Paul P. Mealing

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Sunday, 1 March 2015

Chaos – nature’s preferred means of evolution and dynamics

Ian Stewart is a highly respected mathematician and populariser of mathematics. He has the rare ability to write entire books on the esoteric side of mathematics with hardly an equation in sight. The ‘new edition’ of Does God Play Dice? has the subtitle, The New Mathematics of Chaos, and that’s what the book is all about.  The first edition was published in 1989, the second edition in 1997, so not that new any more. Even so, he gave me more insights and knowledge into the subject than I knew existed. I’d previously read Paul Davies’ The Cosmic Blueprint, which does a pretty good job, but Stewart’s book has more depth, more examples, more explanations and simply more information. In addition, he does this without leaving me feel stranded in the wake of his considerable intellect.

For a start, Stewart puts things into perspective, by pointing out how chaos pervades much of the natural world – more so than science tends to acknowledge. In physics and engineering classes we are taught calculus and differential equations, which, as Stewart points out, are linear, whereas most of the dynamics of the natural world are non-linear, which make them ripe for chaotic analysis. We tend to know about chaos through its application to systems like weather, fluid turbulence, population dynamics yet its origins are almost purely mathematical. Throughout the book, Stewart provides numerous examples where the mathematics of chaos has been applied to physics and biology.

Historically, he gives special attention to Poincare, whom he depicts almost as the ‘father of chaos’ (my term, not his) which seems appropriate as he keeps returning to ‘Poincare sections’ throughout the book. Poincare sections are hard to explain, but they are effectively geometrical representations of periodic phenomena that have an ‘attractor’.  That’s an oversimplification, but ‘attractors’ are an important and little known aspect of chaos, as many chaotic systems display an ability to form a stable dynamical state after numerous iterations, even though, which particular state is often unpredictable. The point is that the system is ‘attracted’ to this stable state. An example, believe it or not, is the rhythmic beat of your heart. As Stewart explains, ‘the heart is a non-linear oscillator’.

Relatively early in the book, he provides an exposition on ‘dynamics in n-space’. Dimensions can be used as a mathematical concept and not just a description of space, which is how we tend to envisage it, even though it’s impossible for us to visualise space with more than 3 dimensions. He gives the example of a bicycle, something we are all familiar with, having numerous freedoms of rotation, which can be mathematically characterised as dimensions. The handle bars, each foot pedal as well as the wheels all have their own freedom of rotation, which gives us 5 at least, and this gives 10 dimensions if each degree of freedom has one variable for position and one for velocity.

He then makes the following counter-intuitive assertion:

What clinches the matter, though, is the way in which the idea of multi-dimensional spaces fit together. It’s like a 999-dimensional hand in a 999-dimensional glove.

In his own words: ‘a system with n degrees of freedom – n different variables – can be thought of as living in n-space.’ Referring back to the bicycle example, its motion can be mathematically represented as a fluid in 10 dimensional space.

Stewart then evokes a theorem, discovered in the 19th Century by Joseph Liouville, that if the system is Hamiltonian (meaning there is no friction) then the fluid is incompressible. As Stewart then points out:

…something rather deep must be going on if the geometric picture turns dynamics not just into some silly fluid in some silly space, but renders it incompressible (the 10-dimensional analogue of ‘volume’ doesn’t change as the fluid flows).

The reason I’ve taken some time to elaborate on this, is that it demonstrates the point Stewart made above – that an abstract n-dimensional space has implications in reality –  his hand-in-glove analogy.

Again, to quote Stewart:

I hope this brings you down to Earth with the same bump I always experience. It isn’t an abstract game! It is real!

Incompressibility is such a natural notion, it can’t be coincidence. Unless you agree with Kurt Vonnegut in Cat’s Cradle, that the Deity made the Universe as an elaborate practical joke.


The point is that the relationships we find between mathematics and reality are much more subtle than we can imagine, the implication being that we’ve only scratched the surface.

Anyone with a cursory interest in chaos knows that there is a relationship between chaos and fractals, and that nature loves fractals. What a lot of people don’t know is that fractals have fractional dimensions (hence the name) which can be expressed logarithmically. As Stewart points out, the relationship with chaos is that the fractal dimension ‘turns out to be a key property of an attractor, governing various quantitative features of the dynamics.’

I won’t elaborate on this as there are more important points that Stewart raises. For a start, he spends considerable time and space pointing out how chaos is not synonymous with randomness or chance as many people tend to think. Chaos is often defined as deterministic but not predictable which reads like a contradiction, so many people dismiss it out-of-hand. But Stewart manages to explain this without sounding like a sophist.

