Paul P. Mealing

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Friday, 25 December 2009

The Origins of Mathematics

Note the plural in the title, because the mathematics we use today comes from a number of different sources, geographically and culturally. We have a very Eurocentric outlook on mathematics that belies its global heritage.

I came across a small, well-presented volume in my local bookshop: The Bedside Book of Algebra by Michael Willers, a Canadian high school teacher of my vintage, going by the pop-cultural references he sprinkles throughout. When I thumbed through it, my first impression was that it contained nothing I didn’t already know, but I liked the presentation and I realised that it gave a history of mathematics as well as an exposition. It would be an excellent book for anyone wanting a grasp of high school mathematics, as it covers most topics, except calculus and matrices. The presentation is excellent, as he delivers his topics in 2 page bites, and provides examples that are easy to follow. I read it from cover to cover, and learnt a few new things as well as reacquainting myself with old friends like Pascal’s triangle. In fact, Willers revealed a few things about Pascal’s triangle that I didn’t know, like its relationship with the Fibonacci sequence and its generation of fractal patterns using a ‘tiling’ algorithm developed by Polish mathematician, Waclaw Sierpinski in 1915.

I already knew that the Chinese had discovered Pascal’s triangle some 500 years before Pascal (11th Century, Jia Xian), but I didn’t know that the earliest known reference was from an Indian mathematician, Varahamihira, in the 6th Century, or that it appeared in 10th Century Persia, thanks to Al-Karaji. (Blaise Pascal lived 1623-1662.)

Fibonacci (1170-1250) is most famously remembered for the arithmetic sequence that bears his name and also the ‘Golden Ratio’, which can be generated from the sequence. Both the Fibonacci sequence and the Golden Ratio can be found in nature – for example, flower petals are invariably a Fibonacci number and the height of a person’s navel to their height is supposedly the Golden Ratio, but I’m unsure if that is true or just wistful thinking on the part of renaissance artists. Because the Fibonacci sequence is derived by the sum of the previous 2 numbers in the sequence, there are some natural events that follow that rule, like the unchecked population growth of rabbits (that is provided as an example in Willers’ book) and was apparently the original example that Fibonacci used to introduce it.

But we all owe Fibonacci a great debt, because it was he who introduced the Hindu-Arabic numeral system to the Western world in a popular format that has made life so much easier for accountants, engineers, economists, mathematical students and anyone who has ever had to deal with numbers, which is all of us. When I was a child I was told that we used ‘Arabic’ numerals, and I only learned recently that they originated in India. The 7th Century Indian mathematician, Brahmagupta, formulated the first known mathematical concepts that treated zero as a number as well as a place holder (to paraphrase Willers).

Zero and negative numbers were treated with suspicion by the ancient Greeks and Romans, as they preferred geometrical over arithmetical analysis. Because there were no negative areas or negative volumes, the idea of a negative number was considered ‘absurd’. (I have to admit I had the same problem with ‘imaginary’ numbers, when I first encountered them, but I’m getting off the track, and I’ll return to imaginary numbers later.) Likewise, there was no place for a number that represented nothing, but once one introduces negative numbers, zero becomes inevitable, because a negative plus a positive of the same amount must give zero. But zero as a place holder is even more important, because it facilitates all arithmetical computations. As Willer says, imagine trying to do basic arithmetic with Roman numerals, let alone anything esoteric.

Willers quotes Pierre-Simon Laplace (1749-1827): “It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit.”

Willers is the first author I’ve read who makes a genuine attempt to give the Indians and the Persians their due credit for our mathematical heritage. Like the Chinese, the Indians discovered Pythagoras’s triangle before the Pythagoreans, though many people believe Pythagoras actually learnt it from the Babylonians. The Indians also investigated the square route of 2, as well as π (pi), around the same time as the ancient Greeks. In the middle ages, a succession of Indian scholars worked on quadratic equations.

But it is Brahmagupta (598-670), who lived in northwestern India (now Pakistan), to whom Willers devotes one of his 2-page treatises, because he argues that Brahmagupta had the biggest influence on Western mathematics. He lists Brahmagupta’s 14 laws, all dealing with the arithmetic ‘rules’ applicable to zero and negative numbers that, in modern times, we all learn in our childhood.

Willers also gives special attention to two Islamic mathematicians, Al-Khwarizmi (born around 780) and Omar Khayam (1048-1122). Al-Khwarizmi came from Khwarezm (present-day Uzbekistan) and worked in the ‘House of Wisdom’ (see below). He gave us two of our most common mathematical terms: algebra and algorithm. Algebra came from the title of a book he wrote, Hisab Al-jabr w’Al-Muqabala, derived from the word, ‘Al-jabr’. Significantly, he developed methods for deriving the roots of quadratic equations.

The word, algorithm, also comes from the title of a book, Algoritmi de Numero Indorum, which is a Latin translation of one of Al-Khwarizmi’s Arabic texts, now lost. And, according to Willers, algorithm ‘means a number of steps or instructions to be followed.’ Of course, this word is now associated with computer programmes (software). This modern incarnation began with Alan Turing’s famous ‘thought experiment’ of a ‘Universal Turing machine’, as the first iconic example of the modern use of algorithm, which is literally a set of instructions, otherwise known as ‘code’. All modern computers are universal Turing machines, by the way, so it’s much more than a thought experiment now, and algorithms are the code, or software, that drives them.

Omar Khayam is probably better known as a poet, from his authorship of The Rubaiyat, a collection of 600 quatrains (4 line poems). But he also authored a number of books on mathematics, including Treatise on Demonstration of Problems of Algebra (1070), in which he solves cubic equations geometrically by intersecting conic sections with a plane. If one cuts a cone with a plane it describes a curve on the plane. Depending on the angle of the plane with the cone, one can get a circle, an ellipse, a parabola, or, using two cones, two mirror hyperbolae. In another text, that Khayam references, but has since been lost, he writes about Pascal’s triangle, though, obviously, he called it something else.

Omar Khayam provides the best quote in Willers’ book, taken from the above text:

The majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes.

As Willers points out, Plato’s academy closed in 529 and Fibonacci came on the scene in Pisa at the end of the 12th Century, and it wasn’t until the renaissance that Western science, art and philosophy really gained the ascendancy again. The interim period is known colloquially as the ‘dark ages’, because knowledge and scientific progress seemed to stagnate. As Willers says: “From that point [the closure of Plato’s Academy] until the thirteenth century the mathematical centre of the world was in the East.”

According to Willers, The House of Wisdom was established in Baghdad by Harun Al-Rashid (763-809) and translated works from Persia, Greece and India. It was a centre for education in humanities and sciences until it was destroyed by the Mongols in 1258. Without this Islamic connection over that period, the Greek and Roman knowledge in the sciences, philosophy, mathematics and literature, which, today, we consider to be our Western heritage, may have been lost.

As well as providing this historical context, in more detail than I can render here, and that most of us don’t even know about, Willers gives us excellent exposition on a number of topics: permutations and combinations, probability theory, logarithms, trigonometry, quadratics, complex algebra, the binomial theorem, and others.

He treats all these exemplarily, but I would like to say something about complex arithmetic and imaginary numbers, because it was a personal stumbling block for me, and, in hindsight, it shouldn’t have been. The set of imaginary numbers contains only one number, i, which is the square route of -1 (some texts say the set contains 2 numbers, i and –i, but, being pedantic, I beg to differ). Now all through my childhood, the square route of -1 was considered an impossibility like dividing by zero. So when someone finally came up with i, I believed I’d been conned – it was a convenience, invented to overcome a conundrum, and, from my perspective, it should have remained an impossibility. Part of the problem, as Willers points out, is that it’s called an ‘imaginary’ number, when it’s just as real as any other number, and I think that’s a very good point.

When one thinks that the Pythagoreans had serious problems accepting irrational numbers, and then the Greek and Roman mathematicians who followed them, had conceptual issues with zero and negative numbers, the concept of i is no different. It’s a number and it opens up an entirely new world in mathematics that includes fractals, the famous Mandelbrot set and quantum mechanics. If one doesn’t explain complex numbers using the complex plane (or Argand diagram) then it won’t make sense, but if one does, everything falls into place. In particular, multiplying by i rotates any graph on the plane through 90 degrees (a right angle), and by i2 (-1) by 180 degrees. In an ordinary number line with positive numbers running right and negative numbers running left, multiplying a positive number by -1 rotates the number through the origin (0) by 180 degrees to its negative equivalent. If you have an i axis running vertically through 0 then multiplying a number by i just rotates it by 90 degrees (half way). If you draw the graph it makes perfect sense.

