Google has honoured Emmy Noether's birthday. Few people know who she is but, amongst other contributions to mathematics, she proved a theorem, known as Noether's theorem, that underpins all of physics because it deals with symmetry and conservation laws of energy and momentum.
The mathematics is well over my head, but I appreciate its ramifications. Basically, it deals with the mathematical relationships between symmetry in space and conservation of momentum, symmetry in time and conservation of energy and symmetry of rotation and conservation of angular momentum. This applies in particular to quantum mechanics, though conservation laws are equally relevant in relativity theory.
Symmetry, in this context, is about translation: translations in space, translations in time and translations in rotation. Richard Feynman gives a good exposition in Six Not-So-Easy Pieces, where I came across it for the first time, and he describes it thus: ...a most profound and beautiful thing, is that, in quantum mechanics, for each of the rules of symmetry there is a corresponding conservation law; there is a definite connection between the laws of conservation and the symmetry of physical laws.
You can read about it in some detail in Wikipedia, though I confess it's a bit esoteric.
Noether died relatively young in America at age 53, 2 years after escaping Nazi Germany, and Einstein wrote a moving tribute to her in the New York Times (1935). Physicists, Leon M. Lederman and Christopher T. Hill, in Symmetry and the Beautiful Universe, give the following accolade: “..certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics…”
The sad part about her story is that she is virtually unknown and was not given due recognition in her own time, simply because she was a woman.
Addendum: It's also 100 years since Noether developed her seminal theorem - the same year that Einstein developed his General Theory of Relativity, incorporating gravity.
Monday, 23 March 2015
Saturday, 21 March 2015
Citizenfour - aka Edward Snowden
I saw this Oscar-winning movie last weekend, and, tellingly, despite very recently winning the Oscar for Best Documentary, it’s only being shown in one cinema in the whole of Melbourne. And yet everyone with internet access should watch this movie, because its message affects all of us.
For those living under a rock, Edward Snowden famously revealed that the US is capturing all our on-line activities and mobile phone calls, even though, as revealed in the film, various representatives of the government had been openly denying this (even under oath) for some time. The closest we get to acknowledgement is when someone says, in a Senate Committee Hearing, in response to this very question about accessing people’s digital communications: “Not willingly.” Reading between the lines, one could conclude that it could happen ‘accidentally’, which suggests a couple of options: the information is available if they want to access it; or they might accidentally access someone’s data whilst trying to access someone else’s, whom they can legitimately target (via a court order or whatever judicial process is required). As it turns out, thanks to Snowden’s expose, we know it’s the first option.
Ethically, there are 2 distinct but related issues implicit in the same movie. Are the American Government’s activities in this regard, ethical, and is it ethical for Snowden to use his position of privileged information to break his Government’s trust (as well as the law) by revealing them to the rest of the world?
There is a third ethical issue, entwined with the previous 2: is it ethical for the American Government to pursue Snowden with the full force of its law, treating him, effectively, as a traitor and a spy (they are charging him under the espionage act)?
Let’s deal with the first ethical issue first; after all, it’s the one that triggered the other two. As pointed out in the movie, this ‘action’ on the part of the American Government is a consequence of 9/11 and the threat of terrorist attacks anywhere in the Western world. It’s also pointed out in the movie that England have even more comprehensive measures than the US, regarding tracking everyday digital information of its own citizens.
On the same day I saw this film, I heard a news bulletin that here in Australia, the Government is currently debating a bill requiring internet providers to keep all user activity (in Australia) for however many years (I don’t know if there’s a limit). Interestingly, the only proviso the Opposition suggested is that journalists be protected in order to protect their sources. This is a very important point, because it’s only journalists that can keep politicians honest, and journalists’ roles in providing a conduit for Snowden’s ‘leak’ was crucial to his expose. I’ve said before that the health of a democracy can be measured by the freedom that journalists are allowed in criticising their elected leaders. Keeping sources ‘secret’ has been critical (at least, in Australia) in allowing journalists that particular freedom.
There are similarities between this documentary and the not-so-recent movie, Kill the Messenger, because, in both cases, someone exposed the Government or the Government’s agents in activities that the public were unaware of, and, in both cases the Government, or its agencies, effectively destroyed the whistleblowers’ lives.
