Why Beliefs Matter; Reflections on the Nature of Science, by E. Brian Davies, is one of the best books I’ve read on science, philosophy and religion, and I’ve read lots of books in all those fields. Davies is Professor of Mathematics at King’s College London and a fellow of the Royal Society. He gives one of the best arguments I’ve encountered against mathematical Platonism, which is high praise indeed from a self-confessed mathematical Platonist like myself.
There is much in this book to be commended, not least his conscientiousness in separating philosophy from science and of pointing out that ‘beliefs’ like the anthropic principle are, in fact, metaphysical considerations rather than truly scientific (it can’t be tested). He outlines the significant difference between the philosophical and scientific ramifications of quantum mechanics, which I’ve expressed myself in a post on Science, Philosophy, Religion (November 2009).
More than anything else, he reinforces the intellectual reality that philosophy often deals with questions for which there may well be no definite answers. And whilst science can provide answers in the form of empirical evidence as well as mathematically based laws to explicate them, the bigger questions, concerning our existence, the origin of the universe and a potential higher purpose, remain elusive.
The scope of Davies' book includes the history of science, the mind-body problem, induction, determinism, artificial intelligence and the modern day ‘warfare’ between science and religion, especially in America (this is not an exhaustive list). I’ll only cover 2 apparently unrelated topics: mathematical Platonism and religion and science.
Davies has no particular barrow to push, and is candid in his disagreement with his fellows on all topics, expressing bewilderment, bordering on amusement, at the hostility one often encounters concerning questions for which there are no definitive answers. One such topic is the philosophy of mathematics and its various ‘schools of thought’ that borders on religious zeal. He calls himself a mathematical ‘pluralist’ because he can see merit in alternative views. As far as mathematical Platonism goes, he expresses appreciation of its appeal to both mathematicians and physicists without necessarily agreeing with them. In his conclusion he calls it ‘irrelevant’, but only because it doesn’t really provide any theoretical benefit. In other words, being a Platonist won’t give you an advantage in understanding mathematics – it’s purely a philosophical position, with no real practical ramifications in executing formulae or even searching for new ones.
He points out that mathematical Platonism has quasi-religious overtones, which I don’t shy away from. I’ve written at least 3 posts previously on this topic, so I won’t labour the point here. It’s a very good example of a philosophical position based more on a ‘feeling’ or ‘sense’ of abstract reality, which its proponents (like myself) then support with rational argument. One of Davies’ strongest arguments is that we are the only species (that we know of) in the entire universe who can not only appreciate mathematics but make it manifest. Without an intelligence like ours, it remains completely hidden which makes its apparent essentiality questionable.
I have 2 not-unrelated responses to this argument. Firstly, all the laws of the universe, that we have discovered, from quantum mechanics to relativity to thermodynamics to the DNA code, would remain complete secrets in the exact same way, yet the universe, that we observe and exist within, is completely dependent on all these things. Secondly, mathematics is the only way we can quantify and interpret these very same laws, which leads me to contend that the mathematics is just as essential as the laws themselves. DNA is a 4 letter code, by the way, that is completely analogous to computer code, so life entails mathematics at a fundamental level.
The alternative view to this is that mathematics is an intellectual construct, purely of human origin, that has allowed us to unravel some of nature’s deepest mysteries. Roger Penrose, whose Platonist philosophy is discussed in some detail by Davies, manages to incorporate both views in a non-contradictory though paradoxical manner, which is what sold me on mathematical Platonism eventually (see below). In other words, I am a convert who came to this position via Pythagoras. In my early years of studying science, I saw mathematics as a tool that physics had seconded, but even then I struggled to reconcile natural laws with their apparent and deeply enigmatic mathematical precision (more on this below).
Davies postulates a hypothetical that there may be a species somewhere in the universe who can fathom nature’s secrets heuristically without mathematics. I can remember, when I was much younger, contemplating the same scenario and even entertained writing a sci-fi story that incorporated such a species. However, I gave up on the enterprise, when I realised that, philosophically, my world-view had changed. Physics, especially quantum mechanics, is so fundamentally dependent on mathematics for its interpretation, that any other methodology appears impossible, which is not to say that it is. Whilst quantum mechanics remains a conundrum in terms of envisaging the ultimate reality of the universe (or universes), it remains, mathematically, a completely consistent and eminently reliable metatheory.
Of course Davies’ discussion on this topic is much more comprehensive than what I’ve presented. I’ve just re-read my post on Schrodinger’s book, What is Life? and his quote concerning mathematics, “…whose truth is not only unassailable, but is obviously there forever; the relations held and will hold irrespective of our inquiry into them. A mathematical truth is timeless, it does not come into being when we discover it.” Davies also quotes Einstein, who wasn’t a Platonist: “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality.” This neatly encapsulates the paradoxical nature of mathematics that I described above.
Davies disputes the ‘independence’ claim because geometry and arithmetic didn’t evolve independently of the physical world that we’ve investigated since we first learnt to count. He also argues that not all ‘objects of reality’ can be defined or described mathematically, but he’s talking about ‘...the contents of our conversations and the products of our culture, in which mathematics is completely useless.’ No one would argue with that, but it’s obvious that Einstein’s ‘objects of reality’ are physical objects rather than cultural artefacts and mental constructs specific to the human race.
I disagree with Davies on his first point too, because mathematical investigations, at least since Newton, have evolved independently of the physical world only to be married back into it when our science has caught up with our mathematics. The most famous examples would include Riemann’s geometry being married to Einstein’s general theory of relativity and complex algebra being essential to Schrodinger’s equations for quantum mechanics. It’s as if mathematical discoveries precede physical discoveries, and, in fact, necessarily so.
There is a sense that mathematics entails a world (at least in the abstract realm) greater than our world, with multiple dimensions that can extend to infinity, of which string theory represents a potential multitude. In other words, there is more mathematics than we need to describe our physical theories, which is why Max Tegmark argues that all mathematically possible universes could exist in a multiverse. If one takes this at face value, then the mathematical world extends beyond the physical world (as Penrose points out) in the same way that the physical world extends beyond our mental world. Hence the paradox that the mind is an effective subset of the physical world, and even if the physical world is not a subset of the mathematical world (as per Tegmark), it appears, at least, to follow mathematical rules, yet mathematics is a product of the human mind. Penrose represents this relationship between the physical, mental and mathematical (Platonic) worlds as a closed circuit, one being a subset of the one before, just as I’ve described them above. Davies also addresses this aspect of Penrose’s philosophical discourse in his book, but I’ll leave that for the reader to pursue if they’re interested.
In a section called The Human Condition, Davies introduces the subject of ‘induction’ by referring to Aristotle’s ‘four types’ of causes that he recorded and discussed in the 4th Century BC. He did this in reference to a clay pot. The ‘material cause’ is the materials that the pot is made from which is the clay. The ‘formal cause’ is its shape or form which is a pot or vase. The ‘efficient cause’ is the process involving the Potter who made it. And the ‘final cause’ is the whole reason it was made which is to store something.
I have to admit I’d never come across this before, despite having read and studied Aristotle at Uni, and Davies makes particular reference to the 4th ‘final cause’ which has disappeared in the philosophy of science, and is arguably the principal source of friction that lies between science and religion. Davies rightly points out that since Descartes, and even more so after Darwin, final cause has no place in science. This is a particular issue of contention I've had with many fundamentalists, like William Lane Craig (refer The God Hypothesis, December 2008). Even if there is a final cause for the universe, science can't tell us anything about it – it’s purely a metaphysical question.