Imaginary numbers, like multiple dimensions, demonstrate that the mathematical world can go places that the physical world doesn’t necessarily follow, yet these esoteric mathematical entities can have applications in the real world that we don’t anticipate at the time of their discovery. Reimann’s geometry giving us Einstein’s General Theory of Relativity and imaginary numbers giving us the key to quantum mechanics are two cases in point, both barely a century ago.

I’m one of those who sees mathematics as an abstract territory that only an intelligent species can navigate. Personally, I would like to think that we are not the only ones in the universe who can, and maybe there is at least one other species somewhere who can navigate even further than we can. It’s a sobering yet tantalising thought.


Addendum: I've since written an exposition on imaginary numbers and the complex plane, for those who are interested.


Monday, 7 December 2009

Tim Flannery’s The Weather Makers

This is a timely post, considering the Copenhagen Summit on Climate Change is starting tonight (my time). I’ve just read Tim Flannery’s The Weather Makers, published in 2005. Tim Flannery is unknown outside of Australia, but he was awarded the Australian of the Year title in 2007. Considering that our Prime Minister of the day, John Howard, was a self-confessed climate-change sceptic, that’s quite an achievement. He was also awarded Australian Humanist of the Year in 2005. I have to admit that I didn’t even know the award existed until I read about it on the back fly cover of his book.

Tim Flannery is a scientist, and the scope and erudition of his book reflects that. Bill Bryson’s endorsement on the cover says it all and is no exaggeration: “It would be hard to imagine a better or more important book.” But reading Flannery’s book, I see the problem that the scientific community is faced with, when it comes to communicating a message. This book is largely aimed at people like myself, who read scientific magazines like New Scientist and Scientific American, and who like to immerse themselves in the scientific challenges of the day, but vicariously, without having to do the research or know all the esotericism of the subject. But most people, and this includes politicians, really aren’t that interested, despite the fact that writers as good as Flannery can engage readers outside of academia. What most people want, politicians included - some may say, politicians especially - is a neat one-liner that summarises the entire subject into a sound-bite. Of course, as soon as you give them this, all the data and all the arguments and all the research is left behind, and then every armchair-critic in the world can challenge its veracity.

Flannery faces this dilemma himself, because I’ve seen him defend his position in the media when he’s been misquoted or misrepresented for his honesty. Evolutionary biologists face exactly the same problem when they have to defend their honesty: that we don’t know all the answers to all of nature’s mysteries. But I get angry when politicians really believe that they know more than the scientists, or look for scientists on the fringe who will support their position. We’ve seen this with the tobacco industry, the cosmetics industry, the invitro-fertilisation industry, and, now, with climate-change, the fossil-fuel industry.

Science is different to any other discipline or endeavour. It’s highly dependent on data, research and the work of diverse groups over long periods of time. It suffers from its own rigor for reporting truth. Scientists need to be conservative when extrapolating or speculating into the future, which, in the case of climate-change, is an imperative. This leads them open to challenges by anyone who is a doubter or believes their immediate interests are in jeopardy. In the case of climate-change, this includes the entire Western and Developing world. Economically, entire nation states are in jeopardy, but, so is the very planet if climate-change is a reality. Risk evasion or risk management has never been circumvented by doing nothing. In many cases - the recent economic crisis being a case in point - doing nothing is often the greatest risk of all. Unless people take that into account, they are not practicing risk-management, they are practicing ignorance and denial.

Flannery’s book covers all the bases. He covers the entire living history of the planet throughout all the geological ages, which puts our current, most recent age in perspective. He explains how evidence from ice cores, fossils and other geological and biological sources from all over the world, comprehensively build a picture that is compelling and believable. Flannery provides the science behind Al Gore’s film, An Inconvenient Truth, and has the advantage, as a book, of being able to expound in detail all of his arguments, providing sources and revealing the evidence that has been accumulating over decades. One of the book’s strength is that Flannery demonstrates how climate-science is not a new invention arising from a perceived threat, but goes back at least half a century to Milutin Milankovich’s Canon of Insolation of the Ice-Age Problem published in 1941, describing, for the first time, the relationship between the ice ages and the Earth’s inherent precession on its axis (wobble). The apparent relationship between sunspot activity and the Earth’s temperature goes back centuries. Flannery also explains the relationship between climate change and the world’s great extinction events, including the near-loss of our own species “around 100,000 years ago when humans were as rare as gorillas are today.” Few people know that we nearly didn’t make it to the end of our current evolutionary branch – a very sobering thought indeed.

Flannery’s expertise is in zoology, and it’s his detailed exposition on the impact of climate-change on ecosystems, especially in both polar regions (where the impacts are different yet equally catastrophic) and in coral reefs, that I found most compelling and most depressing. Compelling because it’s already evident and depressing because the bulk of humanity is both unaware and uncaring. Yet it will be truly disastrous to the planet if entire food chains disappear in this century, and that’s the alarm bell that Flannery is ringing. At the very least, biodiversity will be decimated and the long term consequences to us is unknown. The fact that we’ve gone from 1 billion to 6 billion in the last century doesn’t bode well for this century, nor the long term health of the planet.

He spends an entire chapter explaining how the extinction rate of frog and toad species in all parts of the world are probably the most accurate harbingers of climate change, and how it’s been happening and being recorded since the 1980s.

But of all the arguments and evidence that Flannery presents, it’s the ‘time-gates’ of 1976 and 1998 that leave one in no doubt that climate-change is already occurring and we are fools to ignore it. By time-gates he’s referring to events that have become permanent and will not switch back to previous norms. In other words the norms for global climate have already changed. Obviously, it’s future time-gates that we are now attempting to avoid. All of our policies should be based on working backwards from predicted time-gates, and this is the hardest argument to sell. But if more people (politicians in particular) recognised the time-gates we’ve already passed through in the last 3 decades, one would expect the argument to be a very soft sell indeed.

The 1976 ‘climate gate’ relates to the well know El-Nino effect, and data collected in the central Pacific.

“Between 1945 and 1955 the temperature of the surface of the tropical Pacific commonly dipped below 19.2C, but after the magic gate opened in 1976 it has rarely been below 25.3C.”

The El-Nino La-Nina cycles have since become longer: “one would expect such long cycles only once in several thousand years”.

“The 1998 magic gate is also tied up with the El-Nino La-Nino cycle, a two to eight-year-long cycle that brings extreme climate events to much of the world.”

“The 1997-98 El-Nino year has been immortalised by the World Wide Fund for Nature (now the WWF) as ‘the year the world caught fire’.”

In Australia, we have witnessed the effects of this first-hand. As Flannery once wrote in New Scientist (approximately 2 years ago) Australia is witnessing climate-change in advance of the rest of the world. Despite this, our conservative opposition party is literally split down the middle between climate-change-deniers and climate-change-proponents. As recently as last week, this split resulted in a leadership change with the sceptics now in the ascendant.

But, according to his book, it’s Africa that has possibly suffered most from climate change to date, especially, what he calls ‘the Sahelian catastrophe’ in the Dafur region of Western Sudan. And this goes back 4 decades to the 1960s, when Western governments and Western media believed it was all a problem of the local inhabitants’ own making. Flannery argues that, with hindsight and climatology research, it’s Western induced greenhouse gases that have created the Sahelian catastrophe as early as the 1960s.

“The Sahelian climate shift is emblematic of the situation faced by the world as a whole, for in it we see the West focusing on religion and politics as the problem, rather then the well-documented and evident environmental catastrophe that is its ultimate cause.”

Computer modeling for the future has created the most controversy as I alluded to in my introduction, but one of the factors, that few climatologists disagree on, is that there is a residual effect of 5 decades from CO2. In other words, the full effects of the current status of CO2 in the atmosphere will not be experienced until 2050. This is why climatologists are arguing for immediate action. No one expects us to cut our emissions to zero, yet we can’t remove what we’ve already put in, and we have to wait another 2 generations before the full effects of current levels are known in reality. It’s even more serious when one realises that: “half of the energy generated since the Industrial Revolution has been consumed in just the last twenty years.” Business as usual is not a morally responsible option. I don’t expect corporations to be morally responsible, because they’re not – one only has to look at the way they behave in third world countries – but I expect governments to be.

