Is the world continuous or discrete?

There is an excellent series on YouTube called ‘Closer to Truth’, where the host, Richard Lawrence Kuhn, interviews some of the cleverest people on the planet (about existential and epistemological issues) in such a way that ordinary people, like you and me, can follow. I understand from Wikipedia that it’s really a television series started in 2000 on America’s PBS.

In an interview with Gregory Chaitin, he asks the above question, which made me go back and re-read Chaitin’s book, Thinking about Godel and Turing, which I originally bought and read over a decade ago, and then posted about on this blog, (not long after I created it). It’s really a collection of talks and abridged papers given by Chaitin from 1970 to 2007, so there’s a lot of repetition but also an evolution in his narrative and ideas. Reading it for the second time (from cover to cover) over a decade later has the benefit of using the filter of all the accumulated knowledge that I’ve acquired in the interim.

More than one person (Umberto Eco and Jeremy Lent, for examples) have wondered if the discreteness we find in the world, and which we logically apply to mathematics, is a consequence of a human projection rather than an objective reality. In other words, is it an epistemological bias rather than an ontological condition? I’ll return to this point later.

Back to Chaitin’s opus, he effectively takes us through the same logical and historical evolution over and over again, which ultimately leads to the same conclusions. I’ll summarise briefly. In 1931, Kurt Godel proved a theorem that effectively tells us that, within a formal axiom-based mathematical system, there will always be mathematical truths that can’t be solved. Then in 1936, Alan Turing proved, with a thought experiment that presaged the modern computer, that there will always be machine calculations that may never stop and we can’t predict whether they will or not. For example, Riemann’s hypothesis can be calculated using an algorithm to whatever limit you like (and is being calculated somewhere right now, probably) but you can never know in advance if it will ever stop (by finding a false result). As Chaitin points out, this is an extension of Godel’s theorem, and Godel’s theorem can be deduced from Turing’s.

Then Chaitin himself proved, by inventing (or discovering) a mathematical device, (Ω) called Omega, that there are innumerable numbers that can never be completely calculated (Omega gives a probability of a Turing program halting). In fact, there are more incomputable numbers than rational numbers, even though they are both infinite in extent. The rational Reals are countably infinite while the incomputable Reals are uncountably infinite. I’ve mentioned this previously when discussing Noson Yanofsky’s excellent book, The Outer Limits of Reason; What Science, Mathematics, and Logic CANNOT Tell Us. Chaitin claims that this proves that Godel’s Incompleteness Theorem is not some aberration, but is part of the foundation of mathematics – there are infinitely more numbers that can’t be calculated than those that can.

So that’s the gist of Chaitin’s book, but he draws some interesting conclusions on the side, so-to-speak. For a start, he argues that maths should be done more like physics and maybe we should accept some unproved theorems (like Riemann’s) as new axioms, as one would in physics. In fact, this is happening almost by default in as much as there already exists new theorems that are dependent on Riemann’s conjecture being true. In other words, Riemann’s hypothesis has effectively morphed into a mathematical caveat so people can explore its consequences.

The other area of discussion that Chaitin gets into, which is relevant to this discussion is whether the Universe is like a computer. He cites Stephen Wolfram (who invented Mathematica) and Edward Fredkin.

According to Pythagoras everything is number, and God is a mathematician… However, now a neo-Pythagorean doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information. In other words, now everything is software, God is a computer programmer, not a mathematician, and the world is a giant information-processing system, a giant computer [Fredkin, 2004, Wolfram, 2002, Chaitin, 2005].

Carlo Rovelli also argues that the Universe is discrete, but for different reasons. It’s discrete because quantum mechanics (QM) has a Planck limit for both time and space, which would suggest that even space-time is discrete. Therefore it would seem to lend itself to being made up of ‘bits’. This fits in with the current paradigm that QM and therefore reality, is really about ‘information’ and information, as we know, comes in ‘bits’.

Chaitin, at one point, goes so far as to suggest that the Universe calculates its future state from the current state. This is very similar to Newton’s clockwork universe, whereby Laplace famously claimed that given the position of every particle in the Universe and all the relevant forces, one could, in principle, ‘read the future just as readily as the past’. These days we know that’s not correct, because we’ve since discovered QM, but people are arguing that a QM computer could do the same thing. David Deutsch is one who argues that (in principle).