It’s impossible to predict because all chaotic phenomena are sensitive to the ‘initial conditions’. Mathematically, this means that the initial conditions would have to be determined to an infinitesimal degree, meaning an infinitely long calculation. However the behaviour is deterministic in that it follows a path determined by those initial conditions which we can’t cognise. But in the short term, this allows us to make predictions which is why we have weather forecasts over a few days but not months or years and why climate-forecast modelling can easily be criticised. In defence of climate-forecast modelling, we can use long term historical data to indicate what’s already happening and project that into the future. We know that climate-related phenomena like glaciers retreating, sea temperature rise and seasonal shifts are already happening.

On the other hand, a purely random behaviour like a coin toss or roulette wheel can’t be predicted from one toss to the next, and this is what distinguishes it from chaos, especially where ‘attractors’ are involved.

This short term, long term difference in predictability varies from system to system, including the solar system. We consider the solar system the most stable entity we know, because it’s existed in its current form well before life emerged and will continue for aeons to come. However, computer modelling suggests that its behaviour will become unpredictable eventually. Jacque Laskar of the Bureau des Longitudes in Paris has shown that ‘the entire solar system is chaotic’.

To quote Stewart:

Laskar discovered… for the Earth, an initial uncertainty about its position of 15m grows to only 150m after 10 million years, but over 100 million years the error grows to 150 million kilometres.

So nothing to worry about on that front.

In the last chapter, Stewart attempts to tackle the question posed on the front cover of his book. For anyone with a rudimentary knowledge of physics, this is a reference to Einstein’s famous exhortation that he didn’t believe God plays dice, and Stewart even cites this in the context of the correspondence where Einstein wrote it down.

Einstein, of course, was referring to his discomfort with Bohr’s ‘Copenhagen interpretation’ of quantum mechanics; a discomfort he shared with Erwin Schrodinger. I’ve written about this at length elsewhere when I reviewed Louisa Gilder’s excellent book, The Age of Entanglement. Stewart takes the extraordinary position of suggesting that quantum mechanics may be explicable as a chaotic phenomenon. I say extraordinary because, in all my reading on this subject, no one has ever suggested it and most physicists/philosophers would not even consider it.

I have come across some physicist/philosophers (like David Deutsch) who have argued that the ‘many worlds’ interpretation of quantum mechanics can, in fact, explain chaos. A view which I’m personally sceptical about.

Stewart resurrects David Bohm’s ‘hidden variables’ interpretation, preferred by Einstein, but generally considered disproved by experiments confirming Bell’s Inequality Theorem. It’s impossible for me to do justice to Stewart’s argument but he does provide the first exposition of Bell’s theorem that I was able to follow. The key is that the factors in Bell’s Inequality (as it’s known) refer to correlations that can be derived experimentally. The correlations are a statistical calculation (something I’m familiar with) and the ‘inequality’ tells you whether the results are deterministic or random. In every experiment performed thus far, the theorem confirms that the results are not deterministic, therefore random.

Stewart takes the brave step of suggesting that Bell’s Inequality can be thwarted because it relies on the fact that the results are computable. Stewart claims that if they’re not computable then it can’t resolve the question. He gives the example of so-called ‘riddled basins’ where chaotic phenomena can interact with ‘holes’ that allow them to find other ‘attractors’. Again, an oversimplification on my part, but as I understand it, in these situations, which are not uncommon according to Stewart, it’s impossible to ‘compute’ which attractor a given particle would go to.

Stewart argues that if quantum mechanics was such a chaotic system then the results would be statistical as we observe. I admit I don’t understand it well enough to confer judgement and I don’t have either the mathematical or physics expertise to be a critical commentator. I’ll leave that to others in the field.

I do agree with him that the wave function in Schrodinger’s equation is more than a ‘mathematical fiction’ and it was recently reported in New Scientist that a team from Sydney claim they have experimentally verified its reality. But I conjecture that ‘Hilbert space’, which is the abstract space where the wave function mathematically exists, may be what’s real and we simply interact with it, but there is no more evidence for that than there is for the ‘multiple universes’ that is currently in favour and gaining favour.

Towards the very end of the book, Stewart hypothesises on how different our view of quantum mechanics may be today if chaos theory had been discovered first, though he’s quick to point out the importance of computers in allowing chaos to be exploited. But he makes this interesting observation in relation to the question on the cover of his book:

Now, instead of Einstein protesting that God doesn’t play dice, he probably would have suggested that God does play dice. Nice, classical, deterministic dice. But – of course – chaotic dice. The mechanism of chaos provides a wonderful opportunity for God to run His universe with deterministic laws, yet simultaneously to make fundamental particles seem probabilistic.

