions for the representation.
So, when a politician burbles and blathers about the effect of some x making some y worse, we need to take a step back and ask two very important questions :
How, exactly, are things getting worse? Runaway, linear, or slowing down?
What other inputs (variables) affect the output? Have they been included?
When it comes to climate change, these are fantastically important questions because our future wellbeing and our ability to live free and decent lives is on the line. Some politicians are wanting to impose limits so that you can only eat 300g of meat per week, and buy only 3 new items of clothing every year, to name just two things they want to control.
There are other functions we might want to consider, such as ‘threshold’ type functions like the sigmoid function, but we can see that just a little bit of knowledge about math functions helps us to zone in on the important questions.
"What other inputs (variables) affect the output? Have they been included?"
Seeing this at the end of the article reminded me that I'm sure I read/heard somewhere (possibly the Ukranian professor of physics whose name eludes me) that the climate alarmists don't really take into account the effect of the sun, so I tend to assume all their outputs are pretty worthless.
"Mathematicians love to take a simple idea and turn it into something impossibly complicated."
Kind of the reverse of what economists do then.
Getting flashbacks to fourth grade reading this, that was when functions and equations used to be introduced in schools here way back when. Now, they do it 7th-8th grade. Or whenever the teacher deems the class as whole up to speed. Or when the teacher feels like it. Sounds messy? The consequence of "liberalising" (think Thatcherite neoliberalist policies) the school system in the early 1990s - total chaos, unpredictability and grades that mean squat diddley.
You could say the decline in knowledge and the increase in cost is a direct function of liberalising the system.
I remember doing first semester Macro, Micro and national economics at university - math skills were so deteriorated even then that they needed an extra zero credit compulsory class for people to be able to understand the formulae they'd be using, instead of just being able to feed data into machines preprogrammed with the relevant formulae. And I'm not a math-head, it bores me to tears. Even ghostwritten biographies of politicians are more interesting.
Then again, how someone /wants/ things to be almost always trumps how things /are/; accepting that and learning to recognise it in oneself (or one's self, maybe?) is key to "figgering fings owt". I want to grow pumpkins. I can either insist it should be doable, or I can figure out how to fix the soil, how to maintain at least +10C temp all the time, protection against grubs and creepy-crawlies, and so on.
Which one is the path of least resistance?
And there's the problem - no (selection) pressure on eggspurts, politicians, banks, copro-rations, and so on to do good or be good (as per Aristotle's reasoning). Lots of incentive for doing the opposite. Let the dog nick stuff from the table, it won't ever stop doing it.