There is a fundamental issue with all this that everyone seems to have either forgotten or ignored. Prior to the last century, a man called Henri Poincare discovered some mathematical gremlins that seemed of little relevance to reality, but eventually led to a physics discipline which became known as chaos theory.

So after re-reading Chaitin’s book, I decided to re-read Ian Stewart’s erudite and deeply informative book, Does God Play Dice? The New Mathematics of Chaos.

Not quite a third of the way through, Stewart introduces Chaitin’s theorem (of incomputable numbers) to demonstrate why the initial conditions in chaos theory can never be computed, which I thought was a very nice and tidy way to bring the 2 philosophically opposed ideas together. Chaos theory effectively tells us that a computer can never predict the future evolvement of the Universe, and it’s Chaitin’s own theorem which provides the key.

At another point, Stewart quips that God uses an analogue computer. He’s referring to the fact that most differential equations (used by scientists and engineers) are linear whilst nature is clearly nonlinear.

Today’s science shows that nature is relentlessly nonlinear. So whatever God deals with… God’s got an analogue computer as versatile as the entire universe to play with – in fact, it is the entire universe. (Emphasis in the original.)

As all scientists know (and Yanofsky points out in his book) we mostly use statistical methods to understand nature’s dynamics, not the motion of individual particles, which would be impossible. Erwin Schrodinger made a similar point in his excellent tome, What is Life? To give just one example that most people are aware of: radioactive decay (an example Schrodinger used). Statistically, we know the half-lives of radioactive decay, which follow a precise exponential rule, but no one can predict the radioactive decay of an individual isotope.

Whilst on the subject of Schrodinger, his eponymous equation is both linear and deterministic which seems to contradict the very idea of QM discrete and probabilistic effects. Perhaps that is why Carlo Rovelli contends that Schrodinger’s wavefunction has misled our attempts to understand QM in reality.

Roger Penrose explicates QM in phases: U, R and C (he always displays them bold), depicting the wave function phase; the measurement phase; and the classical physics phase. Logically, Schrodinger’s wave function only exists in the U phase, prior to measurement or observation. If it wasn’t linear you couldn’t add the waves together (of all possible paths) which is essential for determining the probabilities and is also fundamental to QED (which is the latest iteration of QM). The fact that it’s deterministic means that it can calculate symmetrically forward and backward in time.

My own take on this is that QM and classical physics obey different rules, and the rules for classical physics are chaos, which are neither predictable nor linear. Both lead to unpredictability but for different reasons and using different mathematics. Stewart has argued that just maybe you could describe QM using chaos theory and David Deutsch has argued the opposite: that you could use the multi-world interpretation of QM to explain chaos theory. I think they’re both wrong-headed, but I’m the first to admit that all these people know far more than me. Freeman Dyson (one of the smartest physicists not to win a Nobel Prize) is the only other person I know who believes that maybe QM and classical physics are distinct. He’s pointed out that classical physics describes events in the past and QM provides future probabilities. It’s not a great leap from there to suggest that the wavefunction exists in the future.

You may have noticed that I’ve wandered away from my original question, so maybe I should wonder my way back. In my introduction, I mentioned the epistemological point, considered by some, that maybe our employment of mathematics, which is based on integers, has made us project discreteness onto the world.

Chaitin’s theorem demonstrates that most of mathematics is not discrete at all. In fact, he cites his hero, Gottlieb Leibniz, that most of mathematics is ‘transcendental’, which means it’s beyond our intellectual grasp. This turns the general perception that mathematics is a totally logical construct on its head. We access mathematics using logic, but if there are an uncountable infinity of Reals that are not computable, then, logically, they are not accessible to logic, including computer logic. This is a consequence of Chaitin’s own theorem, yet he argues that is the reason it’s not reality.

In fact, Chaitin would argue that it’s because of that inacessability that a discrete universe makes sense. In other words, a discrete universe would be computable. However, chaos theory suggests God would have to keep resetting his parameters. (There is such a thing as ‘chaotic control’, called ‘Proportional Perturbation Feedback’, PPF, which is how pacemakers work.)

Ian Stewart has something to say on this, albeit while talking about something else. He makes the valid point that there is a limit to how many decimals you can use in a computer, which has practical limitations:

The philosophical point is that the discrete computer model we end up with is not the same as the discrete model given by atomic physics.