Aristotle’s final cause refers to a human artefact, and it’s not difficult to see how God became an anthropomorphic equivalent who created the universe, life and us, which means we are the final cause. I really don’t have a problem with this, purely from a philosophical viewpoint, because it makes God dependent on us rather than the other way round. If we are the final cause then, without us, there is no reason for God to exist. Few people appreciate the reverse logic that this argument entails: it doesn’t make sense for God to exist without a purpose, and the only purpose we can come up with is us.
In a recent post (Cycles of Time, last month) I gave considerable space to the exposition of entropy, aka the 2nd law of thermodynamics. A corollary to the 2nd law is that the universe is not teleological and by inclusion neither is evolution. I would suggest that this, and not the Book of Genesis, is the main philosophical difference between science and religion. Religion infers that the universe has a purpose and science infers that it doesn’t.
Davies expounds at length on the indeterminism inherent in chaos theory as well as quantum mechanics. Another Davies (Paul Davies), when he still resided in Australia, wrote an excellent book on chaos theory called The Cosmic Blueprint. The significance of chaos theory, and its particular relationship with entropy, is that very small changes can lead to huge differential consequences. In a not-so-recent issue of New Scientist (16 October 2010) their feature article described how chaos theory appears to rule evolution. In particular, evolution is fractal in the same way that branching blood vessels are in the human body. Fractal relationships appear everywhere in nature; the best example being a coastline (Davies’ example in Cosmic Blueprint). The Mandelbrot set is fractal and so are Pollock’s paintings (like Blue Poles hanging in the Sydney Art Gallery). Fractals demonstrate the same relationship at all scales, which means, in evolutionary terms, that speciation branches appear in similar ratios at all levels. The article explains how, over 65 million years, major climatic events, major tectonic events and major evolutionary events all follow the same ‘chaotic’ patterns, though ‘...connections between them are hard to discern.’
Brian Davies, like Schrodinger (What is Life?), explains how radioactivity is statistically highly predictable whilst individualistically it is impossible to predict. In fact, Schrodinger begins his book with an exposition on how almost everything in physics is statistically determined: from magnetism to the photo-electric effect to the behaviour of gases and fluids. It’s only at a macro scale that physics appears predictable. The point is that between chaos theory, the 2nd law of thermodynamics and quantum phenomena, the universe is a lottery. As Stephen Jay Gould famously said, if you were to rerun the universe you’d get a completely different result. This flies in the face of all religious philosophy.
The last 60 pages of Davies’ 240 page book (so 25%) is devoted to a section titled, Science and Religion. He starts off with a philosophical aphorism: “We must learn to live with the fact that some disagreements cannot be resolved.” Throughout his book he places terse statements in bold type like the following:
Christian theologians ignore the fundamentalist challenge at their own peril. It is the greatest threat to rational thought and toleration at the present time.
To outsiders like myself, America appears to be one of the most polarised societies in the Western world: politically, intellectually and religiously. The all-consuming debate between evolutionary science and fundamentalist religion really doesn’t exist anywhere else in the world, certainly not to the same degree of hostility and, dare-I-say-it, desperation. It’s only taken on a global perspective because American culture is so pervasive, especially on the internet.
Davies points out that humanist philosophy goes back even further than Christianity, citing Socrates, Aristotle, Plato and even Confucius. Confucius is the earliest known philosopher (500 BC) to evoke a fundamentally empathetic approach to ethics: ‘Don’t do to others what you wouldn’t want done to yourself.’ He acknowledged the importance of trust between rulers and their subjects, arguing that trust was the last commodity a ruler could afford to lose. (Someone should point this out to Egypt’s Mubarak.) Davies argues that Mill's utilitarian philosophy has probably been the biggest influence on Western democracy, because it’s inherent in civil rights and feminist movements witnessed in the last half of the last century. Even though no one invoked Mill as the model to follow; utilitarianism is concerned with the greatest benefit to the greatest number.
At the end of his book, Davies discusses the religious views of famous scientific figures, both historical and contemporary. He is not afraid to criticise Dawkins’ The God Delusion, even though he obviously is not completely at odds with Dawkins’ philosophy. Dawkins polarises people almost like no one I know, yet he’s neither a villain nor a hero. He has a demeanour not unlike an Australian politician: provocative, rhetorically aggressive, disputatious, uncompromising and unapologetic. On the blogosphere, if you criticise Dawkins, as I have done a few times, you suddenly become a Christian apologist to his supporters. It’s a sign of insecurity that people can’t deal with criticism without adopting an extreme position. Davies, like myself, takes Dawkins to task for treating all religions and all religious followers the same. It doesn’t help his cause to alienate people who would otherwise support him. ‘The worst feature of Dawkins’ book is its failure to get to grips with the variety of religious belief. Dawkins’ real enemy is fundamentalism, but he attacks religion indiscriminately.’ I agree completely.
Davies ends with a poem by William Cecil Dampier, from which I’ll quote the last verse:
And Nature smiles – still unconfessed
The secret thought she thinks –
Inscrutable she guards unguessed
The Riddle of the Sphinx
Davies follows with these words:
The riddle of our place in the universe may never be solved, and I am content that this should be so. The struggle to divine the meaning of life is a part of being human.
Science can’t solve this riddle either; in fact it tells us that our existence is a completely arbitrary phenomenon built upon an accumulation of arbitrary phenomena. The end result (so far) is mind and mind seeks its own purpose because that’s its nature.
Addendum 1: I need to point out, in all fairness to Davies, that his discourse on mathematics is far more erudite than mine, which is not apparent from my presentation above. I attempted to address this in a later post, Metaphysics in mathematics revisited.
Addendum 2: In March 2012, I give a more definitive response to the question of teleology after re-reading Paul Davies' Cosmic Blueprint and blogging about it.
Sunday, 6 February 2011
Saturday, 29 January 2011
Be afraid, be very afraid
This video was attached to the following email:
Drone Controllers
For non-pilots, these controllers are in Nevada and are each flying a drone thousands of miles away in the combat zone in Iraq and Afghanistan.
Their left hand is on the throttle controlling the drone's engine.
Note all the buttons which perform various tasks without removing the hand from the throttle.
The right hand is flying the plane.
Welcome to the new world order. This is modern warfare.
Today's headline: 'Missiles fired from Nevada controlled drone aircraft kill Taliban leader'
Watch how it's done. Turn the speakers on & watch in full screen.
ALSO NOTICE THE COMFORT FROM WHICH THE "FLYERS" OPERATE.
I don’t know if this is a simulation or the real thing, but I commented on the deployment of military drones in a post I wrote last November, titled: We have to win the war against stupidity first.
If it’s the real thing then it makes me and anyone else who watches it something of a voyeur. I refuse to watch videoed assassinations because it feeds their purpose, but is this any different?
There are a lot of pertinent issues here, not least the implication that this is how wars will be fought in the future, but let’s start with the most obvious one: how is this perceived by non-Western eyes?
Let’s reverse the scenario: how would people in the West respond if this technology was adopted by Iran or North Korea or even Russia or China? At present I believe that only America and Israel actually deploy it. Is this a case of might is right? Those with the best military technology are axiomatically those with the moral prerogative to use it. Because that’s how it appears.
We routinely accuse suicide bombing as an act of cowardice, but is this perceived as any less cowardly by those who are on the receiving end?
Someone once pointed out, in reference to the deployment of U-boats by the Germans in WWI (but it actually applies to all military conflicts), if one’s opponent has a technological advantage then one’s only chance of success is to break the rules – in other words, play dirty. This is why suicide bombing is the weapon of choice by people who believe they are being invaded by a technologically superior force, especially when the superiority is indisputably dominant.
And there are other issues: the scenario is reminiscent of Milgram’s experiment, which demonstrated how easy it is to inflict mortal injuries on a complete stranger who is sight unseen. The couple in the video are so relaxed and detached from the life-and-death consequences of their actions that it makes me wonder if it’s not just a training session.