But this parliamentary debate taking place in Australia reveals an ethical distinction, which, in my view, is worth noting. In Australia, because it has to be passed as Parliamentary law, it cannot be done without the public’s awareness, and, I feel, this is where the American Government went wrong. I don’t have an issue with them keeping all my digital data, most of which, like this blog, is freely available to the public anyway. And I understand how such data is crucial to stopping terrorist attacks, so, as long as the data capture is not used to persecute me personally, I have no problem.
Having said that, the movie points out how data collection on a nation’s citizens is a first step in controlling or oppressing that citizenry. So, hand in hand with this action is an essential ‘trust’ that it will be used only for spoiling terrorist attacks and that the essential democratic character of the nation won’t be compromised in the process. In some ways, this is a moot point because, after watching this film, I couldn’t help but feel that this will never be reversed. We already live in a quasi-Orwellian environment where our entire lives can be tracked digitally, if someone so requires. We effectively have no secrets regarding our on-line activity (including mobile phone communications).
Towards the end of the movie, there is a news clip of President Obama saying that we needed ‘to have this debate’, implying that it could have occurred without Snowden’s revelations. However, this is contradicted by earlier video footage (alluded to above) that, even under oath, no one was going to admit to this whilst the public remained ignorant.
And this brings to the fore Snowden’s ethics. There are some similarities here between Snowden and Assange, both of whom are now living in exile, because both had sensitive material that embarrassed the American Government in particular. However, Snowden is probably more like Chelsea (Bradley) Manning, who had access to and leaked the relevant files, in that both men acted on their conscience at considerable personal cost.
But, in the case of Snowden, he demonstrates neither the naivety of Manning nor the ego of Assange. It is clear from the outset that Snowden understood fully the consequences of his actions, and is remarkably calm throughout his entire dealings with the specific journalists he colluded with in order to make public what the American Government preferred to remain covert. This is the crux of the issue for me: not that the American Government is collecting all our communications data, but that they did it behind our backs and are now enraged that some individual dared to let everyone know.
There is some irony that Snowden now lives in exile with his partner in Russia, a country not renown for honouring freedom of speech or freedom of the press. If Snowden had done to Russia, what he’s done to America, he probably would have been assassinated. As it is, in America, they will lock him up and throw away the key, just like they’ve done to Manning, assuming they ever catch him.
For those living under a rock, Edward Snowden famously revealed that the US is capturing all our on-line activities and mobile phone calls, even though, as revealed in the film, various representatives of the government had been openly denying this (even under oath) for some time. The closest we get to acknowledgement is when someone says, in a Senate Committee Hearing, in response to this very question about accessing people’s digital communications: “Not willingly.” Reading between the lines, one could conclude that it could happen ‘accidentally’, which suggests a couple of options: the information is available if they want to access it; or they might accidentally access someone’s data whilst trying to access someone else’s, whom they can legitimately target (via a court order or whatever judicial process is required). As it turns out, thanks to Snowden’s expose, we know it’s the first option.
Ethically, there are 2 distinct but related issues implicit in the same movie. Are the American Government’s activities in this regard, ethical, and is it ethical for Snowden to use his position of privileged information to break his Government’s trust (as well as the law) by revealing them to the rest of the world?
There is a third ethical issue, entwined with the previous 2: is it ethical for the American Government to pursue Snowden with the full force of its law, treating him, effectively, as a traitor and a spy (they are charging him under the espionage act)?
Let’s deal with the first ethical issue first; after all, it’s the one that triggered the other two. As pointed out in the movie, this ‘action’ on the part of the American Government is a consequence of 9/11 and the threat of terrorist attacks anywhere in the Western world. It’s also pointed out in the movie that England have even more comprehensive measures than the US, regarding tracking everyday digital information of its own citizens.
On the same day I saw this film, I heard a news bulletin that here in Australia, the Government is currently debating a bill requiring internet providers to keep all user activity (in Australia) for however many years (I don’t know if there’s a limit). Interestingly, the only proviso the Opposition suggested is that journalists be protected in order to protect their sources. This is a very important point, because it’s only journalists that can keep politicians honest, and journalists’ roles in providing a conduit for Snowden’s ‘leak’ was crucial to his expose. I’ve said before that the health of a democracy can be measured by the freedom that journalists are allowed in criticising their elected leaders. Keeping sources ‘secret’ has been critical (at least, in Australia) in allowing journalists that particular freedom.
There are similarities between this documentary and the not-so-recent movie, Kill the Messenger, because, in both cases, someone exposed the Government or the Government’s agents in activities that the public were unaware of, and, in both cases the Government, or its agencies, effectively destroyed the whistleblowers’ lives.