Continuity uses calculus, as in the case of Schrodinger’s equation (referenced above) but also Einstein’s field equations, and calculus uses infinitesimals to maintain continuity mathematically. A computer doing calculus ‘cheats’ (as Stewart points out) by adding differences quite literally.

This leads Stewart to make the following observation:

Computers can work with a small number of particles. Continuum mechanics can work with infinitely many. Zero or infinity. Mother Nature slips neatly into the gap between the two.

Wolfram argues that the Universe is pseudo-random, which would allow it to run on algorithms. But there are 2 levels of randomness, one caused by QM and one caused by chaos. (Chaos can create stability as well, which I‘ve discussed elsewhere.) The point is that initial conditions have to be calculated to infinity to determine chaotic phenomena (like weather), but it applies to virtually everything in nature. Even the orbits of the planets are chaotic, but over millions, even billions of years. So at some level the Universe may be discrete, even at the Planck scale, but when it comes to evolutionary phenomena, chaos rules, and it’s neither computably determinable (long term) nor computably discrete.

There is one aspect of this that I’ve never seen discussed and that is the relationship between chaos theory and time. Carlos Rovelli, in his recent book, The Order of Time, argues that ‘time’s arrow’ can only be explained by entropy, but another physicist, Richard A Muller, in his book, NOW; The Physics of Time, argues the converse. Muller provides a lengthy and compelling argument on why entropy doesn’t explain the arrow of time.

This may sound simplistic, but entropy is really about probabilities. As time progresses, a dynamic system, if left to its own devices, progresses to states of higher probability. For example, perfume released from a bottle in one corner of a room soon dissipates throughout the room because there is a much higher probability of that then it accumulating in one spot. A broken egg has an infinitesimally low probability of coming back together again. The caveat, ‘left to its own devices’, simply means that the system is in equilibrium with no external source of energy to keep it out of equilibrium.

What has this to do with chaos theory? Well, chaotic phenomena are time asymmetrical (they can't be repeated, if rerun). Take weather. If weather was time reversible symmetrical, forecasts would be easy. And weather is not in a state of equilibrium, so entropy is not the dominant factor. Take another example: biological evolution. It’s not driven by entropy because it increases in complexity but it’s definitely time asymmetrical and it’s chaotic. In fact, speciation appears to be fractal, which is a chaos parameter.

Now, I pointed out that the U phase of Penroses’s explication of QM is time symmetrical, but I would contend that the overall U, R, C sequence is not. I contend that there is a sequence from QM to classical physics that is time asymmetrical. This infers, of course, that QM and classical physics are distinct.


Addendum 1: This is slightly off-topic, but relevant to my own philosophical viewpoint. Freeman Dyson delivers a lecture on QM, and, in the 22.15 to 24min time period, he argues that the wavefunction and QM can only tell us about the future and not the past.

Addendum 2 (Conclusion): Someone told me that this was difficult to follow, so I've written a summary based on a comment I gave below.

Chaitin's theorem arises from his derivation of omega (Ω), which is the 'halting probability', an extension of Turing's famous halting theorem. You can read about it here, including its significance to incomputability.

I agree with Chaitin 'mathematically' in that I think there are infinitely more incomputable Reals than computable Reals. You already have this with transcendental numbers like π and e, which are incomputable. Chaitin's Ω can be computed to whatever resolution you like, just like π and e, but (of course) not to infinity.

I disagree with him 'philosophically' in that I don't think the Universe is necessarily discrete and can be reduced to 0s and 1s (bits). In other words, I don't think the Universe is like a computer.

Curiously and ironically, Chaitin has proved that the foundation of mathematics consists mostly of incomputable Reals, yet he believes the Universe is computable. I agree with him on the first part but not the second.

Addendum 3: I discussed the idea of the Universe being like a computer in an earlier post, with no reference to Chaitin or Stewart.

Addendum 4: I recently read Jim Al-Khalili's chapter on 'Laplace's demon' in his book, Paradox; The Nine Greatest Enigmas in Physics, which is specifically a discussion on 'chaos theory'. Al-Khalili contends that 'the Universe is almost certainly deterministic', but I think his definition of 'deterministic' might be subtly different to mine. He rightly points out that chaos is deterministic but unpredictable. What this means is that everything in the past and everything in the future has a cause and effect. So there is a causal path from any current event to as far into the past as you want to go. And there will also be a causal path from that same event into the future; it's just that you won't be able to predict it because it's uncomputable. In that sense the future is deterministic but not determinable. However (as Ian Stewart points out in Does God Play Dice?) if you re-run a chaotic experiment you will get a different result, which is almost a definition of chaos; tossing a coin is the most obvious example (cited by Stewart). My point is that if the Universe is chaotic then it follows that if you were to rerun the Universe you'd get a different result. So it may be 'deterministic' but it's not 'determined'. I might elaborate on this in a separate post.