I imagine a lot of people would avoid this book because it makes depressing reading, but, for a start, it should be compulsory reading for all politicians. Flannery discusses 3 possible tipping points, all of which have occurred in the past: the shutting down of the Gulf Stream, the collapse of the Amazon rain forest and the release of methane from the ocean floor, which created the greatest mass extinction ever, 55 million years ago, when an estimated 90% of the planet’s species (that’s species not individual plants and animals) became extinct. Of the three, the last is the most unlikely, and the other two would take the rest of this century to become fully evident, yet, once started, possibly in half that time, they could not be reversed.

But it would take less extreme events to create shortages of food, water and energy, which are already being predicted. Flannery discusses this and the logical outcome is genocidal warfare, because that’s what humans do. This is the scenario that we should all be trying to avoid, yet we don’t even contemplate it, let alone imagine the consequences. It’s human nature to be optimistic and ignore worst-case-scenarios, but we’ve all seen the results of this thinking (in America alone in recent years) with Hurricane Katrina and the subprime mortgage debacle. Unless we consider worst-case-scenarios they will overtake us and cause calamity. In the case of climate-change, this will occur on an unprecedented global scale in human-recorded history.

In the last 2 sections of the book, Flannery talks about solutions, both current and future. He starts with a discussion of the Kyoto protocol; to date, a complete failure compared to the Montreal protocol for the banning of CFCs that saved us from ozone depletion. He concentrates on Australia, partly because its his home and partly because, like the US, it refused to ratify Kyoto and produced spurious arguments to defend its position.

However: “…documents came to light under Australia’s Freedom of Information Act revealing how it [MEGABARE, Australia’s economic model for negotiation] had been funded, to the tune of $400,000, by the Australian Aluminium Council, Rio Tinto, Mobil and other like-minded groups, all of whom had received a seat on the study’s steering committee.” In other words, our position had been determined by representatives in the energy industry rather than climate scientists, even though CSIRO (Australia’s esteemed scientific research establishment) had done considerable research in this area, especially considering Australia’s extensive history of droughts, fires and floods.

Not surprisingly, however, Flannery saves his most scathing criticism for the United States, in particular, the role of the second Bush administration and the energy industries:

“The fact that in the 1970s the US was a world leader and innovator in energy conversation, photovoltaics and wind technology, yet today is a simple follower is testimony to their success [the energy industries]. It is impossible to overestimate the role these industries have played over the last two decades in preventing the world from taking serious action to combat climate change.’

Flannery meticulously documents the role of coal companies, in particular, both in America and Australia, in fighting and funding propaganda warfare against climate-change policy. In both countries, members of the industry were given prominent positions in energy sector reviews, effectively censoring genuine scientific debate. He also cites the ‘Global Climate Coalition’, whose stated purpose was to ‘cast doubt on the theory of global warming’. After 11 years of lobbying, it eventually broke up in 2000 because major players like DuPont and BP realised that they were on the wrong side of the debate and left it in 1997, causing others to follow.

In Australia, a conservative politician recently stated publicly that it was all a ‘hoax’, tacitly referring to a well-circulated conspiracy theory, very popular with climate-change-deniers in this country, that academics, the world over, have created climate-change, or exaggerated its potential impact, for no other reason than to maintain their funding and their careers. This is the most cynical of arguments, but it has a lot of currency amongst the most ignorant and intransigent of my country’s politicians.

On the other hand, Flannery cites the UK as a leader in climate-change reform, going back to the Thatcher years, thanks largely, to the lobbying and influence of James Lovelock.

Flannery is critical of carbon geosequestration, seeing it as a waste of public money to allow the coal industry to continue for another 50 years, when the money could be better spent on other alternatives. Most experts agree that coal is the biggest danger to climate change, yet many countries, including Australia and China, are committed to its continued use for economic reasons.

Flannery discusses all the alternatives, including hydrogen cells and nuclear power but plumps for wind and solar, even arguing a case for self-sufficiency independent of the grid. He also believes that geothermal has been under-explored, especially in Australia, where he contends it could provide all our needs for the next 75 years, carbon free.

Flannery leaves the reader in no doubt that climate-change is already happening. The sceptics argue, considering the extreme climate variations in the geological past, that the real question is whether the current climate change is human-induced or natural. But the correlation between the industrial revolution and consequential global changes in the past century, especially with the 2 significant ‘climate-gate’ changes in the last 3 decades, is compelling evidence.

But if there is any lingering doubt, Flannery added the following postscript to his book:

“As this book was going to press the journal Science published proof positive of global warming. A study by James Hansen and colleagues revealed that Earth is now absorbing more energy, an extra 0.85 watts per square metre, than it’s radiating to space.”

As for the sceptics, it’s an over-eager optimism combined with a reluctance to face a global economic challenge that motivates their opposition. It’s not a coincidence that it’s the political conservatives, in all nations, who are questioning the science. It’s the conservatives who want to maintain the status quo, who believe that change is inherently unwise, yet fail to appreciate that we could well create change on a biblical scale, in this very century, just by doing nothing at all.

Flannery has filled his book with quotations from people as diverse as James Lovelock, William Shakespeare and indigenous people like Aboriginal Elder, Big Bill Neidjie, Gagadju Man. But I thought the best and most relevant quote was from Alfred Russel Wallace, who concurrently discovered the law of natural selection (yes, it’s a law, not a theory) with Charles Darwin.

It is among those nations that claim to be the most civilised, those that profess to be guided by a knowledge of laws of nature, those that most glory in the advance of science, that we find the greatest apathy, the greatest recklessness, in continually rendering impure this all-important necessity of life… (from Man’s Place in the Universe, 1903).

Friday, 20 November 2009

Science, Philosophy, Religion

In a not-so-recent discussion I had on Stephen Law’s blog, I had trouble convincing some of the participants that, not only is there a difference between science and philosophy, but the distinction is an important one.

In a comment on my last post, Timmo made a reference to Richard Feynman’s book, The Character of Physical Law, which got me re-reading it. You may wonder how these 2 issues are related. Well, in my last post I discussed some of Erwin Schrodinger’s philosophy, and the aforementioned Feynman’s book is probably his most philosophical. Together, they highlight the fact that Feynman’s philosophical musings probably couldn’t be more different than Schrodinger’s, yet I doubt that they would disagree on the science. The same is true of contemporary physicists. For example, Roger Penrose and Stephen Hawking, even though they have collaborated scientifically and even won a joint prize in physics, are philosophically miles apart on the nature of mind. In his book, Shadows of the Mind, Penrose actually invited Hawking to provide a counter-philosophical point of view, which, of course, he did. Likewise, Albert Einstein and Kurt Godel were very good friends, when they were both fellows at the Princeton Institute for Advanced Study, but held philosophically divergent views: Godel was a mathematical Platonist and Einstein was not; yet I’m sure they didn’t disagree on the mathematics of each other’s theories.

As a general rule, philosophy deals with questions, the answers for which are not certain, and in many cases, may never be; whereas science deals with questions, where the answers will decide the ultimate truth, and the limits of truth, for a particular theory. Bertrand Russell made the observation that, in philosophy, there may be no right or wrong answers, but the questions, when addressed in the right spirit, are the bulwark against dogmatism and the conservative resistance we find to genuine questing for knowledge. A corollary to this approach is to beware of those who claim they have answers of certainty to questions of profundity.

You may wonder where religion fits into all this. Well, religion is philosophy taken to the metaphysical extreme, but is often confounded by politics to the extent that some people don’t delineate one from the other. In fact, religion is often confounded with ideology, because, for many people, religion and ideology are unassailable truths. But truth is arguably the most elusive concept in the human world, and in this context is an abuse.

I have 2 ways of defining science. Firstly, a general definition is that science is the study of the natural world in all its manifestations. So this leaves out many aspects of knowledge that are human-based, or what is generically called the humanities: all the arts, and topics like ethics and justice. Arguably, psychology crosses the boundary, and I discussed this briefly in another post, Is psychology a science? (Nov. 08). But the topic of ‘mind’, that was raised by Schrodinger, certainly falls into a category where science, psychology and philosophy all merge, but I don’t want to get too far off the track, so I will return to ‘mind’ later. Interestingly, philosophy is generally considered a humanities subject.