But this parliamentary debate taking place in Australia reveals an ethical distinction, which, in my view, is worth noting. In Australia, because it has to be passed as Parliamentary law, it cannot be done without the public’s awareness, and, I feel, this is where the American Government went wrong. I don’t have an issue with them keeping all my digital data, most of which, like this blog, is freely available to the public anyway. And I understand how such data is crucial to stopping terrorist attacks, so, as long as the data capture is not used to persecute me personally, I have no problem.
Having said that, the movie points out how data collection on a nation’s citizens is a first step in controlling or oppressing that citizenry. So, hand in hand with this action is an essential ‘trust’ that it will be used only for spoiling terrorist attacks and that the essential democratic character of the nation won’t be compromised in the process. In some ways, this is a moot point because, after watching this film, I couldn’t help but feel that this will never be reversed. We already live in a quasi-Orwellian environment where our entire lives can be tracked digitally, if someone so requires. We effectively have no secrets regarding our on-line activity (including mobile phone communications).
Towards the end of the movie, there is a news clip of President Obama saying that we needed ‘to have this debate’, implying that it could have occurred without Snowden’s revelations. However, this is contradicted by earlier video footage (alluded to above) that, even under oath, no one was going to admit to this whilst the public remained ignorant.
And this brings to the fore Snowden’s ethics. There are some similarities here between Snowden and Assange, both of whom are now living in exile, because both had sensitive material that embarrassed the American Government in particular. However, Snowden is probably more like Chelsea (Bradley) Manning, who had access to and leaked the relevant files, in that both men acted on their conscience at considerable personal cost.
But, in the case of Snowden, he demonstrates neither the naivety of Manning nor the ego of Assange. It is clear from the outset that Snowden understood fully the consequences of his actions, and is remarkably calm throughout his entire dealings with the specific journalists he colluded with in order to make public what the American Government preferred to remain covert. This is the crux of the issue for me: not that the American Government is collecting all our communications data, but that they did it behind our backs and are now enraged that some individual dared to let everyone know.
There is some irony that Snowden now lives in exile with his partner in Russia, a country not renown for honouring freedom of speech or freedom of the press. If Snowden had done to Russia, what he’s done to America, he probably would have been assassinated. As it is, in America, they will lock him up and throw away the key, just like they’ve done to Manning, assuming they ever catch him.
Saturday, 14 March 2015
Today is π Day
I only found about this yesterday, via COSMOS online. Actually, it's Pi Day tomorrow in the US where it originated because, using American date nomenclature, March 14 (3/14) gives the first 3 digits of pi. But this year is special because the date is 3/14/15, which are the first 5 digits and it will be the only time in the whole century.
π, as everyone knows, is the ratio of the circumference of a perfect circle to its diameter, irrespective of the size of the circle. But it's a very nerdy number, because it turns up in the most unexpected places, like quantum mechanics and Euler's famous formula: eiπ + 1 = 0; earning the sobriquet, 'God's own equation'. A simple derivation can be found here.
Pi is the best known so-called transcendental number and, of course, it has an infinite string of digits that appear to be truly random (refer COSMOS link above). COSMOS also explain that if you toss a coin 2n times, and n is large enough, then the probability of getting equal number of heads and tails is 1/√(nπ). Mathematics contains many hidden formulae like this that give unexpected relationships relevant to the real world.
Addendum: It should be pointed out that today is also Albert Einstein's birthday and this year is the centenary of his masterpiece, the General Theory of Relativity, not to be confused with the Special Theory, which he penned 10 years earlier in 1905.
π, as everyone knows, is the ratio of the circumference of a perfect circle to its diameter, irrespective of the size of the circle. But it's a very nerdy number, because it turns up in the most unexpected places, like quantum mechanics and Euler's famous formula: eiπ + 1 = 0; earning the sobriquet, 'God's own equation'. A simple derivation can be found here.
Pi is the best known so-called transcendental number and, of course, it has an infinite string of digits that appear to be truly random (refer COSMOS link above). COSMOS also explain that if you toss a coin 2n times, and n is large enough, then the probability of getting equal number of heads and tails is 1/√(nπ). Mathematics contains many hidden formulae like this that give unexpected relationships relevant to the real world.
Addendum: It should be pointed out that today is also Albert Einstein's birthday and this year is the centenary of his masterpiece, the General Theory of Relativity, not to be confused with the Special Theory, which he penned 10 years earlier in 1905.
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.
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.
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.
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