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.

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 while chaos is 'deterministic', it's computably indeterminable, which is why it's 'unpredictable'. I've written another post on that specific topic.

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 have neither the mathematical nor 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, in the real world, dice are chaotic because the outcome of a throw is subject to the sensitivity of the initial conditions, which is the throw itself. The same with a coin toss. So each throw has its own initial conditions, which creates the randomness from throw to throw that we observe.

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, because it assumes that God has a plan that 'He' needs to keep interfering with. I prefer to think that God is simply the laws of the Universe (a la Einstein’s God) and they will run their course.

Chaos may be 'deterministic' but you can't rerun a chaotic phenomenon and get the same result - that's how chaos was discovered. The Universe obeys 'strange attractors', which provides stability to some systems while still being ultimately unpredictable. We don't know enough to know why the Universe turned out the way it did. Every age has its own sphere of ignorance, but chaos suggests that the future cannot be ultimately known. In other words, there appears to be a limit to what it's possible to know and not just a limit dependent on our cognitive abilities.

How chaos drives the evolution of the universe and life

The Cosmic Blueprint is the very first book of Paul Davies I ever read nearly a quarter of a century ago, and I’ve read many others since. I heard him being interviewed about it on a car trip from Melbourne to Mulwala (on the Victorian, New South Wales border) and that was the first time I’d heard of him. The book was published in 1987, so it was probably 1988.

Davies received the Templeton Foundation Prize in 1995, though not the wrath of Dawkins for accepting it. He’s also received the 2002 Michael Faraday Prize from the Royal Society and the 2001 Kelvin Medal and Prize from the UK Institute of Physics. He was resident in Australia for a couple of decades but now resides in the US where he’s an astro-biologist at the University of Arizona.

In America, Davies has been accused of being a ‘creationist in disguise’ by people whose ignorance is only out-weighed by their narrow-mindedness (they think there are atheists and there are creationists with nothing in between). The 2004 edition of this book is published by the Templeton Foundation and the first word in the opening chapter is ‘God’ as part of a quote by Ilya Prigogine, who features prominently in the book. But anyone who thinks this is a thesis for Intelligent Design will be disappointed; it’s anything but. In fact, one of the book’s great virtues is its attempt to explain complexity in the universe and evolution as a natural occurrence and not a Divine one.

I’ve long believed that Davies writes about science and philosophy better than anyone else, not least because he seems to be equally erudite in the disciplines of physics, cosmology, biology and philosophy. He’s not a member of the ‘strong atheist’ brigade, which puts him offside with many philosophers and commentators, but his argument against ID in The Goldilocks Enigma (2006) was so compelling that Stephen Law borrowed it for himself.

I remember The Cosmic Blueprint primarily as introducing me to chaos theory; it was the new kid on the block in popular consciousness with fractals and Mandelbroit’s set just becoming conspicuous in pop culture. Reading it now, I’m surprised at how much better it is than I remember it, but that’s partly due to what I’ve learnt in between. A lot of it would have gone over my head, which is not to say it still doesn’t, but less so than before.

More than any other writer on science, Davies demonstrates how much we don’t know and he doesn’t shy away from awkward questions. In particular, he is critical of reductionism as the only method of explanation, especially when it explains things away rather than explicating them; consciousness and life’s emergence being good examples.

I like Davies because his ideas reflect some of my own ruminations, for example that natural selection and mutations can’t possibly explain the whole story of evolution. We think we are on the edge of knowing everything, yet future generations will look back and marvel at our ignorance just as we do with our forebears.

There is an overriding thesis in The Cosmic Blueprint that is obvious once it’s formulated yet is largely ignored in popular writing. It’s fundamentally that there are two arrows of time: one being the well known 2^nd law of thermodynamics or entropy; and the other being equally obvious but less understood as the increase in complexity at all levels in the universe from the formation of galaxies, stars and planets to the evolution of life on Earth, and possibly elsewhere. Both of which demonstrate irreversibility as a key attribute.  And whilst many see them as contradictory and therefore evidence of Divine intervention, Davies sees them as complementary and part of the universe’s overall evolvement.

