Mathematics


One of the big topics in the skeptical community at the moment (like everywhere else I guess) is the climate change issue. It’s a subject that is as fraught with debate as that of Evolution vs Creationism, and indeed, has many of the hallmarks of that particular tussle. What makes it particularly volatile in this setting, though, is that many of the people who claim that there is no need to worry about global warming paint themselves as climate change skeptics, and take the position that they offer a rational approach to the debate. What they are in fact doing is voicing opinions that are in contradiction to MOST of the world’s knowledgeable climate scientists. Though they like to think of themselves as skeptics, this stubborn entrenchment in a belief system has earnt them, instead, the badge of climate change deniers.

I pretty much stay out of the climate change argument, just as I stay out of the Creationism debate. It’s not that I don’t have a strong view on global warming. I think the scientific evidence is conclusive that we have a looming disaster on our hands, and that it’s a disaster of our own making. Bothering to argue with the deniers though is the mental equivalent of jabbing a sharp pencil repeatedly into the back of your hand – a sensible person stops doing it pretty quickly.

The main problem is that, as with evolution, climate science deals with concepts that don’t come easily to the natural human way of thinking. With evolution it has to do with vast amounts of time (which we’re not good at comprehending) and the complexity of the vectors that come to bear on natural selection. With climate science, it’s all in the maths. I’m going to attempt in this post to show you why, even if you haven’t kept up with all the marginalia of the climate discussion, you should be afraid of what we’re doing to the planet.

At the outset I will state that my essay takes one idea as a given: that global warming is a human-instigated phenomenon. You should understand that a cornerstone of the denier’s ‘argument’ is that it isn’t, but I will stand behind the overwhelming scientific viewpoint on this matter.(i)

OK. We’re going to talk about math in this, but you don’t need to understand numbers. And I promise you, it won’t be dull. This is a very scary story. I’m going to divide it into three chapters.

Chapter 1: Boiling the Frog

There is an old fable – it’s probably apocryphal but for our purposes it doesn’t matter – that says that if you take a frog and put it in a bowl of water over a burner and slowly raise the heat, the frog, unable to feel the very slow rise in temperature will make no effort to leave the water and happily sit there until it is boiled to death. In other words, it either doesn’t realise there is a problem, or, by the time it does, it’s too late.

The story illustrates a psychological phenomenon called ‘creeping normalcy’ (or in science, the ‘shifting baseline’ problem). Put simply, it says that if you have changing reference points, you can only judge what is ‘normal’ by what you’re familiar with at any given time. In this way, familiarity changes the baseline of ‘normal’ to whatever you get used to, and if things change slowly enough, ‘normal’ can wander an awfully long way from ‘acceptable’.

The first step towards understanding why the climate issue is so deadly is to understand that humans think like this as a default. Our brains don’t work well on timescales in excess of a few years. Our horizons are small. I’m not the first to mention the Boiling Frog concept in relation to the climate change situation, so its appearance here is no big revelation. But you need to keep it in mind as we head off to chapters 2 & 3:

Chapter 2: The Big Clock

I recently saw a comment on an article in The Conversation from one John Dodds, a ‘retired engineer’:

First a philosophical point: Climate Change is claimed to be complex. I claim it is NOT. It is simple physics – add more energy and the world warms up.

Mr Dodds’ opinion typifies the way in which most people believe that the planet’s climate system behaves – something like a Big Clock. A wheel here, a cog there, a spring yonder – all ticking away in a simple predictable manner that can be completely described if you do the right calculations. Most people think, therefore, that if we’ve caused some kind of problem with the climate, then all we need to do is to ‘oil the gears’ on the clock and everything will go back to the way it was. They believe that the problem is proportionate to the actions we take to correct it.