The other definition, which is effectively a working definition, is that science is a dialectic between theory and experimentation or observation. Questions that can’t be answered by experimental analysis generally remain philosophical until they can. An example is AI (artificial intelligence). Will AI ever be sentient? Providing we can agree on a definition of sentience, this question will probably one day be resolved. Until that day, it will remain a philosophical question. But there are other philosophical questions that may never be decided by science. An example is the so-called multiverse (multiple universes) theory. If they exist, we may never find any evidence of them, though one should be careful of never saying never. Metaphysical questions like: does the universe have a purpose? (See my post on this topic, Oct. 07) is an example of a subtly different nature. This is a question that science can’t answer, although almost anyone who gives an answer, one way or the other, uses their scientific knowledge to support it. And this is why the distinction is important. Using science to support a philosophical point of view doesn’t turn philosophy into science, though many people, when lost in their own rhetoric, may infer that it does, whether intentionally or not.

On the subject of the dialectic in science, Feynman, in his book, The Character of Physical Law, gives excellent examples, whilst discussing the evolution of the Universal Theory of Gravitation: specifically, how astronomical observations forced changes to theory and then confirmed theory. In other words, without experimentation and observation, we would have just continued to bark up the wrong tree.

His opening chapter on The Law of Gravitation, an example of Physical Law provides one of the best expositions of this dialectic, including descriptions of the experiments that Galileo performed to show gravity’s universality on Earth. And how Tycho Brahe’s unprecedented accuracy in tracking planetary motion gave Johannes Kepler the key to his 3 laws, which ultimately led Newton to the Universal Theory of Gravity we have today. Yes, it’s been modified by Einstein, as Feynman explains, but Newton was able to marry Kepler’s laws to his calculus that not only clinched the theory but eventually led to predictions of another planet perturbing Neptune’s orbit. The ultimate test of a theory is when it predicts hitherto unobserved events.

String Theory is an example of a theory without the dialectic, so we have innumerable variants of which none can be validated by reality. String Theory is not exactly philosophy either – it’s a mathematical adventure. I would describe it as a mathematical model looking for an experiment to make it a scientifically valid theory. I’m not an expert on the subject, but I provide a review of Peter Woit’s book, Not Even Wrong, in a post I wrote earlier this year (Nature’s Layers of Reality, May 09).

And this leads to the significance of mathematics. No one who discusses physics and philosophy can avoid discussing the role of mathematics, and this includes Feynman. In the edition of Feynman’s book that I have (1992), Paul Davies has written an Introduction. He not only acknowledges Feynman’s influence, unorthodoxy and brilliance as a communicator, but relates a dialogue he once had with him on the philosophy of mathematics.

“…Feynman had an abiding suspicion of philosophers. I once had occasion to tackle him about the nature of mathematics… whether abstract mathematical laws could be considered to enjoy an independent Platonic existence. He gave a spirited and skilful description of why this indeed appears so but soon backed off when I pressed him to take a specific philosophical position. He was similarly wary when I attempted to draw him out on the subject of reductionism.”

Feynman devotes an entire chapter (lecture) to the topic, The Relation of Mathematics to Physics, describing it as a language with reasoning, and sees it as an intellectual construct based on axioms. He doesn’t address Godel’s Incompleteness Theorem, because it’s not strictly relevant to his topic: mathematics in physics. He refers to Newton’s calculus as an ‘invention’, whereas Platonists would call it a ‘discovery’.

But more relevant to this discussion is that he describes 3 different ways of looking at the Universal Theory of Gravity, even though they are all mathematically equivalent. One is ‘action at a distance’ or force mediated by the inverse square law; two is by a ‘potential field’; and three is by the ‘least action’ principle, which is Feynman’s personal favourite, and I discuss it in 2 other post ( Nature’s Layers of Reality, May 09 and The Laws of Nature, Mar.08). The point is that these are philosophical interpretations that would determine how a scientist may investigate a phenomenon further. Feynman prefers the ‘least action’ principle because it applies to the refraction of light as well, and therefore suggests a universal principle.

So there is philosophy within science as well as philosophy outside of science, and, once again, I think the distinction is important. Philosophy within science is more likely to be eventually resolved because it generally leads to new avenues of investigation. Feynman says of this: “…every theoretical physicist who is any good knows six or seven different theoretical representations of exactly the same physics.” By ‘exactly the same physics’ he means the mathematics is equivalent (this will become more evident when I discuss quantum mechanics). In other words, it contributes to the dialectic between theory and empirical evidence. Philosophy outside of science is generally removed from the dialectic, which is why it remains philosophy and not science. Philosophy within science remains philosophy until it can evolve into theory. In quantum mechanics (as I discuss below) theory is effectively deadlocked and has been for many decades. At least, that is the impression I get from what I’ve read on the subject by people who know it.

As an aside, the abovementioned quote was once construed by a philosophical writer (Michael Frayn in The Human Touch) as evidence that theoretical physicists effectively make things up because "nature doesn’t have six or seven different ways to represent itself, or even one." But it’s obvious to me that, even though Feynman referred to theories as ‘guesses’ in his usual cavalier manner, he didn’t doubt the validity of nature’s laws. In the cases he’s referring to, the mathematics is solid, but the philosophical interpretations are not (I elaborate on this below).

Elsewhere in the book, Feynman alludes to a view that we will eventually understand all the laws of physics. This is a philosophical position and one I’ve argued against in the past. My reason is history. We never know what we are going to discover and every resolution of a mystery in science has only revealed more mysteries. I find it hard to imagine that this will ever stop, but I also admit that I don’t want it to stop. Feynman, on the other hand, argues that we will eventually run out of finding new laws: either, because of the limit of our ability to reveal them or the limit of their actual existence. He believes that the 20th Century was a golden age of discovery in physics, and no one can deny that. But each age has uncovered new intellectual territory and nature appears far from revealing all its secrets.

On a related note, I quote Feynman in my post, Nature’s Layers of Reality, (cited by Peter Woit, Not Even Wrong) where he is scathing about String Theory. I’m not in a position to judge String Theory, but I don’t think it’s the scientific Holy Grail as some commentators do, and it does reveal how much we still don’t know. String Theory is an example of where people hope to find a ‘Theory of Everything’. It’s one of the reasons I’m a sceptic, but I could be proven wrong.

In previous posts (specifically Quantum Mechanical Philosophy, Jul.09) I describe how the philosophical implications of quantum mechanics are not resolved, yet as a meta-theory, it is arguably the most empirically successful ever. Paul Davies makes exactly the same point in The Goldilocks Enigma. Quantum mechanics demonstrates, more strikingly than any other endeavour, the fundamental differences that lie between science and philosophy. Philosophically, there is the Copenhagen interpretation (Neils Bohr), the Many Worlds interpretation (Hugh Everitt) and the Hidden Variables interpretation (David Bohm). And there are variations amongst these, which I discuss to some extent in the aforementioned post. These are not just different theories; they all have philosophical implications on how we perceive reality. Epistemologically, it can’t get more serious than that.

The Copenhagen interpretation is generally considered to be the conventional interpretation, but as Feynman says in his book: “…I think I can safely say that nobody understands quantum mechanics”. What he means is that no one can explain quantum phenomena in plain language without creating cognitive or logical contradictions. Schrodinger created a thought experiment, popularly known as Schrodinger’s Cat, that encapsulates this conundrum perfectly, where, theoretically, a cat can be dead and alive at the same time. Ironically, Schrodinger also created (he would say discovered) the mathematical equations that have made quantum mechanics the most successful theory ever.

Mathematically, there are no contradictions or conundrums – Schrodinger’s wave mechanical equations and their derivatives, especially the famous Dirac equation, have not only confirmed existing observed phenomena but predicted new ones. Dirac’s equation not only prescribed quantum electron ‘spin’ as an inherent feature of the equation, but predicted the electron’s anti-particle (the positron) and therefore anti-matter. As Feynman says, the best theories, by far, are those where we get more out than what we've put in. More relevant to this discussion, quantum mechanics demonstrates explicitly that science deals in answers and philosophy deals in questions, and sometimes one is not resolved by the other as we might expect.