In the 1960s I can still remember reading a MAD magazine that satirically showed 2 chess opponents facing each other off with ballistic missile launchers instead of chess pieces and consequently destroying each other, the chess board and the room in which they were playing. It was a commentary on the cold war mentality of the time and the threat of intercontinental ballistic missiles, which could render the planet virtually uninhabitable without any army taking the field.
We no longer see that as a threat, but the idea of waging war without committing ground troops (which is theoretically the same scenario we have in the video) has strong political appeal despite the obvious moral issues that it raises.
There are 2 fundamental issues, one of which was addressed in my post last November. Firstly, the entire operation is dependent on ‘intelligence’ that the ‘target’ is the enemy. In Vietnam, the CIA used ‘assassination squads’ made up of local tribesmen to target specific enemies. Barry Petersen, an Australian seconded to the CIA in that conflict, fell out with his superiors when he refused to use Montagnard tribesmen, loyal to him, as assassination squads, despite their commendable military record (Frank Walker, The Tiger Man of Vietnam). His reasoning was that they would be used to settle personal vendettas, creating distrust and secondary enmity that would not help win the war. In a tribal environment, like Afghanistan and Iraq, this type of abuse of ‘intelligence’ can also occur.
But it’s the psychological component of this type of warfare that makes it most unpalatable, at least, to me. Unfortunately, intervention by Western military units have shown extraordinary lack of cultural sensitivity in the countries they become involved in. This was true in Vietnam, in Iraq, and, I suspect, Afghanistan. Sometimes military leaders on the ground recognise this when their political leaders don’t. America, in particular, doesn’t have a good record in this area.
If one insists on waging a war without face to face involvement then the consequences will be dire for everyone concerned. The psychological impact on the civilians of a country being attacked by robotic planes can not be overstated. It will foster hate, resentment and a stubborn will to reek vengeance. All you have to do is put yourself in their shoes.
Sunday, 16 January 2011
Cycles of Time – a new theory of cosmology
Cycles of Time, subtitled An Extraordinary New View of the Universe, is a very recent book by Roger Penrose; so recent that I pre-ordered it. Anyone who has followed my blog over the last few years will know that I’m a big fan of Penrose. Along with Paul Davies and Richard Feynman, I think he’s one of the top physics writers for laypeople ever. John Gribbin and James Gleick are also very good but not quite in the same league in my opinion. Davies, Feynman and Penrose all have different strengths so comparisons are not entirely fair. Feynman was the great communicator of some of the most esoteric theories in physics and if you want to grasp the physics, he’s the best. Davies is, in my view, the best philosophical writer and also covers the widest field: covering topics like astrophysics, the origin of life, cosmology, chaos theory, the nature of time and in The Goldilocks Enigma the meaning of life, the universe and everything.
Penrose is actually a mathematician and made significant contributions to tessellation (tiles, map boundaries etc), but he’s also won at least one award in physics (1988 Wolf Prize jointly with Stephen Hawking) and his dissertations on the subject of consciousness reveal him as an erudite and compelling polymath.
My favourite book of his is The Emperor’s New Mind(1989) where he first tackled the subject of consciousness and challenged the prevailing view that Artificial Intelligence would herald in a new consciousness equivalent to or better than our own. But the book also covers almost the entire field of physics, argues cogently for a Platonic view of mathematics, explains the role of entropy on a cosmic scale, and devotes an entire chapter to the contingent nature of ‘truth’ in science. A must-read for anyone who thinks we know everything or are on the verge of knowing everything.
Now I’m the first to admit that I can quickly get out of my depth on this topic, and I can’t defend all the arguments that Penrose delivers, because, quite frankly, I don’t understand all the physics that lay behind them, but he’s one of the few people, with the relevant intellectual credentials, who can challenge the prevailing view on our universe’s origins and not lose credibility in the process.
For a start, reading this book makes one realise how little we do know and how speculative some of our theories are. Many commentators treat theoreticians who challenge string theory, and its latest incantation, M theory, as modern-day luddites, which is entirely unfair considering that string theory has no experimental or observational successes to its name. In other words, it’s a work of mathematical genius that may or may not reflect reality. Penrose’s CCC (Conformal Cyclic Cosmology) is also a mathematically consistent theory with no empirical evidence to either confirm or deny it. (Penrose does suggest avenues of enquiry to rectify that however.)
I first came across CCC in a book, On Space and Time (2008), a collection of ‘essays’ by people like Alain Connes, Shahn Majid, Andrew Taylor and of course Sir Roger Penrose. It also included John Polkinghorne and Michael Heller to provide a theological perspective. Personally, I think it would have been a better book if it stuck to the physics, because I don’t think metaphysical philosophies are any help in understanding cosmology, even though one could argue that mathematical Platonism is a metaphysical philosophy. I don’t mind that people want to reconcile scientific knowledge with their personal religious beliefs, but it’s misleading to imply that religion can inform science. And science can only inform religion if one conscientiously rejects all the mythology that religions seem to attract and generate. Putting that personal caveat aside, I can highly recommend this book, edited by Shahn Majid, for an overview of current thinking on cosmology and all the mysteries that this topic entails. This is true frontier-science and that perspective should never be lost in any such discussion.
Getting back to Penrose, his latest book tackles cosmology on the grandest scale from the universe’s Big Bang to its inevitable demise. Along the way he challenges the accepted wisdom of inflation amongst other prevailing ideas. He commences with a detailed description of entropy because it lies at the heart of the conundrum as he sees it. It’s entropy that makes the Big Bang so very special, and he spends almost half the book on expounding why.
Penrose describes specific aspects of time that I referred to in a post last year (The enigma we call time, July 2010). He gives the same example I did of an egg falling off a table demonstrating the inherent relationship between entropy (the 2nd law of thermodynamics) and the arrow of time we are all familiar with. He even cites a film running backwards showing an egg reconstituting itself and rising from the floor as an example of time reversal and a violation of the 2nd law of thermodynamics acting simultaneously, just as I did. He also explains how time doesn’t exist without mass, because for photons (light rays), which are massless, time is always zero.
The prevailing view, according to almost everything I read on this subject via science magazines, is that we live in a multiverse where universes pop out like exploding bubbles, of which the Big Bang and its consequent ‘inflation’ was just one. In the Christmas/New Year edition of New Scientist (25 December 2010/1 January 2011, p.9) there is an article that claims we may have ‘evidence’ of ‘bruising’ in the CMB (Cosmic Microwave Background) resulting from ‘collisions’ with other universes. (The cosmic background radiation was predicted by the Big Bang and discovered purely by accident, which makes it the best evidence we have that our universe did indeed begin with the Big Bang.)
Some people also believe there is an asymmetry to the universe, implying there is an ‘axis’, which would be consistent with us being ‘joined’ to a ‘neighbouring universe’. But be careful with all these speculative scenarios fed by inexplicable and potentially paradigm-changing observations – they just confirm how little we really know.
The multiverse in conjunction with the ‘anthropic principle’ appears to be the most widely accepted explanation for the how, why and wherewithal of our hard-to-believe existence. Because we live in possibly the only universe of an infinite number then naturally it is the only universe we have knowledge of. If all the other universes, or almost all, are uninhabitable then no one will ever observe them. Ergo we observe this universe because it’s the one that produced life, of which we are the ultimate example.
Paul Davies, in The Goldilocks Enigma, spends a page and a half discussing both the virtues and pitfalls of the multiverse proposition. In particular, he discusses what he calls ‘...the extreme multiverse model proposed by Max Tegmark in which all possible worlds of any description really exist…’ In other words, whatever mathematics allows can exist. Quoting Davies again: ‘The advantage of the extreme multiverse is that it explains everything because it contains everything.’ However, as he also points out, because it explains everything it virtually explains nothing. As someone else, a theologian (I can’t remember who), once pointed out, in a discussion with Richard Dawkins, it’s no more helpful than a ‘God-of-the-gaps’ argument, which also explains everything and therefore ultimately explains nothing.