This is a massive and perilous failure of understanding. It’s a mechanical Newtonian notion of the way things work that is fine for pipes and balls and clocks, but breaks down catastrophically when applied to something like climate behaviour. To grasp why, we have to venture into the frightening, mind-bending and completely unintuitive world of complex systems.

clock

First, let’s consider the pendulum in our Big Clock. As physical systems go, this is about as unadorned as you can get. A swinging pendulum exhibits what is known as simple harmonic motion(ii) and it is a very reliable behaviour that allows us to build a clock that will behave predictably and dependably. A simple pendulum is mathematically very straightforward. Its properties can be described completely in terms of the length of the ‘rod’ of the pendulum, gravity, the mass of the ‘bob’ on the end of the pendulum and the angle of swing. If you know these things, you can predict exactly how this pendulum will behave. This uncomplicated mechanism works great for a clock, and it’s fairly tolerant of perturbations in the system: if you push the pendulum a little hard, it will dampen down to its normal swing pretty quickly. You need to be pretty violent to cause the clock to have problems big enough to effect its function.

This is the kind of path we could expect the bob on the simple pendulum in our clock to trace. Every time:

Unfortunately for us, the climate system isn’t driven by a simple pendulum.

Let’s consider a physical system only a tiny step away from our Big Clock’s single pendulum: the double pendulum. A double pendulum makes one small alteration to the simple pendulum model – instead of a simple bob at the end of the pendulum, you add another pendulum. This very unassuming variation has sudden and profound effects.

Here’s a computer simulation of the path traced by the tip of a double pendulum:

If that looks weird and science fictiony to you, let me assure you that double pendulums behave exactly like that in reality. There are dozens of YouTube videos that show them in action.

You can see how this one small change to our pendulum quickly throws a simple harmonic oscillation into a volatile and complex motion. The double pendulum system can be very easily described,(iii) but its ultimate behaviour cannot. Each time you set it swinging its bob will trace a completely different path in space because, crucially, a double pendulum is very sensitive to initial conditions. Unlike our clock’s pendulum, we can’t accidentally give it a bit too much of a shove and have it simply settle back into its predictable ol’ groove.

Imagine, now, that you have a pendulum with n arms, each with a bob of mass n, n points of articulation and variable gravity. It doesn’t take much of a leap of imagination to understand how wildly such a device will behave. In fact (and this is where most people fall off the bike), for surprisingly small values of n, no amount of computing power in the universe can ever predict the path of motion it will describe!

Well, the Earth’s climate is exactly such a system.

Unfortunately one thing that tends to be a little confusing with this is that climate scientists often speak of ‘climate modelling’ and to many people this sounds again like they’re talking about some kind of Big Clock: you stick in all the variables into your computer and ‘ping’ – out comes the behaviour that the Big Clock will exhibit. If it were only that easy.

When you look up a weather report on your i-Device of choice, you’re seeing climate modelling at work. One thing I probably don’t have to tell you, is that you shouldn’t rely on the information more than a few days ahead. That’s the state of the art in climate modelling. We’re just not very good at predicting the behaviour of complex systems (like weather) even a few days in advance. Here’s the kicker: it’s not our fault! These systems are inherently unpredictable. Even if we had super-super-super computers, we couldn’t do it. Even if we had a computer that could take ALL the variables – and that’s a HUGE amount of variables – and then run the simulation in real time to see what it did, it would do us no good – we would get a different outcome every time we ran the program. Just like its very simple distant relative, the double pendulum, a complete detailed model of a complex system like the climate is critically dependent on initial conditions. (We actually do have such a computer – it’s called ‘Reality’. The only accurate simulation of what the climate will do is the climate itself).

So, when you hear scientists talk about modelling the climate, you should not understand that to mean they are trying different kinds of wood for the clock case, or a new type of oil to make the gears run smoother. They mean they are making their best educated guess at the Big Picture of what might happen if they picked enough of the right factors to plug into their equations. Just like you understand the weather man to be doing when he tells you that in a week’s time it looks like rain (are you starting to get nervous yet? No? Then you’re not following me).

So what’s the problem, right? We don’t know what the weather will do – why is that different from any other period in our history? Why are we suddenly worrying now? Well, one of the things that modelling can predict pretty confidently is trends. Just as we can say that a double pendulum pushed gently is unlikely to do the crazy loop-the-loops that we see in the same system dropped from a higher angle, models can tell us that when we change something in the climate system too much, we’re likely to see unpredictable behaviour. In recent times (the last few million years or so) the climate has been ticking along like a gently-pushed double pendulum; little flurries here, little irregularities there, but for the most part, predictable enough for life-forms to have evolved strategies to cope. Things do change, but they change slowly. The system keeps itself in check through millions of years of self-modification that has allowed it to reach a relatively stable, though delicately balanced, equilibrium. The evidence is clear, though, that over the last few hundred years (a VERY short period by geological standards) humans are swinging the pendulum’s arc wider and wider by the simple act of burning things. We’re taking carbon that has been for eons locked up in the biosphere and chucking it into the atmosphere where it has started to imprison the Earth’s heat. We can, therefore, state with a high degree of confidence (based on an enormous amount of accumulated data) that the planet is heating up monumentally faster than it ever has before, and that that heating-up is concomitant with the technological period of humans.(iv)