Davies explains how complexity and self-organisation can occur when dynamic systems are pushed beyond equilibrium with an open source of energy. Entropy, on the other hand, is a natural consequence of systems in equilibrium.

In the early pages, Davies explains chaotic behaviour with a simple-to-follow example that’s purely mathematical. In particular, he demonstrates how the system is completely deterministic yet totally unpredictable because the initial conditions are mathematically impossible to define. This occurs in nature all the time, like coin tosses, so that the outcome is totally random but only because the initial conditions are impossible to determine, not because the coin follows non-deterministic laws. This is a subtle but significant distinction.

A commonly cited example is cellular automata that can be generated by a computer programme. Stephen Wolfram of the Institute for Advanced Study, Princeton, has done a detailed study of one-dimensional automata that could give an insight into evolution. Davies quotes Wolfram:

“…the cellular automaton evolution concentrates the probabilities for particular configurations, thereby reducing entropy. This phenomenon allows for the possibility of self-organization by enhancing the probabilities of organized configurations and suppressing disorganized configurations.”

Wolfram is cited by Gregory Chaitin, in Thinking about Godel and Turing, as speculating that the universe may be pseudo-random and chaos theory provides an innate mechanism: deterministic laws that can’t be predicted. However, it seems that the universe’s innate chaotic laws provide opportunities for a diverse range of evolutionary possibilities, and the sheer magnitude of the universe in space and time, along with a propensity for self-organisation, in direct opposition to entropy, may be enough to ensure intelligent life as an outcome. The truth is that we don’t know. (Btw, Davies wrote the forward to Chaitin’s book.)

Davies calls this position ‘predestiny’ but he’s quick to qualify it thus: ‘Predestiny is a way of thinking about the world. It is not a scientific theory. It receives support, however, from those experiments that show how complexity and organization arise spontaneously and naturally under a wide range of conditions.’

This view is mirrored in the anthropic principle, which Davies also briefly discusses, but there are two version, as expounded by Frank Tipler and John Barrow in The Anthropic Cosmological Principle: the weak anthropic principle and the strong anthropic principle; and ‘predestiny’ is effectively the strong anthropic principle.

Roughly twenty years later, in The Goldilocks Enigma, Davies elaborates on this philosophical viewpoint when he argues for the ‘self-explaining universe’ amongst a critique of all the current ‘flavours’ of universe explanations: ‘I have suggested that only self-consistent loops capable of understanding themselves can create themselves, so that only universes with (at least the potential for) life and mind really exist.’ This is effectively a description of John Wheeler’s speculative cosmic quantum loop explanation of the universe’s existence – it exists because we’re in it. Davies argues that such a universe is ‘self-activating’ to avoid religious connotations: ‘…perhaps existence isn’t something that gets bestowed from outside…’

Teleological is a word that most scientists avoid, but Davies points out that the development of every organism is teleological because it follows a ‘blueprint’ or ‘plan’ entailed in its DNA. How this occurs is not entirely understood, but Davies makes an analogy with software which is apposite, as DNA provides coded instructions that ultimately result in fully developed organisms like us. He explores a concept called ‘downward causation’ whereby information can actually ‘cause’ materialistic events and software in computers provide the best example. In fact, as Davies hypothesises, one could imagine a software programme that makes physical changes to the computer that it’s operating on. Perhaps this is how the ‘mind’ works, which is similar to Douglas Hofstadter’s idea of a ‘strange loop’ that he introduced in Godel Escher Bach (which I reviewed in Feb. 2009) and later explored in another tome called I am a Strange Loop (which I haven’t read).

Davies introduces the concept of ‘downward causation’ in his discussion on quantum mechanics because it’s the measurement or observation that crystallises the quantum phenomenon into the real world. According to Davies, Wheeler speculated that ‘downward causation’ in quantum mechanics is ‘backwards in time’ and suggested a ‘delayed-choice’ thought experiment. To quote Davies: ‘The experiment has recently been conducted, and accords entirely with Wheeler’s expectations. It must be understood, however, that no actual communication with the past is involved.’

It’s impossible to discuss every aspect of this book, covering as it does: chaos theory, fractals, cosmological evolution, biological evolution, quantum mechanics and mind and matter.