Stephen Hawking has also come out with a new book with Leonard Mlodinow titled The Grand Design, which I haven’t read but read reviews of, in particular Scientific American. Someone in America (Dale, who has a blog, Faith in Honest Doubt) put me onto a radio podcast by some guys under the name, Reasonable Doubts, who ran a 3-part series on Buddhism. At the end of one of their programmes they took Hawking to task for making what they saw as the absurd claim that the universe could be ‘something from nothing’.
I left a comment on their blog that this was not a new idea:
I'm not sure why you got in a tiz about Hawkings' position, though I haven't read his latest book, but I read an editorial comment in Scientific American under the heading, Hawking vs God. The idea that the universe could be 'something for nothing' is not new. Paul Davies discussed it over 20 years ago in God and the New Physics (1983) in a chapter titled: Is the universe a free lunch? He says almost exactly what Hawking is credited with saying (according to Scientific American): the universe (according to the 'free lunch' scenario) can account for itself, the only thing that is unaccountable are the laws of nature that apparently brought it about. Davies quotes physicist, Alan Guth: "It's often said that there is no such thing as a free lunch. The universe, however, is a free lunch."
Davies, Hawking and Penrose are not loonies – they are all highly respected physicists. We’ve learned from Einstein and Bohr that nature doesn’t obey rules according to our common sense view of the world, and, arguably, the universe’s origin is the greatest of all unsolved mysteries. Why is there something instead of nothing? And is there any reason to assume that there wasn’t nothing before we had something?
What, may you ask, has any of this to do with Penrose’s CCC theory? It’s just a detour to synoptically describe the intellectual landscape that his theory inhabits.
As I alluded to earlier, Penrose focuses on the biggest conundrum in the universe, being entropy, and how it makes the Big Bang so ultra-ultra special. Few discussions I’ve read on cosmology even mention the role of entropy, yet it literally drives the entire universe’s evolution – Paul Davies doesn’t shy away from it in God and the New Physics - but otherwise, only Penrose puts it centre stage from my reading experience.
Both Davies and Penrose discuss it in terms of ‘phase space’ which is really hard to explain and really hard to envisage without thinking about dimensional space. But effectively the equation for entropy is the logarithm of a volume of phase space multiplied by Boltzmann’s constant: S = k log(V). The use of a logarithm allows one to differentiate between entropies in a dynamic system. Significantly, one can only ‘take away’ entropy by adding it to somewhere else that’s external to the ‘closed’ environment one is studying. The most obvious example is a refrigerator that keeps cold by dumping heat externally to the ambient air in a room (the fridge loses entropy by adding it externally). As Penrose points out, the only reason the Sun’s energy is ‘useful’ to us is because it’s a ‘locally’ hot spot in an otherwise cold space. If it was in thermal equilibrium with its environment it would be useless to Earth. ‘Work’ can only be done when there is an imbalance in energy (usually temperature) between a system and its environment.
But more significantly, to decrease the entropy in a ‘closed’ system (like a refrigerator or Earth) there must be an increase in entropy externally. So ultimately the entire universe’s entropy must always be increasing. The corollary to that is that the universe must have started with a very small entropy indeed, and that is what makes the Big Bang so very special. In fact Penrose calculates the ultimate phase space volume of the entire universe as e raised to the power of 10 raised to the power of 123, (e10)123, or, if it’s easier to comprehend, take 10 raised to the power of 10 (10 plus 10 noughts) raised to the power of 123 (10 x 123 noughts). So That’s 1 with 123 x 10 noughts after it. To reverse this calculation, it means that the precision of the big bang to create the universe that we live in is one part in 10 to the 10 to 123, (1-10)-123. So that’s a precision of 0.00…(123x10 0’s)1.
Penrose takes the universe in its current state and extrapolates it back to its near-origin at the so-called inflationary stage between 10-35 and 10-32 seconds from its birth. He also extrapolates it into its distant future, making some assumptions, and finding that the two states are ‘conformally’ equivalent. One of his key assumptions is that the universe is inherently hyperbolic so it has a small but positive cosmological constant. This means that the universe will always expand and never collapse back onto itself. Penrose provides good arguments, that I won’t attempt to replicate here, that a ‘Big Bounce’ scenario could not produce the necessary entropic precision that we appear to need for the Big Bang. In other words, it would be a violation of the 2nd law of thermodynamics.
Penrose’s future universe assumes that the universe would consist entirely of black holes, many of which exist at the centre of all known galaxies. As these black holes become ‘hotter’ than the space that surrounds them, they will evaporate through Hawking radiation, so that eventually the entire universe will be radiation in the form of electromagnetic waves and gravitons. Significantly there will be virtually no mass therefore no clocks, and, from what I can understand, that’s what makes the universe conformal. It will have a ‘conformal boundary’. Penrose’s bold hypothesis is that this conformal boundary will become the conformal boundary that we envisage at the end of the inflationary period of our universe. Hence the death of one universe becomes the birth of the next.
What of the conundrum of the 2nd law of thermodynamics? Penrose spends considerable time discussing whether or not information is lost in black holes, which is a contentious point. Hawking once argued that information was lost, but now argues otherwise. Penrose thinks he should have stuck to his guns. Many scientists believe it’s a serious flaw in cosmological thinking to consider that information could be lost in black holes. Many scientists and philosophers argue that ‘everything’ is information, including us. There’s an argument that teleportation is theoretically achievable, even on a macro scale, because everything is just information at base. I’ve never been convinced of that premise, but leaving that aside, I think that information could be lost in black holes and so does Penrose. If this is true then all information regarding our universe will no longer exist after all the black holes evaporate, and, arguably, entropy will be reset, along with time. I’ve simplified this part of Penrose’s treatise, so I may not be doing him justice, but I know that the loss of information through multiple black hole evaporation is crucial to his theory.
When I first came across this thesis in On Space and Time I admit that it appealed to me philosophically. The idea that the end of the universe could be mathematically and physically equivalent to its beginning, and therefore could recycle endlessly is an intellectually attractive idea. Nature is full of beginnings and endings on all sorts of scales, why not on the cosmological scale? Infinity is the scariest concept there is if you think about it seriously – the alternative is oblivion, nihilism effectively. We have a life of finite length that we are only aware of while we are living it, yet we know that existence goes on before we arrive and after we’re gone. Why should it be any different for the universe itself?
I admit I don’t understand all the physics, and there still seems to be the issue of going from a cold universe of maximum entropy to a hot universe of minimum entropy, yet Penrose seems to believe that his ‘conformal boundary’ at both ends allows for that eventuality.
Penrose is actually a mathematician and made significant contributions to tessellation (tiles, map boundaries etc), but he’s also won at least one award in physics (1988 Wolf Prize jointly with Stephen Hawking) and his dissertations on the subject of consciousness reveal him as an erudite and compelling polymath.
My favourite book of his is The Emperor’s New Mind(1989) where he first tackled the subject of consciousness and challenged the prevailing view that Artificial Intelligence would herald in a new consciousness equivalent to or better than our own. But the book also covers almost the entire field of physics, argues cogently for a Platonic view of mathematics, explains the role of entropy on a cosmic scale, and devotes an entire chapter to the contingent nature of ‘truth’ in science. A must-read for anyone who thinks we know everything or are on the verge of knowing everything.
Now I’m the first to admit that I can quickly get out of my depth on this topic, and I can’t defend all the arguments that Penrose delivers, because, quite frankly, I don’t understand all the physics that lay behind them, but he’s one of the few people, with the relevant intellectual credentials, who can challenge the prevailing view on our universe’s origins and not lose credibility in the process.