But when climate modelling scientists make a ‘prediction’ that the temperatures will rise 3 or 4 degrees by the end of the century, you should not think of that as a jolly nice warming of the winter months, and the odd extra scorcher in July (or January, depending in which hemisphere you live). You should instead interpret it to mean ‘We figure the whole system is going to heat up, but how it delivers that heat, and to whom, depends on the swing of the double pendulum…’ What you should expect is periods when the weather seems just as it always has, interspersed with occasional outbursts of extreme behaviour. For a while this will seem normal, and you will be as happy as a frog in a warm pond. But this extreme behaviour itself will start to interfere with the system – it’s another phenomenon of mathematics which those in the know approach with respect: feedback. And that feedback will almost certainly affect the system in ways which we can’t even imagine.(v)

This coupling of complex behaviour and feedback is the thing which frightens the scientists, because it’s something with which the world of science has become very familiar in the last fifty or sixty years. We know that a complex system exhibiting instability and feedback can suddenly and capriciously become chaotic. That is, the system is likely to reach a point where even modelling is completely useless – it just goes completely berserk.

Trust me when I say that we really don’t want to see our climate system go chaotic. If we hit that point, it is likely that the great majority of the human race will suffer.(vi)

Chapter 3: Jenga

clock

The kind of critical instability that I’ve just described is a lot like the game of Jenga. The Jenga tower will remain upright as long as the system is stable around its centre of gravity. If you lived on top of the Jenga tower, you would probably be aware of nothing at all as pieces are removed. Maybe the tower might wobble a bit, but, hey, things look pretty normal. Every removal of a Jenga tile is exactly the same kind of small effort, but each one of these small efforts moves the system closer and closer to critical instability. When the Jenga system reaches this point, the collapse into chaos happens rapidly and catastrophically, with little warning.

Well, that’s where we are right now. The tower is wobbling a bit, but everyone is saying ‘Hey, the tower has wobbled before and we were OK – what’s the problem? Worse, we continue to slide out the pieces, because that’s what we’ve been doing for years and it’s been just fine.

Unfortunately, this kind of situation is the very worst sort of thing to try to get resolved by a ‘popular vote’. When you combine the Boiling Frog situation with the Big Clock scenario and stir in a whole lot of poorly educated(vii) points of view, you just get lots of personal assessments of the problem – or debates about even whether or not there IS a problem – and a bucketload of total inaction. The grim truth is that it’s a state of affairs that seriously needs everyone on the planet to be in complete agreement, or we will, without doubt, plod our way into extinction.

The way it stands at the moment is that the vast majority of people are either uninterested or confused, a small minority is in denial and beset by superstition and petty agendas, and another small minority is informed but frightened, frustrated and powerless. I think that what we are seeing here are the ramifications of a massive failure as a species to improve ourselves by putting emphasis on the capacity to understand our world through observing it properly. That is, through science. It is of no use to put an appropriate course of action to a popular vote in this instance, because the holders of a popular vote aren’t equipped to understand what it is they’re voting on. And frankly, I think we’ve run out of time to get them up to speed. Added to that is the negative influence that whatever we need to do will, most likely, cause great inconvenience to a large number of people, and will include increased poverty, loss of jobs, deprivation of luxuries (and maybe even necessities for many) and a general willingness to just suck it up and take a beating. It doesn’t take much insight to see that we’re never going to get people to volunteer for that, unless they become very afraid indeed (by which time – I emphasize once more – it will be way too late).

If ever there was a time for the leaders of our nations to listen to the science, and act decisively and quickly for the good of human race, this is it.

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Image of the Earth courtesy NASA and the Visible Earth project.