And now I must come to ‘mind’ because it’s the one topic that really does cross boundaries (including religion). Feynman doesn’t discuss it, because it’s not relevant to his lectures on physics, but Schrodinger did (see previous post), and so does Penrose, who has written 3 books on the subject that I have read. I haven’t read Daniel Dennett’s Consciousness Explained but I’ve read John Searle’s Mind, and it’s the most accessible I’ve found on the subject thus far. I’ve discussed this in previous posts (Subjectivity: The Mind’s I, June 09) and of course in my last post on Schrodinger. I think Schrodinger makes a couple of salient points, which I’ve alluded to previously. In particular, that there is a subjective aspect to consciousness that makes it ontological as well as epistemological. Searle makes this point as well, in his aforementioned book, as does the Dalai Lama in his book, The Universe in a Single Atom.

Schrodinger, in particular, explains how phenomena like light and sound can be measured and analysed by instruments, and we can even analyse how they are transcribed into nerve impulses in our bodies, but all the instruments and analysis in the world can’t describe or explain the actual experience we have of light and sound. This is a contentious point, but people forget that this is what consciousness is, first and foremost: an experience. And if each and every one of us didn’t have this experience, science would no doubt tell us that it doesn’t exist, in the same way that science tells us that free will doesn’t exist. It is still the greatest enigma in the universe, and is likely to remain that way, possibly for ever.

And this leads to Schrodinger’s second salient point: without ‘mind’ the universe would be meaningless. In an earlier post (The Existential God, Sep.09) I reviewed Don Cupitt’s book, Above Us Only Sky, who goes further and says that without language, there would be no meaning and no ‘truth’. I won’t revisit Cupitt, but one should not confuse meaning with reality, nor ontology with epistemology. To quote Einstein: “The most incomprehensible thing about the universe is that it’s comprehensible.” There are various ways one can interpret that statement but mine is: The greatest mystery of the universe is that it created the ability to understand itself. Paul Davies takes this head-on in The Goldilocks Enigma and elaborates on a philosophical premise proposed by John Wheeler. Wheeler effectively argued that the universe exists as the result of a cosmological-scale quantum loop. Because we observe it, it exists. I’m not going to argue one way or the other with Wheeler, but I agree with Schrodinger that without ‘mind’ there is no point to the universe’s existence, and Davies makes a similar point. At the end of The Goldilocks Enigma he summarises all the philosophical viewpoints that are in currency (including ID, the multiverse and the ‘absurd universe’, probably better known as the accidental universe) ending with Wheeler’s, which he calls The self-explaining universe. To quote: “I have suggested that only self-consistent loops capable of understanding themselves can create themselves, so that only universes with (at least the potential for) life and mind really exist.”

In a way I’ve returned to a point I alluded to much earlier: does the universe have a purpose? This is a philosophical question, as I said, but it leads into religion and religious belief. Paul Davies obviously believes it does, and says so, but he’s quick to point out that this does not axiomatically lead to a belief in God. Feynman, whom I’m almost certain was an atheist, makes only one reference to God in his book, when he discusses the hierarchical nature of nature. He explains how the laws of physics can have consequences at a higher level that are unforeseeable yet totally necessary for the universe’s existence as we know it. The example he gives is Hoyle’s and Salpeter’s prediction concerning carbon 12, which arises from the unlikely combination of 3 helium atoms creating a specific new energy level that allows the rest of the elements in the periodic table to exist. Feynman doesn’t make anything metaphysical of this, but he makes the point that nature’s laws at one level have consequences at a higher level of existence that are not readily apparent.

He invokes God (metaphorically, as he’s quick to point out) as either the progenitor of the laws or the ultimate end result; at opposite ends of reality. In an uncharacteristically poetic moment, in another part of the book, he says: “Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry.” He’s indirectly invoking the implication in the title of the Dalai Lama’s book on science and religion The Universe in a Single Atom. The laws of nature are the threads and the tapestry is the universe in all its complexity.

There are no objective religious truths, contrary to what fundamentalists tell us, but there are mathematical truths. And the more we learn about the universe, the more mathematics plays a role. Every book I’ve read on nature’s laws illustrates this fundamental premise. Feynman, Einstein and Hawking would suggest that the mathematics is human reason, but others, like Penrose, Schrodinger and Godel, would argue that mathematics is independent of human thought, albeit we only know it through human thought. Pythagoras and Plato might have argued that God exists in the mathematics and Schrodinger might have argued that God is the ultimate unity of mind (refer my last post). Like Feynman’s metaphorical attribution, they represent opposite ends of reality. At the end of the day, God becomes a metaphor and a projection for what we don’t know, whichever end of reality we posit that projection.

Religion is mind’s quest to find meaning in its own existence. If we were to accept that simple premise without the urge to create an edifice of mythology and political ideology around it, maybe we could all accept each other’s religion.

Sunday, 15 November 2009

Schrodinger’s philosophy (absolutely nothing to do with cats)

I mentioned in my last post, a recent acquisition: What is Life? by Erwin Schrodinger. This is a book that I’d heard about on more than a few occasions, so expectations were high, and I can honestly say it doesn’t disappoint. It says a lot that the copy I have is the eighteenth edition published in 2008, and it was originally published in 1944.

It started off as a series of lectures for the Dublin Institute for Advanced Studies at Trinity College, Dublin in February 1943. He then added an epilogue, On Determinism and Free Will. This segues into another essay (book really) called Mind and Matter which is another set of lectures delivered at Trinity College, Cambridge in 1956. The collection is bookended by a one-page Forward written by Roger Penrose in 1991, and Autobiographical Sketches, written as a virtual appendix by Schrodinger himself in November 1960.

Erwin Schrodinger is most famously known for the set of equations that bear his name, formulated in 1925/6 and for which he was awarded a Nobel Prize in 1933. They are the fundamental equations for quantum mechanics, arguably no less important than Einstein’s equations of relativity that I discussed in my previous post.

This collection is essentially a book on philosophy, that starts off by explaining the role of statistics in physics, then the role of quantum mechanics in evolutionary biology, then a philosophical discourse on mind that leads to a discussion on religion and finally epistemology. It’s a slim volume, a little over 150 pages long (leaving aside his autobiographical sketches). Yet I would recommend that all philosophers and students of philosophy should read it. To quote Paul Davies on the back cover:

“In these little books [Schrodinger] set down … most of the great conceptual issues that confront the scientist who would attempt to unravel the mysteries of life. This combined volume should be compulsory reading for all students…”

According to Penrose’s Forward, this book influenced J.B.S. Haldane and Francis Crick. Considering that it’s over 60 years old and was written before the discovery of DNA, it gives a remarkable insight into the role of mutations in evolutionary biology. Not only that, but Schrodinger explains the role of quantum mechanics in creating mutations. But he begins by explaining how virtually all of physics is statistical, giving examples ranging from Brownian motion, to the physics of magnetism, to radioactivity. His salient point is that, in each of these cases, no one can say when an individual element (atom) might change or react, but statistically they all follow strict mathematical rules. This is a mystery that struck me when I studied physics in high school, and here is a Nobel Prize winning physicist confirming what I thought then: it’s a facet of nature that defies our intuitive logic yet it’s been proven in virtually every arena of physics. We can’t predict the outcome of an individual element but we can predict the overall outcome with preternatural accuracy. He also explains the role of scale: the magnitude of numerical atoms or molecules that make up the smallest physical entities, which is what gives statistical power to many of nature’s dynamics.

Schrodinger then addresses the fundamentals that make life unique, including the fact that every cell contains the ‘code’, effectively the ‘blueprint’ that determines every facet of an organism like us. Remember, this is decades before the structure of DNA was discovered, yet Schrodinger explains how ‘isomers’ can create a code analogous to the way Morse code can be created by just dots and dashes, and this code determines how a life form functions, appears and grows.