For a start, reading this book makes one realise how little we do know and how speculative some of our theories are. Many commentators treat theoreticians who challenge string theory, and its latest incantation, M theory, as modern-day luddites, which is entirely unfair considering that string theory has no experimental or observational successes to its name. In other words, it’s a work of mathematical genius that may or may not reflect reality. Penrose’s CCC (Conformal Cyclic Cosmology) is also a mathematically consistent theory with no empirical evidence to either confirm or deny it. (Penrose does suggest avenues of enquiry to rectify that however.)
I first came across CCC in a book, On Space and Time (2008), a collection of ‘essays’ by people like Alain Connes, Shahn Majid, Andrew Taylor and of course Sir Roger Penrose. It also included John Polkinghorne and Michael Heller to provide a theological perspective. Personally, I think it would have been a better book if it stuck to the physics, because I don’t think metaphysical philosophies are any help in understanding cosmology, even though one could argue that mathematical Platonism is a metaphysical philosophy. I don’t mind that people want to reconcile scientific knowledge with their personal religious beliefs, but it’s misleading to imply that religion can inform science. And science can only inform religion if one conscientiously rejects all the mythology that religions seem to attract and generate. Putting that personal caveat aside, I can highly recommend this book, edited by Shahn Majid, for an overview of current thinking on cosmology and all the mysteries that this topic entails. This is true frontier-science and that perspective should never be lost in any such discussion.
Getting back to Penrose, his latest book tackles cosmology on the grandest scale from the universe’s Big Bang to its inevitable demise. Along the way he challenges the accepted wisdom of inflation amongst other prevailing ideas. He commences with a detailed description of entropy because it lies at the heart of the conundrum as he sees it. It’s entropy that makes the Big Bang so very special, and he spends almost half the book on expounding why.
Penrose describes specific aspects of time that I referred to in a post last year (The enigma we call time, July 2010). He gives the same example I did of an egg falling off a table demonstrating the inherent relationship between entropy (the 2nd law of thermodynamics) and the arrow of time we are all familiar with. He even cites a film running backwards showing an egg reconstituting itself and rising from the floor as an example of time reversal and a violation of the 2nd law of thermodynamics acting simultaneously, just as I did. He also explains how time doesn’t exist without mass, because for photons (light rays), which are massless, time is always zero.
The prevailing view, according to almost everything I read on this subject via science magazines, is that we live in a multiverse where universes pop out like exploding bubbles, of which the Big Bang and its consequent ‘inflation’ was just one. In the Christmas/New Year edition of New Scientist (25 December 2010/1 January 2011, p.9) there is an article that claims we may have ‘evidence’ of ‘bruising’ in the CMB (Cosmic Microwave Background) resulting from ‘collisions’ with other universes. (The cosmic background radiation was predicted by the Big Bang and discovered purely by accident, which makes it the best evidence we have that our universe did indeed begin with the Big Bang.)
Some people also believe there is an asymmetry to the universe, implying there is an ‘axis’, which would be consistent with us being ‘joined’ to a ‘neighbouring universe’. But be careful with all these speculative scenarios fed by inexplicable and potentially paradigm-changing observations – they just confirm how little we really know.
The multiverse in conjunction with the ‘anthropic principle’ appears to be the most widely accepted explanation for the how, why and wherewithal of our hard-to-believe existence. Because we live in possibly the only universe of an infinite number then naturally it is the only universe we have knowledge of. If all the other universes, or almost all, are uninhabitable then no one will ever observe them. Ergo we observe this universe because it’s the one that produced life, of which we are the ultimate example.
Paul Davies, in The Goldilocks Enigma, spends a page and a half discussing both the virtues and pitfalls of the multiverse proposition. In particular, he discusses what he calls ‘...the extreme multiverse model proposed by Max Tegmark in which all possible worlds of any description really exist…’ In other words, whatever mathematics allows can exist. Quoting Davies again: ‘The advantage of the extreme multiverse is that it explains everything because it contains everything.’ However, as he also points out, because it explains everything it virtually explains nothing. As someone else, a theologian (I can’t remember who), once pointed out, in a discussion with Richard Dawkins, it’s no more helpful than a ‘God-of-the-gaps’ argument, which also explains everything and therefore ultimately explains nothing.
Stephen Hawking has also come out with a new book with Leonard Mlodinow titled The Grand Design, which I haven’t read but read reviews of, in particular Scientific American. Someone in America (Dale, who has a blog, Faith in Honest Doubt) put me onto a radio podcast by some guys under the name, Reasonable Doubts, who ran a 3-part series on Buddhism. At the end of one of their programmes they took Hawking to task for making what they saw as the absurd claim that the universe could be ‘something from nothing’.
I left a comment on their blog that this was not a new idea:
I'm not sure why you got in a tiz about Hawkings' position, though I haven't read his latest book, but I read an editorial comment in Scientific American under the heading, Hawking vs God. The idea that the universe could be 'something for nothing' is not new. Paul Davies discussed it over 20 years ago in God and the New Physics (1983) in a chapter titled: Is the universe a free lunch? He says almost exactly what Hawking is credited with saying (according to Scientific American): the universe (according to the 'free lunch' scenario) can account for itself, the only thing that is unaccountable are the laws of nature that apparently brought it about. Davies quotes physicist, Alan Guth: "It's often said that there is no such thing as a free lunch. The universe, however, is a free lunch."
Davies, Hawking and Penrose are not loonies – they are all highly respected physicists. We’ve learned from Einstein and Bohr that nature doesn’t obey rules according to our common sense view of the world, and, arguably, the universe’s origin is the greatest of all unsolved mysteries. Why is there something instead of nothing? And is there any reason to assume that there wasn’t nothing before we had something?
What, may you ask, has any of this to do with Penrose’s CCC theory? It’s just a detour to synoptically describe the intellectual landscape that his theory inhabits.
As I alluded to earlier, Penrose focuses on the biggest conundrum in the universe, being entropy, and how it makes the Big Bang so ultra-ultra special. Few discussions I’ve read on cosmology even mention the role of entropy, yet it literally drives the entire universe’s evolution – Paul Davies doesn’t shy away from it in God and the New Physics - but otherwise, only Penrose puts it centre stage from my reading experience.
Both Davies and Penrose discuss it in terms of ‘phase space’ which is really hard to explain and really hard to envisage without thinking about dimensional space. But effectively the equation for entropy is the logarithm of a volume of phase space multiplied by Boltzmann’s constant: S = k log(V). The use of a logarithm allows one to differentiate between entropies in a dynamic system. Significantly, one can only ‘take away’ entropy by adding it to somewhere else that’s external to the ‘closed’ environment one is studying. The most obvious example is a refrigerator that keeps cold by dumping heat externally to the ambient air in a room (the fridge loses entropy by adding it externally). As Penrose points out, the only reason the Sun’s energy is ‘useful’ to us is because it’s a ‘locally’ hot spot in an otherwise cold space. If it was in thermal equilibrium with its environment it would be useless to Earth. ‘Work’ can only be done when there is an imbalance in energy (usually temperature) between a system and its environment.
But more significantly, to decrease the entropy in a ‘closed’ system (like a refrigerator or Earth) there must be an increase in entropy externally. So ultimately the entire universe’s entropy must always be increasing. The corollary to that is that the universe must have started with a very small entropy indeed, and that is what makes the Big Bang so very special. In fact Penrose calculates the ultimate phase space volume of the entire universe as e raised to the power of 10 raised to the power of 123, (e10)123, or, if it’s easier to comprehend, take 10 raised to the power of 10 (10 plus 10 noughts) raised to the power of 123 (10 x 123 noughts). So That’s 1 with 123 x 10 noughts after it. To reverse this calculation, it means that the precision of the big bang to create the universe that we live in is one part in 10 to the 10 to 123, (1-10)-123. So that’s a precision of 0.00…(123x10 0’s)1.