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Footnotes:

  1. If the deniers are right on this and global warming is an inevitable natural process, then we’re in a handcart to hell anyway, and it doesn’t matter what we do. So we may as well make efforts to ameliorate the situation as not. An argument of financial imperative (‘it will ruin our economy’) is quite irrelevant because in a hundred years there won’t be an economy. []
  2. For small angles of swing. As the angular acceleration increases things become a little more complicated, but for our purposes we can assume true simple harmonic motion. []
  3. We still know the lengths of the rods, the mass of the bobs and the gravity coefficient. []
  4. We’re excluding events that happened in geological times of many hundreds of millions of years ago, where lots of weird climate events happened. They are not relevant to our argument because we weren’t involved. If we had been, we’d be dead, which is of course the issue at hand. []
  5. We don’t really have much of an idea of the way the climate system is held in such delicate check anyway – global atmospheric behaviour is without doubt one of the most complex systems we know. All we can say for certain is that that if it changes much, we are in trouble. []
  6. It should be understood here – because I often think that it’s not – that the planet is indifferent to this problem. You hear climate deniers putting forward ideas like ‘Well there have always been periods of global warming’ or that ‘Sea levels have changed many times though the Earth’s history’. Well, sure. But mostly, there were no, or few, humans around, and other creatures were affected by these events, often in the form of species-wide extinctions. The Earth was once a giant greenhouse, covered with plants. But WE could never have lived in it. The planet would probably survive quite extreme results of our global warming efforts – it’s just that we wouldn’t. []
  7. I say ‘poorly educated’ because I think that even the great majority of people who are literate do not have a good grasp on science, nor on rational ways of thinking. Any of you who have been reading TCA for a significant period of time will understand exactly what I mean here. []

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Quantum physics is clearly at work in the new Trinfinity8: Energy-On-The-Go iPhone and Android phone app. This unique app streams natural mathematical algorithms to the user via earbuds to help balance energy, feel better, and even boost sexual libido. And all that for only $3.99.

How many of them did you count, Faithful Acowlytes? You know what I mean: woo buzzwords. Let me see, there was ‘quantum’, ‘natural’, ‘balance’ and ‘energy’. No mention of fractals or magnets, but I’m sure that if we visit the Trinfinity8 website we’ll get at least one of those…

This unique software program was developed as a direct result of information brought back from a near death experience by Dr. Kathy Forti. Trinfinity8 is the first system of its kind to use a personal computer to deliver non-invasive rejuvenation programs based on mathematical codes, vibrational energies, and fractal formulations that are in harmony with core energetics that encompass all of nature.

Haha. Well, look at that. Aside from fractals we got a bonus ‘vibrational’, as well as ‘harmony’ and ‘energetics’. It’s a veritable treasure trove of woo buzzword bingo.

I will save you the interminable chore of wading through the addled pseudoscience that makes up Trinfinity8 by telling you that it is software (clearly using quantum physics) that plays musical drones and movies of fractal patterns that pretend to have all kinds of magical powers. How does it work? I’m glad you asked:

Trinfinity8’s unique technology allows for streams of coded data to be transmitted through your computer’s USB port.

As opposed to what’s normally transmitted through your computers USB port which would, of course be… oh, that’s right – streams of coded data!

A digital translator device then sends information to the body via specially designed hand-held quartz crystal transmitter/receiver rods

Bingo! ‘Crystals’! I win.

Trinfinity understands that you may not always have a computer nearby and so have made their software available in a handy iPhone version: Trinfinity8; Energy On-The-Go. What better way to christen my new iPhone 4, dear Cowpokes, than with this app. You know my commitment to exposing all things ridiculous is profound when I am prepared to part with my hard-earned money to bring you personal demonstrations here on The Cow. So I downloaded Trinfinity8; Energy On-The-Go from the Apple Store and set it running:

Here you see me using the Male Libido Boost program. I can assure you I was not at all aroused when I was doing this. Of course, to be fair to the Trinfinity8 instructions I should have had my two thumbs on the screen, rather than a thumb and forefinger. It’s a little difficult to take a photo while doing that, however. Personally, I think that even if I’d had a thumb and my penis on the screen the effect would have been exactly the same.(i) What basically happens is that a little movie of an animated fractal cycles back and forth and a droning tone plays. The drone is supposedly made up of what the Trinfinity8 website calls ‘Solfeggio sound tones’.