Of course, both Stewart’s and Einstein’s reference to a Deity is tongue-in-cheek, but I’ve long thought that chaos provides the ideal mechanism for a Deity to intervene in the Universe. Having said that, I don’t believe in Divine intervention, if for no other reason than it’s axiomatically associated with the concept of God’s will. And people who believe in God’s will behave like they have a licence to do whatever they want, especially when it comes to persecuting their fellow humans, because they can perversely justify their actions as being ‘God’s will’.

13 comments:

Bill MD said...

Hi paul, my name is Bill and I am a physician in the US. I love your posts and I very much enjoyed this one in particular but i am a little disappointed at how you ended it. I am a religious person and i believe in God; I am also a scientist and I am fascinated by math, physics and the theories of gravity, the quantum, multiverses, etc... I have no intention of imposing my view on fellow men. I don't justify violence in the name of God and I am not sure why you would imply that being religious is kind of a slippery slope to fascism. Could you please elaborate on this? Isn't there room for people like me who believes in the constant battle between Good and Evil but also believes in the multiverse? Are they necessarily mutually exclusive? thanks Paul, look forward to your response.

Paul P. Mealing said...

Hi Bill,

Thanks for commenting. I'm away from home at present, staying with family and friends.
I'll give you a more considered response when I get back home.

Regards, Paul.

Paul P. Mealing said...

Hi Bill,

Yes, I thought at the time I wrote it, that last paragraph is one that some of my theistic friends might take exception to.

My views on theism are rather unorthodox, to say the least, though I’m not anti-theist per se, and I wouldn’t even call myself an atheist although I admit I argue like one.

I do have an issue when people talk about ‘God’s will’ because, even if you believe in God (of whatever nomination) you can’t possibly know what God’s will is, so when people reference it they’re really talking about ‘their will’, but by giving it Divine authority, they make it unassailable to logic or moral reasoning.

So I would say to someone that it is alright to talk about ‘God’s will’ as long as they know and acknowledge that they’re really talking about their own will.

My views on theism can be found here and here.

Regards, Paul.

Bill MD said...

Right, I understand and agree. I read both of your posts on theism and, by and large, agree with your views (although I think Evil is very real and not a projection of our psyche. Would love to have a discussion about that with you going forward). My personal view is that, at the end of the day, us humans will always need a creator to explain/ understand the very beginning of the universe; no matter how sophisticated our theory of everything, one will always ask: "but how did these formulas get to exist?" And even if, as Hawkins proposes the formulas are so beautiful and elegant that they compel themselves to exist, the question would then be how did the math that the formulas are written in get created. I guess some would then say that math has no beginning, that it is infinite, but they would just be replacing God with Math, they would be giving math deity status. All that said, I still think we must continue to advance and explore scientifically and develop ever more complex models of the Universe/s, but we are just trying to "..understand the mind of God".

Bill MD said...

BY the way, Paul, love your paragraph: "..At the end of the day, what matters is how one perceives and treats their fellows, not whether they are theists or atheists.." Rings a familiar tone: "do unto to others as you would like done to you", doesn't it? .. what brings me to (yet) another question for you (sorry) .. who do you think/ believe Jesus Christ was? Alien from advanced civilization? Son of God? Gifted preacher? Thanks Paul

Paul P. Mealing said...

Hi Bill,

As I explicate in one or both of those posts I linked, God is something people find in themselves – in other words, it’s totally subjective. It’s obvious to me that everyone’s idea of God is unique to the individual – it’s something deeply personal and not something that can be shared. And this contradicts institutionalised religions’ fundamental tenet (at least for the monotheistic ones) that everyone’s belief in God is the same.

This also leads to the epistemological dilemma that the Creator is a different type of god. In other words, it’s a bit of a conceit to think that the personal God people find in their heads is also the creator of the entire universe and any other universes that may exist that we’re not cognisant of. Now this brings us back to the issue of assuming that God is like colour – something we all experience subjectively but is determined by an external source. This may be true but it doesn’t explain why so many people have so many different ideas of what God is or represents. But as someone recently pointed out (an American clergyman called Pong?) the obvious answer to this is that we simply project our own prejudices onto our version of God.

And, personally, I’m not adverse to the idea that God as Creator could be replaced by mathematics, in as much as the laws of the universe seem mathematically dependent. What’s more, if such a creator God exists then he can’t change mathematics any more than we can. For example, God can’t make prime numbers non-prime or change the value of pi.