Penrose takes the universe in its current state and extrapolates it back to its near-origin at the so-called inflationary stage between 10-35 and 10-32 seconds from its birth. He also extrapolates it into its distant future, making some assumptions, and finding that the two states are ‘conformally’ equivalent. One of his key assumptions is that the universe is inherently hyperbolic so it has a small but positive cosmological constant. This means that the universe will always expand and never collapse back onto itself. Penrose provides good arguments, that I won’t attempt to replicate here, that a ‘Big Bounce’ scenario could not produce the necessary entropic precision that we appear to need for the Big Bang. In other words, it would be a violation of the 2nd law of thermodynamics.
Penrose’s future universe assumes that the universe would consist entirely of black holes, many of which exist at the centre of all known galaxies. As these black holes become ‘hotter’ than the space that surrounds them, they will evaporate through Hawking radiation, so that eventually the entire universe will be radiation in the form of electromagnetic waves and gravitons. Significantly there will be virtually no mass therefore no clocks, and, from what I can understand, that’s what makes the universe conformal. It will have a ‘conformal boundary’. Penrose’s bold hypothesis is that this conformal boundary will become the conformal boundary that we envisage at the end of the inflationary period of our universe. Hence the death of one universe becomes the birth of the next.
What of the conundrum of the 2nd law of thermodynamics? Penrose spends considerable time discussing whether or not information is lost in black holes, which is a contentious point. Hawking once argued that information was lost, but now argues otherwise. Penrose thinks he should have stuck to his guns. Many scientists believe it’s a serious flaw in cosmological thinking to consider that information could be lost in black holes. Many scientists and philosophers argue that ‘everything’ is information, including us. There’s an argument that teleportation is theoretically achievable, even on a macro scale, because everything is just information at base. I’ve never been convinced of that premise, but leaving that aside, I think that information could be lost in black holes and so does Penrose. If this is true then all information regarding our universe will no longer exist after all the black holes evaporate, and, arguably, entropy will be reset, along with time. I’ve simplified this part of Penrose’s treatise, so I may not be doing him justice, but I know that the loss of information through multiple black hole evaporation is crucial to his theory.
When I first came across this thesis in On Space and Time I admit that it appealed to me philosophically. The idea that the end of the universe could be mathematically and physically equivalent to its beginning, and therefore could recycle endlessly is an intellectually attractive idea. Nature is full of beginnings and endings on all sorts of scales, why not on the cosmological scale? Infinity is the scariest concept there is if you think about it seriously – the alternative is oblivion, nihilism effectively. We have a life of finite length that we are only aware of while we are living it, yet we know that existence goes on before we arrive and after we’re gone. Why should it be any different for the universe itself?
I admit I don’t understand all the physics, and there still seems to be the issue of going from a cold universe of maximum entropy to a hot universe of minimum entropy, yet Penrose seems to believe that his ‘conformal boundary’ at both ends allows for that eventuality.
Saturday, 8 January 2011
It's women who choose, not men
Not so recently, I told someone I had a blog and it was called Journeyman Philosopher and they couldn’t stop themselves from laughing. I said, ‘Yes, it is a bit wankerish.’ Especially for an Aussie. But I’m not and never will be the real thing – a philosopher, that is – yet I practice philosophy, by attempting to emulate the credo I have inscribed at the top. The truth is that none of us, who value knowledge for its own sake, ever stop learning, and I’ve made it a lifetime passion. This blog does little more than pass on and share, and occasionally provide insights. But I also attempt to provoke thought, and if I should ever fail at that then I should call it quits.
So this post is one of those thought-provoking ones, because it challenges centuries of culturally accepted norms. I’m a single bloke who’s never married, so I’m hardly an expert on relationships, but this is a philosophical position on relationships garnered both from experience and observation.
Recently, I took part in a discussion on Eli’s blog, Rustbelt Philosophy, whereby I cited Nietzsche from Beyond Good and Evil that most people take a philosophical position on visceral grounds and then rationalise it with an argument. As I commented on Eli’s blog, I think this is especially true for religious arguments, but it has wider applications as well. The more we invest in a theory (for example) the less likely we are to reject it, even in the face of conflicting evidence.
I’m currently reading Roger Penrose’s latest book, Cycles of Time (to be the subject of a future post) and he readily acknowledges his personal prejudices in outlining his iconoclastic theory for the origins of our universe. The point I’m making, and its relevance to this post, is that I too have prejudices that shape my views on this topic.
In the last decade or 2 there has been a strong and popular resurgence in Jane Austen’s novels (through film and TV), which indicates they have strong universal themes. Jane Austen suffered from the prejudices of her day when women were not supposed to earn money, and the class structure, in which she lived, precluded intelligent women, like herself, from attaining fulfilling lives. Everything was dependent on them marrying the right bloke, or more clinically, marrying into the right family. I have to say that I’ve seen examples of that narrow-minded thinking even in my own lifetime. Austen had her novels published through a male intermediary and on her grave there is no mention that she was an author because it was considered a sleight for a woman to admit she had a profession.
But the theme of every Austen novel that I’ve seen (I haven’t read any of them) is that the woman finds the right bloke despite the obstacles that her society puts in her way. And the right bloke is the one who demonstrates that he’s a genuine friend and not someone who is playing the social game according to the rules of their society. Austen was an iconoclast in her own right, and the fact that her stories still ring true today, indicates that she was revealing a universal truth.
Somewhere in my childhood I realised that women are in fact the stronger sex, and that whilst men can’t live without women, they can live without us. But this is only one reason that I believe women should do the choosing and not the men. The mechanics of courtship also indicate that it is the woman who chooses even though the bloke thinks it’s him. I remember seeing a documentary on speed-dating once, and the facilitator made the exact same observation. Personally, I wasn’t surprised.
In many respects, I think the best analogy in the animal kingdom is with birds. The male really just wants to have sex, so what does he do? He sings or he flashes colourful plumage or he performs a dance or he builds a bower, and then the female chooses the one she thinks is best, not the other way round. Now, this is an analogy, but I think it applies to humans just as well. Whilst it is the woman who might arguably wear the plumage, she does the selecting, and it is the men who perform. We show off our wit and conversation, we drive flash cars and buy big houses and use whatever talents we may have to impress. I read somewhere recently (Scientific American Mind) that in mixed company it is the men who tell the jokes and the women who do the laughing.
So my argument is that we woo but women select. I believe this is the natural order and centuries of cultural, religious and political control have attempted to overturn it. All our institutions have been patriarchal and marriage is arguably the most patriarchal of them all.
And this is where my argument reflects the sentiment expressed by Nietzsche, because I have a rational justification to support my intuitively-premised prejudice. It is the woman who has most to lose in a relationship because she’s the one who gets pregnant. So what I’m arguing is that it should be her choice all the way down the line. It is the woman who should determine the parameters and limits of a relationship. It is she who should decide how intimate it should become and whether marriage is an option, not the bloke. I would even argue that men cope with rejection better than women. Our sex drive is like a tap, easy to turn on, not so easy to turn off, but that’s what masturbation is for.
Anyone who has read my book, Elvene, will recognise a feminist theme that pretty well reflects the philosophy I’ve outlined above. It wasn’t intentional, and it was only afterwards that I realised that I had encapsulated that theme into my writing. Considering it’s set in the future, not the past, it has little in common with Jane Austen. As one of its reviewers pointed out, the book also deals with relationship issues like respect, honesty and generosity of spirit.