These pure sacred tones have been used since ancient times to awaken a natural expansion of consciousness in order to bring about transformation. In Trinfinity8, the tones act as a carrier wave to further strengthen the transmission of digital data to the cellular system.

Of course they do. Let’s do a Search™ on solfeggio tones. Crikey. It’s a Google Jackpot of WooWoo sites. Looks like Solfeggio tones are the new Schumann Waves. But it’s not just ‘music’ that comes out of your iPhone earpiece, my friends. Oh no! The Trinfinity8 iPhone app mainlines PURE MATH straight into your ears!

The algorithms are delivered through earphones sub-audibly with the music.

Sub-audibly. As in ‘you can’t hear them’. As in ‘if they weren’t there, no-one could tell’. As in ‘total utter fucking bullshit on stilts’.

Oh, really, I can’t go on. The Trinfinity8 website is such an addled and meandering mess of stupid doublespeak and woo weasel language that it’s painful. It’s particularly offensive to me, since, as some of you know, I have a great love for mathematically based image generation (which includes fractal work) and mathematics in nature. Dr Forti, the ‘brains’ behind Trinfinity8 (who you can hear babbling on and on here, if you are so inclined) has had, in effect, a ‘vision’ that told her to bring this half-baked concept to the world. As is the manner of such visions, it makes little sense to anyone who is not taking the same drugs.(ii) Dr Forti quite predictably claims that her system has efficacy in just about any field where a subjective outcome is available: stress reduction; ‘energy’ restoration; libido improvement; skin & hair rejuvenation; slowing down the ‘signs’ of aging (WTF?); weight loss.

The Trinfinity8 site is, in fact, one of the biggest agglomerations of utter crap that I’ve seen in a very long time. Here are some choice snips from around the Trinfinty8 universe:

Trinfinity8™ Energy On-The-Go uses uniquely designed geometric fractal images to amplify the algorithms. Quantum physicists are discovering that enlightenment is the charge you attract when your thought patterns get fractal.

Is that what quantum physicists are discovering! And here was I thinking that they were after the Higgs boson!

A fractal is a geometric shape that can be split into billions of parts, each of which contains an exact copy of the whole.

No dear. That’s a hologram, not a fractal. Like most everything else in your ‘theory’, you have taken a half understood idea and mushed it into another half understood idea, ending up with a completely useless factoid. If you want a proper definition of a fractal you could try: A curve or geometric figure, each part of which has the same statistical character as the whole.(iii) I don’t suppose you even have the remotest clue how that is different from what you wrote.

The Science of Fractal allows phase conjugation and a unified field for waves of people.

Now you don’t even know what that means, do you?

Trinfinity8 incorporates the most sophisticated technology that sends billions of bits of information to the body to help effect positive vibrational and consciousness change that is in alignment with nature.

Wha? Have you been gargling with Special One Drop Liquid, by any chance?

Energy On-The-Go does not contain any subliminal messages or binaural beats. This application contains pure information algorithms.

Well, that’s nice to know. I certainly wouldn’t want subliminal messages working on me, ‘cos that’s, y’know, BALONEY isn’t it! As for binaural beats, well, they’re some kind of hippie bullshit too! Not like the jen-yoo-wine, rooty-tooty, honest-to-goodness fractals and crystals of Trinfinity8!(iv)

Digital messages are sent to help dissolve unwanted fat and cellulite.

Yeah, right. I think maybe you should re-tune them to dissolve unwanted court cases, because with claims like that, you are really asking for them.

(The) fractal resonator amplifies the energetic and mathematical codes for the most powerful remedies, rejuvenators, and body elixirs that have been digitally imbedded into Trinfinity8 to maximize the desired treatment effect.

Jesus H. Christ. Is there anything Trinfinity8 doesn’t do? Or, perhaps, a more pertinent question: is there any ridiculous pseudoscientific buzzword that isn’t used on the Trinfinity8 site? Oh. Magnets. I couldn’t find magnets. Maybe they should fix that.