His book is full of these little treasures – so obvious when he points them out – yet never really contemplated by most of us. One name that keeps appearing throughout this volume is Ludwig Boltzman, whom Schrodinger considered of no less significant to our knowledge of physics than Planck or Einstein. He explains the contribution that Boltzman made to thermodynamics, and entropy in particular, including the simple mathematical equation that encapsulates it. He also explains the role this has on the ‘arrow of time’. Few people appreciate that entropy determines the direction of time in physics, not relativity nor quantum mechanics. This was first pointed out to me by Penrose, in his book The Emperor’s New Mind. But Schrodinger covers it better still (only in the second part on Mind). He explains it by evoking statistical outcomes and the very simple analogy of shuffling a pack of cards. How many times would you need to shuffle a pack to get it in the right order. In effect this is entropy, and it’s like trying to reorganise the molecules of a broken egg to return it to its unbroken state.

On the subject of life and entropy, he addresses the fact that life alone seems to defy the second law of thermodynamics (actually, it doesn’t, otherwise we wouldn’t die). Nevertheless, life has a dynamism unlike non-organic molecular structures that defies our intuition. Schrodinger introduces the term, ‘negative entropy’, to explain how organisms increase the entropy of the environment; effectively the expense they impose for keeping themselves alive, whether they be primates like us, or bees, or trees in a forest.

In his short treatise on Determinism and Free Will, which he writes as an ‘Epilogue’ to the first set of lectures, he ventures, without apology, into the metaphysical, and acknowledges an influence by Aldous Huxley, specifically his The Perennial Philosophy. Early in this essay he says: “…I wish to emphasize that in my opinion , and contrary to the opinion upheld in some quarters, quantum indeterminancy plays no biological relevant role … except perhaps by enhancing their purely accidental character in such events as meiosis, natural and X-ray-induced mutation and so on…”

I find this a strange declaration, since he had just elaborated at length on the role of quantum mechanics in mutations, which are the causal factors in evolutionary biology, with natural selection being nature’s scythe so to speak. In the previous passage to the one quoted, Schrodinger emphasises that quantum mechanics is ‘statistico-deterministic’, which means that determinism is not completely eliminated by quantum phenomena as many people seem to believe. However, mutations are purely chance and, importantly, rare events, which Schrodinger explains in detail in the body of his lectures, so biological evolution is far from deterministic at its root cause.

It’s impossible in the space I’ve allotted myself here, to do justice to Schrodinger’s book, but whilst it’s full of gems from the opening pages, it’s towards the end that it becomes truly philosophical. Schrodinger tackles the problem of mind in a way that one rarely finds. For a start, he points out that we tend to ignore ‘the elephant in the room’, though, of course, he doesn’t use that phrase, whereby it’s only through mind that the universe has any meaning at all. And that, when we examine the universe - exactly in the way he has throughout the book - we effectively pretend that mind is not part of it. I’ve attempted to address this myself in a previous post on Subjectivity: The Mind's I (June 09). Schrodinger uses the term ‘objectivation’ which he’s obviously coined himself to highlight this point. He alludes to religion (specifically the Eastern religion of the Upanishads) by postulating that there is ‘one mind’ not many, without which the universe would not exist, not because it requires a God to create it, but because, there would be no reason for it to exist without mind. I may not be doing him justice here, so I would beg you read his words yourself, but that’s how I interpret him. I can actually see his point, and I’ve made similar arguments myself: without consciousness there is no point to the universe at all. I need to point out, by the way, that Schrodinger rejected orthodox religion early in his life, and he makes almost no reference to God, except, at one point, to acknowledge: "when God is experienced.. he must be missing in the space-time picture" just like our minds are.

He talks at length about 3 philosophers he considers significant: Plato, Kant and Einstein, all relating to epistemology. In regard to Plato, he gives easy-to-follow examples in both geometry and arithmetic to demonstrate “...true relations 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.”

Then he proceeds to Kant, giving one of the best accounts I’ve read concerning Kant’s controversial views on space and time, which leads to the discussion on the ‘arrow of time’ (and Boltzman’s resolution that I referenced earlier). “He [Kant] would show plainly that space was necessarily infinite and believed firmly that it was in the nature of the human mind to endow it with the geometrical properties summarized by Euclid.”

This leads to Einstein who revealed that space and time are not independent as Kant thought, and is not Euclidean either. However, Schrodinger makes the following point: “Einstein has not – as you sometimes hear – given the lie to Kant’s deep thoughts on the idealization of space and time; he has, on the contrary, made a step towards its accomplishment.”

I’m not sure I agree with Schrodinger on this particular point. Space and time do exist outside the human mind – in fact, space-time is arguably the very fabric of the universe – which, on the surface, does put the lie to Kant’s interpretation as I’ve read it in his Critique of Pure Reason. Having said that, Schrodinger does argue that it is only mind that sees time as past, present and future, and that is an insight that is undeniable as it is obvious. It brings us back to the question: what meaning does the universe have without mind? Einstein showed that time is (relativistically) dependent on the observer, and to that extent, one could say relativity theory supports Kant’s contention of time being internal.

Lastly, Schrodinger touches on a point that is at once obvious, yet rarely, if ever, contemplated, which he calls: “The Mystery of the Sensual Qualities”. In particular, he discusses colour and sound, and he discusses both of them in depth, explaining that, whilst they are both frequency-dependent, the means in which we perceive them and they are propagated are entirely different. For example, colours of quite different frequencies can be mixed to produce a new colour of a frequency that is identical to a single colour of the same frequency and we can’t sense the difference. On the other hand, when sounds of different frequencies are mixed, as in music, we have no trouble in delineating them. But that’s not the main point he is making. The main point is that, whilst we can perform experiments with instruments to give ‘objective’ analysis of colours and sounds, we can’t objectively identify the sensing of them – that is entirely a ‘subjective’ affair. I’ve made this point myself in other posts. It’s why I argue that AI will never ‘sense’ colours and sounds like we do. In fact, it’s why I argue that AI will never have ‘mind’. Schrodinger argues this point better than any other author I’ve read. Of course, he makes no mention of AI, even though Turing had already set that ball rolling in Schrodinger’s own time.

In his autobiographical notes, Schrodinger explains how he learnt English (from an Aunt) even before he had learnt to write German. (He also mentions in passing that his mother’s Aunt had ‘six Angora cats’, which is the only reference to cats in the entire volume.) He was conscripted in the first World War, but spent World War 2 in Dublin; in fact, from 1939 through to 1956, for which he considered himself very fortunate. He called it “My Long Exile, but without the bitter association of the word, as it was a wonderful time.”

Schrodinger’s transcribed lectures are provocative, erudite and articulate. He makes you think deeply about topics and philosophical issues that are common place yet are fundamentally and inexplicably profound. It is one of the best philosophical books I’ve read and I’m surprised it’s not a prescribed text, though perhaps it is in some parts of the world. I know I will read it again.

Many people, in fact most, whether they be scientists, philosophers or theologians, will disagree with Schrodinger. But that’s not the point. The point is that he makes you think about issues you believe you have resolved when you almost certainly haven’t.

Sunday, 8 November 2009

Einstein’s Code and Kerr’s Solution

When I started this blog, over 2 years ago now, I never anticipated picking up ‘followers’ and now I feel the need to maintain some sort of standard. For those who do follow this blog, it is obvious that I don’t comment on a regular basis (although I do on other people’s blogs) but that I only write when something especially attracts my attention. It’s becoming increasingly a blog where I want to share rare intellectual discoveries rather than express my opinions, though I do that as well.

Two recent such discoveries, are Cracking the Einstein Code by Fulvio Melia and What is Life by Erwin Schrodinger. The second book is a classic that I’ve wanted to read for a long time, while the first was an unexpected discovery. This post will focus on Melia’s book, subtitled, Relativity and the Birth of Black Hole Physics, and Schrodinger’s tome will probably be a subject for a future post.

Melia’s book is largely concerned with a little-known aspect of Einstein’s General Theory of Relativity (yes, it deserves capitals): a Kiwi called Roy Kerr, in 1963, unlocked the code inherent in Einstein’s 6 field equations that gave a description of space-time for a rotating body, which is the normal reality for massive bodies in the universe, from planets like ours to entire galaxies.

Now I need to say at the start, that whilst I write on esoteric topics, my knowledge is limited in the extreme, unlike the authors whom I read. Anyone following recent comments on this blog will notice that a generous intellect, called Timmo, has made critical comments on 2 of my former posts (Quantum Tunneling, Oct. 09 and Nature's Layers of Reality, May 09). I wish to acknowledge Timmo’s contribution and I welcome someone who really does know what they’re talking about when it comes to physics.