In essence, I think the patriarchal cultural mores that we’ve had for centuries are not only past their use-by-date, but are in conflict with the natural order for human relationships. Our societies would be a lot more psychologically healthy if that was acknowledged.
So this post is one of those thought-provoking ones, because it challenges centuries of culturally accepted norms. I’m a single bloke who’s never married, so I’m hardly an expert on relationships, but this is a philosophical position on relationships garnered both from experience and observation.
Recently, I took part in a discussion on Eli’s blog, Rustbelt Philosophy, whereby I cited Nietzsche from Beyond Good and Evil that most people take a philosophical position on visceral grounds and then rationalise it with an argument. As I commented on Eli’s blog, I think this is especially true for religious arguments, but it has wider applications as well. The more we invest in a theory (for example) the less likely we are to reject it, even in the face of conflicting evidence.
I’m currently reading Roger Penrose’s latest book, Cycles of Time (to be the subject of a future post) and he readily acknowledges his personal prejudices in outlining his iconoclastic theory for the origins of our universe. The point I’m making, and its relevance to this post, is that I too have prejudices that shape my views on this topic.
In the last decade or 2 there has been a strong and popular resurgence in Jane Austen’s novels (through film and TV), which indicates they have strong universal themes. Jane Austen suffered from the prejudices of her day when women were not supposed to earn money, and the class structure, in which she lived, precluded intelligent women, like herself, from attaining fulfilling lives. Everything was dependent on them marrying the right bloke, or more clinically, marrying into the right family. I have to say that I’ve seen examples of that narrow-minded thinking even in my own lifetime. Austen had her novels published through a male intermediary and on her grave there is no mention that she was an author because it was considered a sleight for a woman to admit she had a profession.
But the theme of every Austen novel that I’ve seen (I haven’t read any of them) is that the woman finds the right bloke despite the obstacles that her society puts in her way. And the right bloke is the one who demonstrates that he’s a genuine friend and not someone who is playing the social game according to the rules of their society. Austen was an iconoclast in her own right, and the fact that her stories still ring true today, indicates that she was revealing a universal truth.
Somewhere in my childhood I realised that women are in fact the stronger sex, and that whilst men can’t live without women, they can live without us. But this is only one reason that I believe women should do the choosing and not the men. The mechanics of courtship also indicate that it is the woman who chooses even though the bloke thinks it’s him. I remember seeing a documentary on speed-dating once, and the facilitator made the exact same observation. Personally, I wasn’t surprised.
In many respects, I think the best analogy in the animal kingdom is with birds. The male really just wants to have sex, so what does he do? He sings or he flashes colourful plumage or he performs a dance or he builds a bower, and then the female chooses the one she thinks is best, not the other way round. Now, this is an analogy, but I think it applies to humans just as well. Whilst it is the woman who might arguably wear the plumage, she does the selecting, and it is the men who perform. We show off our wit and conversation, we drive flash cars and buy big houses and use whatever talents we may have to impress. I read somewhere recently (Scientific American Mind) that in mixed company it is the men who tell the jokes and the women who do the laughing.
So my argument is that we woo but women select. I believe this is the natural order and centuries of cultural, religious and political control have attempted to overturn it. All our institutions have been patriarchal and marriage is arguably the most patriarchal of them all.
And this is where my argument reflects the sentiment expressed by Nietzsche, because I have a rational justification to support my intuitively-premised prejudice. It is the woman who has most to lose in a relationship because she’s the one who gets pregnant. So what I’m arguing is that it should be her choice all the way down the line. It is the woman who should determine the parameters and limits of a relationship. It is she who should decide how intimate it should become and whether marriage is an option, not the bloke. I would even argue that men cope with rejection better than women. Our sex drive is like a tap, easy to turn on, not so easy to turn off, but that’s what masturbation is for.
Anyone who has read my book, Elvene, will recognise a feminist theme that pretty well reflects the philosophy I’ve outlined above. It wasn’t intentional, and it was only afterwards that I realised that I had encapsulated that theme into my writing. Considering it’s set in the future, not the past, it has little in common with Jane Austen. As one of its reviewers pointed out, the book also deals with relationship issues like respect, honesty and generosity of spirit.
In essence, I think the patriarchal cultural mores that we’ve had for centuries are not only past their use-by-date, but are in conflict with the natural order for human relationships. Our societies would be a lot more psychologically healthy if that was acknowledged.
Sunday, 2 January 2011
The Number Devil by Hans Magnus Enzensberger
It’s been a while since I’ve written anything really meaty on my blog and an entire year since I last wrote a post that reviewed a book on mathematics.
But what I really like about this particular post is that it renders the near to the global. This arose from a Christmas drink that I had with my neighbour across the road, Sarah, who lent me a book, that she never lends, on the proviso I write it up on my blog. So from my neighbour, who literally lives directly opposite me with her 2 sons, Andre and Emelio, to the blogosphere.
Over a bottle of Aussie red (Barossa Valley Shiraz 2008) – yes that’s worth mentioning because we both agreed that it was a bloody good drop (literally and figuratively) – we somehow got into a discussion on mathematics and the teaching of mathematics in particular, which led us to swapping books the next day.
On Christmas Day 2009, I published a post on The Bedside Book of Algebra (Michael Willers), which is the book I swapped with Sarah. The Number Devil; A Mathematical Adventure covers some of the same material but it’s aimed at a younger audience and it has a different approach. The whole purpose of this book it to reveal to young people that mathematics is a world worth exploring and not just a sadistic intellectual exercise designed by teachers to torment young developing minds. Sarah’s book has 2 bookmarks in it: one for her and one for her 7 year-old son; and her son’s bookmark is further advanced than hers.
It is written in novel-form and the premise of the narrative is very simple: the protagonist, Robert, is having tormenting dreams when he is visited by a devil, who calls himself the ‘Number Devil’ and begins to give him lessons in mathematics. It’s extremely clever, because it’s engaging and contains entertaining and informative illustrations, as well as providing exposition on some of the more esoteric mathematical concepts like infinity, transfinite numbers, combinations and permutations, Pascal’s triangle, Fibonacci numbers, prime numbers and Goldbach’s conjecture.
Whilst Enzensberger reveals the relationship between Pascal’s triangle and Fibonacci numbers, he doesn’t explain the relationship between Pascal’s triangle and the binomial theorem, which I learned in high school. He also explains the relationship between Pascal’s triangle and the combination algorithm, but not the way I learned it, which I think is more intuitive and useful. He uses diagonals (within Pascal’s triangle) whereas I learned it by using the rows.
The cleverness is that he provides these expositions without revealing to the reader how advanced these mathematical ‘lessons’ are. In fact, the reader is introduced to the ‘mysteries’ that have fascinated ‘ancients’ from many cultures across the world. Enzensberger’s inspired approach is to reveal the appeal of mathematics (that most mathematicians only find in adulthood) to young people before they are turned off it forever. He demonstrates that esoteric concepts can be taught without emphasising their esoterica.
Even the idea of a ‘number devil’ is inspired because mathematics is considered to be so devilish, and, in some cultures, mathematicians were considered to be devil’s apprentices (refer my recent post on Hypatia). In the second chapter (chapters are sequential nights of dreaming) Robert finds himself in a cave with the Number Devil, and the illustration is an obvious allusion to Plato’s cave, though no mention is made of this in the text.
At the end, the Number Devil takes Robert to ‘Number Heaven’ and ‘Number Hell’, though they appear to be the same place, where he meets some of the ‘masters’ like Russell, Fibonacci, Archimedes and a Chinese man whose name we don’t learn. We don’t meet Pythagoras who lives in a higher realm altogether, up in the clouds.