Towards the end, Davies reveals some of his own philosophical prejudices, which, unsurprisingly, are mirrored in The Goldilocks Enigma twenty years on.

The very fact that the universe is creative, and that the laws have permitted complex structures to emerge and develop to the point of consciousness – in other words, that the universe has organized its own self-awareness – is for me powerful evidence that there is ‘something going on’ behind it all.

This last phrase elicits the ‘design’ word, many years before Intelligent Design was introduced as a ‘wedge’ tactic for creationists, but Davies has been an outspoken critic of both creationism and ID, as I explained above. Davies strongly believes the universe has a purpose and the evidence supports that point of view. But it’s a philosophical point of view, not a scientific one.

This leads to the logical question: is the universe teleological? I think chaos theory provides an answer. In the same way that chaotic phenomena, which includes all complex dynamics in the universe (like evolution) are deterministic yet unpredictable, the universe could be purposeful yet not teleological. In other words, the purpose is not predetermined but the universe’s dynamics allow purpose to evolve.

Speciation: still one of nature’s great mysteries

First of all a disclaimer: I’m a self-confessed dilettante, not a real philosopher, and, even though I read widely and take an interest in all sorts of things scientific, I’m not a scientist either. I know a little bit more about physics and mathematics than I do biology, but I can say with some confidence that evolution, like consciousness and quantum mechanics, is one of nature’s great mysteries. But, like consciousness and quantum mechanics, just because it’s a mystery doesn’t make it any less real. Back in Nov.07, I wrote a post titled: Is evolution fact? Is creationism myth?

First, I defined what I meant by ‘fact’: it’s either true or false, not something in between. So it has to be one or the other: like does the earth go round the sun or does the sun go round the earth? One of those is right and one is wrong, and the one that is right is a fact.

Well, I put evolution into that category: it makes no sense to say that evolution only worked for some species and not others; or that it occurred millions of years ago but doesn’t occur now; or the converse that it occurs now, but not in the distant past. Either it occurs or it never occurred, and all the evidence, and I mean all of the evidence, in every area of science: genetics, zoology, palaeontology, virology; suggests it does. There are so many ways that evolution could have been proven false in the last 150 years since Darwin’s and Wallace’s theory of natural selection, that it’s just as unassailable as quantum mechanics. Natural selection, by the way, is not a theory, it’s a law of nature.

Now, both proponents and opponents of evolutionary theory often make the mistake of assuming that natural selection is the whole story of evolution and there’s nothing else to explain. So I can confidently say that natural selection is a natural law because we see evidence of it everywhere in the natural world, but it doesn’t explain speciation, and that is another part of the story that is rarely discussed. But it’s also why it’s one of nature’s great mysteries. To quote from this week’s New Scientist (13 March, 2010, p.31): ‘Speciation still remains one of the biggest mysteries in evolutionary biology.’

This is a rare admission in a science magazine, because many people believe, on both sides of the ideological divide (that evolution has created in some parts of the world, like the US) that it opens up a crack in the scientific edifice for creationists and intelligent design advocates to pull it down.

But again, let’s compare this to quantum mechanics. In a recent post on Quantum Entanglement (Jan.10), where I reviewed Louisa Gilder’s outstanding and very accessible book on the subject, I explain just how big a mystery it remains, even after more than a century of experimentation, verification and speculation. Yet, no one, whether a religious fundamentalist or not, wants to replace it with a religious text or any other so-called paradigm or theory. This is because quantum mechanics doesn’t challenge anything in the Bible, because the Bible, unsurprisingly, doesn’t include anything about physics or mathematics.

Now, the Bible doesn’t include anything about biology either, but the story of Genesis, which is still a story after all the analysis, has been substantially overtaken by scientific discoveries, especially in the last 2 centuries.

But it’s because of this ridiculous debate, that has taken on a political force in the most powerful and wealthy nation in the world, that no one ever mentions that we really don’t know how speciation works. People are sure to counter this with one word, mutation, but mutations and genetic drift don’t explain how genetic anomalies amongst individuals lead to new species. It is assumed that they accumulate to the point that, in combination with natural selection, a new species branches off. But the New Scientist cover story, reporting on work done by Mark Pagel (an evolutionary biologist at the University of Reading, UK) challenges this conventionally held view.