What I especially like about Melia’s account is that he acknowledges all the other people who contributed to the success of relativity theory (specifically, the General Theory), most of whom I’d never heard of. Like many people, I thought that Einstein’s theory had come effectively fully-fledged from his own mind. I wasn’t aware that there was a history of significant contributions from its conception right up to 1963, almost a decade after Einstein’s death.

Firstly, there is David Hilbert (who is extraordinarily famous in mathematics) and who had a correspondence with Einstein and helped him to develop his field equations. In fact, according to Melia, Hilbert actually published the equations on 20 November 1915, 5 days before Einstein, which led to an argument over priority. However, Einstein wrote a letter of reconciliation on 20 December in the same year, which Melia quotes from.

But even Hilbert “could not overcome a serious problem – how to demonstrate that energy is conserved in Einstein’s theory.” From a conceptual point of view, this had always troubled me about relativity, and it wasn’t until I read Feynman’s account in Six Not-So-Easy Pieces, that I believe I understood it. What I didn’t know, before reading Melia’s account, is that a woman, Emmy Noether, who worked with Hilbert at Gottingen University, was the one who resolved this issue by introducing symmetries in connection with conservation laws, specifically conservation of momentum and energy. To paraphrase Melia, Newton’s second law relates changes in momentum to a force; Noether’s Theorem shows how a change in our frame of reference and Newton’s second law are effectively the same thing. (Different frames of reference refer to different observers moving about at different velocities – with no absolute frame of reference, conservation of energy and momentum becomes an issue.)

Einstein’s General Theory of Relativity is premised on the ‘Principle of Equivalence’. Standing in a stationary elevator car in earth’s gravity is equivalent to being accelerated in an elevator car ‘vertically’ in gravity-free space (vertical, in this context, means being pulled from above our heads so our feet are pressed against the floor of the car). Gravity is felt as a force, by us on earth, only because we are stopped from falling. In free fall, no one feels a force being exerted on them, whether they are in a space ship orbiting the earth or jumping off a cliff. This is the key conceptual point to grasp about Einstein’s theory of gravity (which is the General Theory of Relativity). In free fall there is no force, even though this is counter-intuitive when you are earth-bound, because we rarely experience free-fall for any meaningful period of time without dying.

I don’t claim to understand Noether’s Theorem, but I understand its significance. 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). Melia quotes physicists, Leon M. Lederman and Christopher T. Hill, from their book, Symmetry and the Beautiful Universe: “..certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics…” And I had never even heard of her.

Likewise, I’d never heard of Roy Kerr before reading Melia’s book, yet his contribution to relativistic physics is arguably no less significant. According to Melia, Kerr’s Theorem is the fundamental methodology used to investigate black holes (theoretically) to this day.

Kerr completed his undergraduate course at Canterbury University in Christchurch NZ (enrolling at the age of 16 and going straight into 3rd year mathematics). Canterbury is also where Ernest Rutherford started his academic career (Rutherford uncovered the secrets of the atom: that it was mostly empty space, amongst other things). Kerr then went on to Cambridge to study pure mathematics. His doctoral thesis supervisor was Professor Philip Hall, “one of the century’s greatest mathematicians”, according to Melia, and “Britain’s greatest algebraist”. Hall realised that Kerr’s abilities were singularly impressive but his knowledge incomplete. He set him 3 problems in ‘group theory’, including the ‘Axiom of Choice’, which is a fundamental component of ‘set theory’. Kerr dealt with this and the second problem with relative ease, but the third problem, called the ‘Burnside Conjecture’ was beyond him. Following his admission of defeat, Hall apparently lectured him on the subject for an hour but didn’t tell him that the problem had never been solved. In fact, a decade later, someone managed to prove that the conjecture was false by counterexample.

Unaware of this (at the time), Kerr decided that pure mathematics wasn’t his forte and so went into applied mathematics instead, specifically relativistic physics. It is well known (amongst people who take an interest in physics) that Karl Schwarzschild was the first to provide a solution to Einstein’s field equations for the simplest, idealised scenario of a completely symmetrical sphere in a vacuum. He was a Professor of Potsdam University but formulated his solution whilst serving on the Russian front in WW1. He became ill soon after and died after returning home, but his name remains forever associated with black holes, which are a natural theoretical consequence of his solution.

Kerr’s solution (known as Kerr’s Theorem) was not realised until 1963 when he was at the University of Texas, Austin, which had major ramifications for relativistic physics, in particular black hole physics, that are still with us today. Kerr’s monumental breakthrough was overshadowed by the discovery of quasars, a source of radio waves of unprecedented energy. In 1963, the Parkes radio telescope (in Australia) was used to employ a method postulated by British astronomer, Cyril Hazard. His method was simple but ingenious: to use the moon eclipsing the radio signal to exactly pinpoint the source in the night sky. This allowed astronomers to locate the ‘light’ source of the radio waves and thus use spectroscopy to determine its distance from us.

Spectroscopy analyses the exact wavelengths of light emitted by a distant star, and from the Doppler shift we are able to determine how fast they retreat from us and thus how far away they are. There is a direct proportional relationship between how fast stars retreat and how far away they are using Hubble’s constant, named after Edwin Hubble who first discovered this phenomenon.

The first quasar, 3C273, was discovered by Maarten Schmidt at the Palomar Observatory in California, but because they were only seen as radio sources, spectroscopic analysis was not possible until a light source could be found to be directly associated with the radio source. Hence Hazard’s brilliant idea, subsequently employed at Parkes, to pinpoint quasar 3C273. And it was Schmidt who did the spectroscopic analysis, revealing that the light was red-shifted by an enormous 16% making it much further away then anyone had imagined.

During the 1960s, Australia was at the forefront of radio astronomy, and I remember in 1966, when satellites first linked up to produce the world’s first global televised transmission, the Beatles sang All You Need is Love to a worldwide audience simultaneously. Australia’s major contribution was to show the furthest known object in the universe being tracked by the Parkes’ radio telescope. This telescope featured in the movie The Dish, an Australian-made comedy starring another Kiwi, Sam Neill, which was a comedic rendition of Parkes’ role in broadcasting the first pictures from the moon landing in 1969. It also played a role in the Apollo 13 rescue mission, being the means to communicate the time to fire its retro rockets which allowed the astronauts to return to earth rather than bouncing off the atmosphere or diving too deep and burning up. The angle of descent was critical to their survival, and the ‘dish’ played a small, and little-known, but crucial role. (Unlike the rest of this exposition, nothing in this paragraph comes from Melia's book)

The curious aspect of Schmidt’s discovery is that, at the very first Texas Symposium on Relativistic Astrophysics in 1963, Kerr gave a 10minute lecture on his Theorem, which was virtually ignored because all the participants were far more interested in the recently discovered quasar. Yet Kerr’s Theorem gave the only relativistic solution to spinning black holes, which is exactly what quasars are.

Melia is meticulous in his coverage of all the people who contributed to our understanding of relativistic physics, both theoretically and experimentally. Not only the well known ones like Karl Schwarzschild, John Wheeler, Roger Penrose and Stephen Hawking, but unknown heroes like Noether and Kerr. He also mentions an Australian, Brandon Carter, who used Kerr’s Theorem to show that a ‘time loop’ (or 'time machine') could theoretically be generated beyond the event horizon of a rotating black hole. (But it only works in a vacuum, which makes it a catch-22 time machine.)

The significance of Kerr’s solution is that every significant physical body we know of in the universe is rotating, so Schwarzschild’s solution would almost certainly never be applicable to reality. Kerr’s solution reveals that there are, in fact, 2 event horizons for a rotating black hole. The event horizon is where the escape velocity from a black hole becomes the speed of light so nothing can escape from it. But a spinning black hole literally drags space-time around with it, which creates an inner and outer event horizon – don’t worry, I don’t understand it either. When a body crosses the first event horizon, the parameters of space and time are reversed: space becomes time-like and time becomes space-like. This is because time freezes at the event horizon for an outside observer and the external time becomes infinite from the inside. Time becomes space-like in that it becomes static and infinite, if I interpret it correctly. When an object crosses the second event horizon they reverse again so that ‘inside’ a spinning black hole, space and time become theoretically normal again. Of course, no one knows how true this really is. The other problem is that these theoretical considerations all assume a vacuum which is not the case if something is actually ‘crossing over’. To this day, there are no solutions to Einstein’s field equations for a non-vacuum – that is, for example, inside an object like the earth or the sun – only for outside in space.