The other point that I feel is lost is that there is no point for God to exist without humankind, if you think about it, and, likewise, we tend to think that there is no point to the Universe existing without humankind. Certainly, as I’ve pointed out many times, the Universe may as well not exist if there was no consciousness. And of course, the corollary to that is that the most remarkable thing about the Universe is that it created the means to understand itself (through us).

But when we talk about God in the sense that you allude to, we are really talking about ‘purpose’. How can the Universe have a purpose without God. Science’s answer is that there is no purpose – the Universe is not teleological – we are just a freak accident. My view is that if there is a purpose it’s not ours to know, but if there is a purpose then we fulfil it simply by living.

Regards, Paul.

Paul P. Mealing said...

Hi Bill,

It occurred to me that this relatively short post, that I wrote in 2009, may give the best perspective of my unusual stance.

Regards, Paul.

Bill MD said...

Ok, I read it Paul, very nice. I also read your post on Evil, very intresting and well balanced. One of the reasons I like your blog is because of your nice contrast between science and religion; your opinions on Good and Evil are always well balanced with explanations of sociological experiments and cultural insights. I tend to agree that Evil can be seen as an aberration of evolutionary traits such as resource hoarding by a group, individual domination to assure reproduction of the strongest member and supremacy of a tribe over the others to guarantee the staying power of its values and customs. But I also think there is a more subtle, individual evil that religion would call temptation. Stealing, lying, cheating, drugs use and in general hurting others with our actions are ways that evil (the devil?) manifests itself. It seems to me that Evil is always trying to undermine the goodness in the world. Perhaps a reflection of a more Universal battle? Could we be part of a universal battle between a creator and a destroyer? And here is another thought, in a multi multi-verse universe, we have to accept that there is a world out there where Hitler won and Nazis rule the world: another Earth where the Middle Ages never ended and torture and oppression by the Inquisition continue to this day. So my question is, is it just a lucky coincidence that we live in a world where the Good always prevails over time? If the answer is yes, then you could argue we are "blessed" then? I guess another explanation for this uniqueness is that the jury is still out and in the future the Earth may end up going belly up, with evil destroying everything. But I (like you) have a more optimistic outlook on humanity. What are your thoughts on this, Paul?. Thank you. Oh and please, let me know if you have blogged on Jesus Christ yet, would be interested to see your take on that subject.

Paul P. Mealing said...

Hi Bill,

You’ve given me a lot to address. Firstly, I’m not a fan of the multiverse theory in its many manifestations, though I’ve read Max Tegmark’s book where he argues for 4 levels of multiverse. I’ve addressed it very briefly in a post I wrote not so long ago on the fine-tuned universe.

Regarding evil, I was tempted to address it in your second-to-last post, but I’d already rambled on enough and it’s quite a big subject. If you run with the belief that evil has a Divine dimension then you have to accept that God created evil, which is effectively what the so-called ‘problem of evil’ is all about. The problem I have with it is that it completely sidesteps the real cause and the real means to counter it. I also have an issue with the mythology of ‘original sin’ and its implications, but that’s another argument.

Evil is fundamentally a psychological phenomenon and almost invariably arises from either a lack or a denial of empathy. It requires a conscious denial of empathy to perform acts that most of us consider evil. Most people who commit evil don’t see themselves as evil, at least whilst they’re doing it. In fact, it’s amazing how people can rationalise and justify it, whether they’re members of ISIS or Hitler’s Gestapo.

The other component of evil is narcissism, which was very evident in the 20th century, but blaming Satan is avoiding the problem. If Satan exists, then like God, he exists in our heads, and, in fact, one person’s God is another person’s Satan.

I have written a post on Jesus, based on a book I read by Don Cupitt, a theologian turned philosopher. I think it’s unfortunate that Jesus’ death has overridden his life, as we can learn more from his life than his death, assuming we can separate his life from the mythology. But his death has been tied to that other mythological event, original sin (mentioned earlier), which is why Christianity, when taken literally, makes no sense at all and has the unfortunate implied message that people’s beliefs are more important than their actions. Now, to be fair, most, if not all the Christians I know, don’t judge me on my beliefs anymore than I judge them, but the teachings I received as a child placed emphasis on a belief in Jesus over the inherent consequences of one’s own actions.

Regards, Paul.

Bill MD said...