Well, that’s all for now my friends. I see, though, that now I have purchased my Trinfinty8 On-The-Go for my iPhone I am entitled to Customer Support. I think I just might take advantage of that offer. That should be… enlightening…

[Travel into the FUTURE and see how I fared with my customer support!]

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A big thanks to Universal Head for bringing the Trinfinty8 madness to The Cow’s attention.

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Footnotes:

  1. And no, I’m not going to take a photo of that, even if I could. []
  2. Seriously, I remember having a conversation once with a guy who was off his face on mescaline, who promoted pretty much the same idea as Dr Forti. The main difference is that he finally came down and Dr Forti is still tripping. []
  3. Thank you Wikipedia for your staunch support in the face of woo. []
  4. What is this tactic? Diss other idiot ideas in promotion of your own unhinged ravings? Do people really fall for that? “Oh, I don’t hold with binaural beats – fractal brain massage is MUCH safer!” []

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My current fixation with soap bubbles came about as a side-effect of some research I’ve been doing for my new animation project. To be specific, I was looking at how transparency and refraction work, and although I can easily see it in glass, I wanted something more organic and also less refractive. Hence bubbles. The animation, though, is more of a watery affair – mysterious and slightly eerie. Here are some stills. I’m in early stages just yet, but I thought you might like a sneak peak.

I also have some exciting news to tell you about one of my other works, but more of that in a bit.

Enjoy!

A little while back (oh my – I see it was just over a year!) I posted up some still images from my short experimental movie Microspore, but as I mentioned at the time, I was having some difficulty making it acceptably viewable in a format other than the very high resolution 10 gigabyte file that it’s rendered to.

I’ve been doing a bit of experimentation and I’m pleased to be able to show it to you now in reasonable quality. Be warned though, this is still a (relatively) massive file (about 250m) so it’s not for the faint-hearted. It’s still only a shadow of the beautiful hi-rez version, but at least it’s now watchable.

Everything you see – the critters themselves, the dirt, the distortion, the colours and the motion – is the result of reasonably simple math algorithms that I’ve constructed. The sound is also mine. If you can be patient, I recommend letting the complete movie load before you hit play – it sort of ruins the floating effect somewhat if you stop & start…

UPDATE: For some puzzling technical reason there’s a problem with the vid. I will investigate and repost.

Haeckel Illustration 1

For our first wedding anniversary (traditionally considered the ‘paper’ anniversary), Violet Towne gave me a beautiful book: Visions of Nature: The Art and Science of Ernst Haeckel.

Haeckel was a biologist and artist and an early subscriber to Charles Darwin’s evolutionary theories. Although he famously made many erroneous assumptions about evolution,* his detailed naturalistic drawings, particularly his intricate observations of the microscopic sea creatures called radiolarians, are entirely accurate and strikingly beautiful.

Haeckel Illustration 2

Haeckel was also fascinated by the obvious mathematical influences that he observed in life-forms, and documented many of their geometrical characteristics in his drawings.

Haeckel Illustration 3

His ornate organic renderings were almost certainly one of the influences that came to bear on the Art Nouveau movement. It’s not hard to understand why – take a look at this beautiful collection of high quality pdfs of some of Haeckel’s astonishing work.

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*Haeckel was a staunch believer in the outmoded ideas of Lamarckism and the now discredited recapitulation theory. Creationists love to wave Haeckel’s name about in reference to errors he made in embryonic illustrations that fulfilled his wishful speculation that ontogeny recapitulates phylogeny. In doing so they are demonstrating (once again) the profundity of their ignorance; Haeckel was never a believer of Darwin’s idea of natural selection, and in his zeal to advance his own preconceptions, some of his drawings became a little more ‘inventive’ than they had any right to be. Haeckel’s fabrications were never endorsed by Darwin, and in time succumbed to the scrutiny of rational examination, as all bad science necessarily must.

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I’m working on a new project with my images, this time an animation called Microspore. I wanted to post the moving version up for you to see, but no matter what I do I can’t get it to look presentable, and the full file at proper resolution is far too big for web streaming. Teh internets are cool, but still w-a-a-a-a-y too slow for serious stuff.

Anyways, here are some stills from the film. You’ll have to imagine that you’re looking through a microscope at little critters drifting slowly past.

Microspore 1

Microspore 2

Microspore 3

Microspore 4

Microspore 5

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