I’d recommend this book to any parent whose children show the slightest mathematical inclination and also adults who want an introduction to this esoteric world. As Sarah said, it’s like a mathematical version of Jostein Gaarder’s Sophie’s World, which is a high enough recommendation in itself.
Oh, I should mention that the illustrations are by Rotraut Susanne Berner; they augment the text perfectly.
But what I really like about this particular post is that it renders the near to the global. This arose from a Christmas drink that I had with my neighbour across the road, Sarah, who lent me a book, that she never lends, on the proviso I write it up on my blog. So from my neighbour, who literally lives directly opposite me with her 2 sons, Andre and Emelio, to the blogosphere.
Over a bottle of Aussie red (Barossa Valley Shiraz 2008) – yes that’s worth mentioning because we both agreed that it was a bloody good drop (literally and figuratively) – we somehow got into a discussion on mathematics and the teaching of mathematics in particular, which led us to swapping books the next day.
On Christmas Day 2009, I published a post on The Bedside Book of Algebra (Michael Willers), which is the book I swapped with Sarah. The Number Devil; A Mathematical Adventure covers some of the same material but it’s aimed at a younger audience and it has a different approach. The whole purpose of this book it to reveal to young people that mathematics is a world worth exploring and not just a sadistic intellectual exercise designed by teachers to torment young developing minds. Sarah’s book has 2 bookmarks in it: one for her and one for her 7 year-old son; and her son’s bookmark is further advanced than hers.
It is written in novel-form and the premise of the narrative is very simple: the protagonist, Robert, is having tormenting dreams when he is visited by a devil, who calls himself the ‘Number Devil’ and begins to give him lessons in mathematics. It’s extremely clever, because it’s engaging and contains entertaining and informative illustrations, as well as providing exposition on some of the more esoteric mathematical concepts like infinity, transfinite numbers, combinations and permutations, Pascal’s triangle, Fibonacci numbers, prime numbers and Goldbach’s conjecture.
Whilst Enzensberger reveals the relationship between Pascal’s triangle and Fibonacci numbers, he doesn’t explain the relationship between Pascal’s triangle and the binomial theorem, which I learned in high school. He also explains the relationship between Pascal’s triangle and the combination algorithm, but not the way I learned it, which I think is more intuitive and useful. He uses diagonals (within Pascal’s triangle) whereas I learned it by using the rows.
The cleverness is that he provides these expositions without revealing to the reader how advanced these mathematical ‘lessons’ are. In fact, the reader is introduced to the ‘mysteries’ that have fascinated ‘ancients’ from many cultures across the world. Enzensberger’s inspired approach is to reveal the appeal of mathematics (that most mathematicians only find in adulthood) to young people before they are turned off it forever. He demonstrates that esoteric concepts can be taught without emphasising their esoterica.
Even the idea of a ‘number devil’ is inspired because mathematics is considered to be so devilish, and, in some cultures, mathematicians were considered to be devil’s apprentices (refer my recent post on Hypatia). In the second chapter (chapters are sequential nights of dreaming) Robert finds himself in a cave with the Number Devil, and the illustration is an obvious allusion to Plato’s cave, though no mention is made of this in the text.
At the end, the Number Devil takes Robert to ‘Number Heaven’ and ‘Number Hell’, though they appear to be the same place, where he meets some of the ‘masters’ like Russell, Fibonacci, Archimedes and a Chinese man whose name we don’t learn. We don’t meet Pythagoras who lives in a higher realm altogether, up in the clouds.
I’d recommend this book to any parent whose children show the slightest mathematical inclination and also adults who want an introduction to this esoteric world. As Sarah said, it’s like a mathematical version of Jostein Gaarder’s Sophie’s World, which is a high enough recommendation in itself.
Oh, I should mention that the illustrations are by Rotraut Susanne Berner; they augment the text perfectly.
Tuesday, 21 December 2010
Ayaan Hirsi Ali
This woman should need no introduction, she’s been in the media in most Western countries I’m sure. I thought this was a really good interview (Tue. 21 Dec. 2010) because it gives an insight into her background as well as a candid exposition of her political and philosophical views.
I haven’t read either of her biographies, but I’ve read second-hand criticism which led me to believe she was anti-Islamic. This is not entirely true, depending on how one defines Islam. To quote her own words: “I have no problem with the religious dimension of Islam.” She’s not the first Muslim I’ve come across to differentiate between religious and political Islam. Most Westerners, especially since 9/11, believe that any such distinction is artificial. I beg to differ.
She makes it very clear that she’s against the imposition of Sharia law, the subjugation of women and any form of totalitarianism premised on religious-based scripture (irrespective of the religion). In short, she’s a feminist. She decries the trivial arguments over dress when there are other issues of far greater import, like arranged marriages, so-called circumcision of women and honour killings. (For an intelligent debate on whether the burqua should be ‘outlawed’ I refer you to this.)
What I found remarkable, and almost unimaginable, was how violent her childhood and upbringing were. There was violence in the school, violence in the home, violence in politics. As she points out it was so pervasive that a peaceful environment was considered unthinkable. One of the most poignant stories she tells was when she went to Holland to seek asylum, and on going to a Police Station to register, the policeman asked her if she would like a cup of coffee or tea. This was a revelation to her: that a man in uniform should offer a woman, a stranger and a foreigner, a cup of coffee or tea was simply mind-blowing.
It is beyond most of us to imagine a childhood where violence is the principal form of interrogation and negotiation between people in all walks of life: home, education and work; yet that was her life. That she can now talk of falling in love and of writing a letter to her unborn child for a hopeful future is close to miraculous.
What resonated with me was her argument that it doesn’t take 600 years to reconcile Islam with the modern secular world, but only 4 generations. I have Muslim friends, both in America and in Australia, and they belie the belief, held by many in Western societies, that Muslims can’t assimilate and yet keep their cultural and spiritual beliefs. They demonstrate to me that Ayaan Hirsi Ali is correct in her fundamental assumptions and philosophical approach.
I haven’t read either of her biographies, but I’ve read second-hand criticism which led me to believe she was anti-Islamic. This is not entirely true, depending on how one defines Islam. To quote her own words: “I have no problem with the religious dimension of Islam.” She’s not the first Muslim I’ve come across to differentiate between religious and political Islam. Most Westerners, especially since 9/11, believe that any such distinction is artificial. I beg to differ.
She makes it very clear that she’s against the imposition of Sharia law, the subjugation of women and any form of totalitarianism premised on religious-based scripture (irrespective of the religion). In short, she’s a feminist. She decries the trivial arguments over dress when there are other issues of far greater import, like arranged marriages, so-called circumcision of women and honour killings. (For an intelligent debate on whether the burqua should be ‘outlawed’ I refer you to this.)
What I found remarkable, and almost unimaginable, was how violent her childhood and upbringing were. There was violence in the school, violence in the home, violence in politics. As she points out it was so pervasive that a peaceful environment was considered unthinkable. One of the most poignant stories she tells was when she went to Holland to seek asylum, and on going to a Police Station to register, the policeman asked her if she would like a cup of coffee or tea. This was a revelation to her: that a man in uniform should offer a woman, a stranger and a foreigner, a cup of coffee or tea was simply mind-blowing.
It is beyond most of us to imagine a childhood where violence is the principal form of interrogation and negotiation between people in all walks of life: home, education and work; yet that was her life. That she can now talk of falling in love and of writing a letter to her unborn child for a hopeful future is close to miraculous.
What resonated with me was her argument that it doesn’t take 600 years to reconcile Islam with the modern secular world, but only 4 generations. I have Muslim friends, both in America and in Australia, and they belie the belief, held by many in Western societies, that Muslims can’t assimilate and yet keep their cultural and spiritual beliefs. They demonstrate to me that Ayaan Hirsi Ali is correct in her fundamental assumptions and philosophical approach.
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