Very insightful Paul, loved your response. I did read again your fine-tuned universe blog.it's excellent, I think that was the first one I came across and started my interest in your blog. I agree that you can disect good and evil down to a psychological level and explain good deeds by their evolutionary advantage to the individual, the tribe and the community. You can probably also explain evil away as psychological disorders and aberrant tribal behaviors that are deleterious to the community. I guess that kind of explains why overall we live in a good world, because if evil had prevailed we probably wouldn't be here to talk about it. Basically, we are explaining good and evil with evolution, acts of goodness and empathy are beneficial to the individual who lives in a society whereas acts of destruction and self-serving acts, basically evil acts, are evolutionary bad for the community. But, let me ask you, don't you think at some level that there is more to the story? I don't know about you, but I have a gut feeling that it is a lot more complex than that. I can't help but feel like the dog who thinks it can understand its master's behavior because most of the time his actions are kind of predictable and "kind of" make sense most of the time. Also, good is part of evolution but, evolution towards what? Why is there evolution to begin with. Why not a dead, lifeless world? If you are against the accident theory of life, then as you say there has to be a purpose to life and evolution. I do agree that, like the dog / master analogy, that purpose is not for us to ever understand I am afraid. Regarding Jesus, I agreed that his life can teach us a lot more than his death. After all, the New Testament is about his life. I do believe he is the son of God but that is at a deep personal level. The scientist in me is very curious about him. If you take away all the religious implications, he comes across as a very special man. What do you think about the theory of him being from another world? A messenger from a more advanced civilization? A traveler from the future? Perhaps the first of an evolutionary type of human, the next Homo sapiens sapiens? Thank you for your insights and I hope I am NOT bothering you with all these questions. Feel free to end this conversation when you feel like doing so. Thank you Paul.

Paul P. Mealing said...

Hi Bill,

I have to admit that you surprise me a bit. I’m obviously aware that your views are quite different to mine yet you are willing to engage me – that’s quite rare.

By the way, does MD stand for Master of Divinity?

Why is there something rather than nothing? Actually, I’ve written a post on that too. Actually, the difference between something and nothing for any individual is consciousness. In fact, without consciousness the universe may as well not exist. So it does seem to have a purpose, even though it’s not teleological. Because it’s chaotic it can’t be teleological, and I’ve already explained why chaos ensures that the future is unpredictable. Some philosophers, like Stephen Law, for example, argue that it’s the wrong question. He made that comment in a debate and I wanted to ask him what the right question was? But I wasn’t the one he was debating, and unfortunately, he didn’t get the chance to elaborate.

I’ve written another post that does address your last comment to some extent: Theory, Knowledge and Truth.

There is also this quote from another post I wrote, titled The Introspective Cosmos, which is really a review cum discussion of Noson Yanosky’s book, The Outer Limits of Reason; What Science, Mathematics, and Logic CANNOT Tell Us.

In his notes on this chapter, Yanofsky makes this point:

Perhaps we can say that the universe is against having intelligent life and that the chances of having intelligent life are, say, 0.0000001 percent. We, therefore, only see intelligent life in 0.0000001 percent of the universe.

This reminds me of John Barrow’s point, in one of his many books, that the reason the universe is so old, and so big, is because that’s how long it takes to create complex life, and, because the universe is uniformly expanding, age and size are commensurate.


Did you read my post on Jesus? No, I don’t think he came from another world, anymore than any one of us. I do think he may have travelled to the East though – I read a very interesting book on that – where they called him Issa. I no longer have the book, as I gave it to someone else. In fact, the book claims he was buried in Srinigar, Kashmir.

Because a universe without consciousness has no meaning and yet the universe is full of meaning, I tend to think that we are its self-culmination. This is effectively a restating of the strong anthropic principle – the universe necessarily created conditions that allows observers to arise (the observers don’t have to be us).

Regards, Paul.

Bill MD said...

Very interesting, as usual Paul. The MD after my name does stand for Medical Doctor. But I am a Christian and I do believe in God. I don't think both things are mutually exclusive at all. I have studied and seen many things as a doctor that have reinforced my faith. I will continue to read your blog and I have really enjoyed your comments. I will read your blog on Jesus soon (real busy at work these days) and I will be sure to comment on it, maybe start a new thread. thanks again.

Paul P. Mealing said...

Just goes to show: I should trust my own judgement. My first thought was that MD was Medical Doctor, but when I told a friend about our conversation, he suggested Master of Divinity, which I thought: fits the context of the discussion.

Here is another quote from my post: The Introspective Cosmos; which is the review of Yanofsky's book, referenced in my last comment.

We are each an organism with a brain that creates something we call consciousness that allows us to reflect on ourselves, individually. And the Universe created, via an extraordinary convoluted process, the ability to reflect on itself, its origins and its possible meaning [via us].

Yanofsky himself commented on that post - he was very impressed with my review.

I also added an addendum to The Fine-Tuned Universe, based on a YouTube interview with Leonard Susskind on the same topic.

Regards, Paul.