To quote Pagel: “I think the unexamined view that most people have of speciation is this gradual accumulation by natural selection of a whole lot of changes, until you get a group of individuals that can no longer mate with their old population.”

Before I’m misconstrued, I’m not saying that mutation doesn’t play a fundamental role, as it obviously does, which I elaborate on below. But mutations within individuals don’t axiomatically lead to new species. This is a point that Erwin Schrodinger attempted to address in his book, What is Life? (see my review posted Nov.09).

Years ago, I wrote a letter to science journalist, John Horgan, after reading his excellent book The End of Science (a collection of interviews and reflections by some of the world’s greatest minds in the late 20th Century). I suggested to him an analogy between genes and environment interacting to create a human personality, and the interaction between speciation and natural selection creating biological evolution. I postulated back then that we had the environment part, which was natural selection, but not the gene part of the analogy, which is speciation. In other words, I suggested that there is still more to learn, just like there is still more to learn about quantum mechanics. We always assume that we know everything that there is to know, when clearly we don’t. The mystery inherent in quantum mechanics indicates that there is something that we don’t know, and the same is true for evolution.

Mark Pagel’s research is paradigm-challenging, because he’s demonstrated statistically that genetic drift by mutation doesn’t give the right answers. I need to explain this without getting too esoteric. Pagel looked at the branches of 101 various (evolutionary) trees, including: ‘cats, bumblebees, hawks, roses and the like’. By doing a statistical analysis of the time between speciation events (the length of the branches) he expected to get a Bell curve distribution which would account for the conventional view, but instead he got an exponential curve.

To quote New Scientist: ‘The exponential is the pattern you get when you are waiting for some single, infrequent event to happen… the length of time it takes a radioactive atom to decay, and the distance between roadkills on a highway.’

In other words, as the New Scientist article expounds in some detail, new species happen purely by accident. What I found curious about the above quote is the reference to ‘radioactive decay’ which was the starting point for Erwin Schrodinger’s explanation of mutation events, which is why mutation is still a critical factor in the whole process.

Schrodinger went to great lengths, very early in his exposition, to explain that nearly all of physics is statistical, and gave examples from magnetism to thermal activity to radioactive decay. He explained how this same statistical process works in creating mutations. Schrodinger coined a term, ‘statistico-deterministic’, but in regard to quantum mechanics rather than physics in general. Nevertheless, chaos and complexity theory reinforce the view that the universe is far from deterministic at almost every level that one cares to examine it. As the New Scientist article argues, Pagel’s revelation supports Stephen Jay Gould’s assertion: ‘that if you were able to rewind history and replay the evolution on Earth, it would turn out differently every time.’

I’ve left a lot out in this brief exposition, including those who challenge Pagel’s analysis, and how his new paradigm interacts with natural selection and geographical separation, which are also part of the overall picture. Pagel describes his own epiphany when he was in Tanzinia: ‘watching two species of colobus monkeys frolic in the canopy 40 metres overhead. “Apart from the fact that one is black and white and one is red, they do all the same things... I can remember thinking that speciation was very arbitrary. And here we are – that’s what our models are telling us.”’ In other words, natural selection and niche-filling are not enough to explain diversification and speciation.

What I find interesting is that wherever we look in science, chance plays a far greater role than we credit. It’s not just the cosmos at one end of the scale, and quantum mechanics at the other end, that rides on chance, but evolution, like earthquakes and other unpredictable events, also seems to be totally dependent on the metaphorical roll of the dice.

Addendum 1 : (18 March 2010)

Comments posted on New Scientist challenge the idea that a ‘bell curve’ distribution should have been expected at all. I won’t go into that, because it doesn’t change the outcome: 78% of ‘branches’ statistically analysed (from 110) were exponential and 0% were normal distribution (bell curve). Whatever the causal factors, in which mutation plays a definitive role, speciation is as unpredictable as earthquakes, weather events and radio-active decay (for an individual isotope).

Addendum 2: (18 March 2010)

Writing this post, reminded me of Einstein’s famous quote that ‘God does not play with dice’. Well, I couldn’t disagree more. If there is a creator-God (in the Einstein mould) then first and foremost, he or she is a mathematician. Secondly, he or she is a gambler who loves to play the odds. The role of chance in the natural world is more fundamental and universally manifest than we realise. In nature, small variances can have large consequences: we see that with quantum theory, chaos theory and evolutionary theory. There appears to be little room for determinism in the overall play of the universe.