That effectively is the limit of my understanding of this subject, even after reading Melia’s book. The story of Kerr himself is no less interesting. One gets the impression that, despite his obvious talents, he was not cut out for high level academia. He did not publish everything he uncovered, and he was not competitive in the sense that he sought to outdo his rivals at every opportunity, nor was he one for self-promotion. He left America in 1971 to take up a position of Head of Mathematics in the University of Canterbury in Christchurch, New Zealand. After his close friend and associate, Alfred Schild, died in 1977, Kerr virtually stopped visiting the US. He received the Hughes Medal from the Royal Society in 1984, the highest accolade he has received.

Roy Kerr writes his own afterward in Fulvio Melia’s book (they are good friends), in which he talks about the difficulties in attempting to get the advanced education he badly needed in 1950s New Zealand. But after going to Cambridge in the UK and the University of Texas in Austin, he believes he was fortunate in that he never had any mentors, as they would have undoubtedly led him away from the path that led to his groundbreaking discovery.

Sunday, 11 October 2009

Watching Watchmen

Yes, I know it was released over 6 months ago, but I’ve just seen it. I don’t normally review movies on this blog – in fact, I’ve only done it once before: Man on Wire (refer The philosophy of Philippe Petit, Oct. 08), and that’s a completely different kettle of cinematic and philosophical fish.

But Watchmen is such a good movie on so many levels, and it encapsulates so much of the American psyche, especially the not-so-recent paranoia of the Cold War, as well as our universal infatuation with violence. And the cinematic references: Apocalypse Now and Dr. Strangelove being the most obvious; both relevant to the cold war era. I am an outsider, regarding America, so my perspective may be different to those who imbibe and live its culture every day.

In Australia it got great reviews, with one notable exception: Evan Williams panned it for its gross glorification of violence as he saw it. Even the pacifists in this movie seem to thrive on violence. One can’t help but compare it to Sin City, which is an iconic movie for technical reasons rather than story content, not to mention a stellar cast. It was clearly influenced by Quentin Tarantino, whereas Watchmen probably owes more to the Matrix trilogy. But both films explore the moral landscape and both are based on comic books or graphic novels. It seems that this genre is becoming increasingly obsessed with violence, with a particular emphasis on the ‘graphic’ in graphic novel, and this is reflected in the movie renditions of the same.

I think the violence in movies has had one tragic consequence in real life. In the last 5 years, in Australia, there has been an increase in alcohol-fueled street violence with people receiving life-long and life-losing injuries. I think movies portray an unrealistic expectation that you can belt the crap out of someone and only inflict superficial injuries. In the case of super-heroes, the story can justify it, but the violence is part of the entertainment in these movies – it doesn’t really condemn it the way we do in real life. Cinema has replaced the Roman gladiators without the leftover corpses upsetting our sense of fun.

One of the characters in the movie, The Comedian, is quite literally a psychopath, yet he is clearly tolerated by his brethren because he’s on the side of 'good'. He is an allegory for the darker side of the American psyche, in particular, what Dick Cheney referred to as the ‘dark side’ of foreign operations. There is a scene in a bar in Vietnam where The Comedian shoots a pregnant woman, apparently pregnant with his child, but first he delivers a diatribe on why he hates her country, even though he’s supposed to be there to ‘help’ it.

Back in America, he continues the same psychopathic behaviour towards what he sees as America’s internal enemies. The Comedian pretty well encapsulates the contradictions that the rest of us wrestle with when we observe the dialectic in America between extreme right wing and liberal politics that inevitably overflows onto the global stage.

I’ve said in a previous post on Storytelling (Jul.09) that comic books are our equivalent to Greek mythology, and, like all mythology, allegory should be its core ingredient. In this regard, I felt Watchmen doesn’t disappoint, especially with the character, Doctor Manhattan. Named after the Manhattan project (as the movie reveals for those who don’t make the connection) which effectively ended WWII with the construction and deployment of the atomic bomb.

One of the advantages of reviewing a film so long after its release is I don’t feel guilty about giving away the ending. Doctor Manhattan effectively becomes an allegory for God, especially when it’s his ability to destroy on a cataclysmic, even biblical, scale that finally achieves world peace. This is a particularly pessimistic view of humanity, exemplified by the Bible in my view. We are inherently self-destructive by nature and only a fear of a superhuman (therefore supernatural) force can stop us from achieving our genetically determined destiny (in biblical terms, original sin). So, in a way, it’s a cautionary tale – but the moral of the tale in my view is that paranoia is what will lead to our self-annihilation and not Divine vengeance.

There are 2 things that make Watchmen an exceptional movie. Firstly, it’s cinematic rendering is close to perfect. A combination of film noir and graphic realisation that sets the standard above anything else I’ve seen, including The Matrix and Sin City. But it’s the rendering of the characters that really sets this movie above the norm for comic book movies. The romance between Nite Owl and Silk Spectre is completely believable. Only Bryan Singer’s Superman Returns compares in the genre and Singer is a master storyteller. But all the characters, in particular, the deeply, psychically wounded Rorschach, have a psychological depth one doesn’t expect in these movies. Again, I would reference Singer’s original X-Men as one of the few comparable movies in the genre, and of course Heath Ledger’s memorable rendition of The Joker in The Dark Knight.

But it’s as allegory for the American psyche, in all its contradictions, that I feel this movie delivers. It competes with Apocalypse Now and Dr. Strangelove on that level, both of which it unashamedly honours, and that’s the highest praise I can give it.


Oh, and I almost forgot to mention the soundtrack - from Philip Glass to Leonard Cohen to Bob Dylan to Jimi Hendrix - what more could one ask for?

Sunday, 4 October 2009

Quantum Tunneling

In some respects this logically follows on from a post I wrote in July this year, Quantum Mechanical Philosophy, which is one of the more esoteric essays I’ve written on this blog. Hopefully, this essay will be less so, as the source material is well written and aimed at the uninitiated.

But I need to recount the gist of that post to make the relevant connection: specifically, the enigmatic Bell’s Theorem or Bell Inequality. To summarise, Bell’s Theorem arose from a thought experiment created by Einstein in an attempt to prove Bohr’s interpretation of quantum mechanics (the famous ‘Copenhagen Interpretation’) wrong.

The thought experiment was elaborated upon by Podolsky and Rosen, so it became known as the Einstein-Podolsky-Rosen or EPR experiment. It examines the purported ‘action-at-a-distance’ phenomenon predicted by quantum physics for certain traits of particles or photons, which Einstein described, quite accurately, as ‘spooky’. If you have 2 particles with a common origin (could be photons with opposite polarisation or subatomic particles like electrons with opposite ‘spins’), then separate them over any distance whatsoever, you will not know what the spin or polarity, or whatever quantum mechanical trait you are measuring, is, until you take the actual measurement. The ‘spooky’ bit is that as soon as you make the measurement the ‘twin’ particle will instantaneously become the opposite. Before the measurement or observation is made the particles are in, what’s called, a ‘superposition’ of states – it can be either one or the other.

Einstein realised that this conjecture contradicted his special theory of relativity, which states that no signal or means of communication between particles of any kind can travel faster than the speed of light, which had already been confirmed by experiment. John Bell developed a mathematical equation that analysed correlations of hypothetical results from the thought experiment that would categorically prove either Einstein or Bohr wrong.

Alain Aspect developed a real experiment to test Bell’s Inequality (made the thought experiment actually happen) and proved Einstein wrong (long after Einstein had died, by the way).

As I point out in that previous post, the upshot of this is that either faster-than-light actions are possible (called non-locality) or there is no objective reality. Non-locality is self-explanatory (you can’t communicate faster than the speed of light) but no objective reality means that the thing doesn’t exist until someone measures it or takes an observation. I discuss this in more detail (lots of detail) in my previous post, but that’s effectively the Copenhagen interpretation of quantum mechanics: at the subatomic scale, particles don’t exist until they are measured or observed. A less extreme and more popular interpretation is that they remain in a superposition of states until they interact with something else. If you want to delve deeper, read my previous post, but you may be none the wiser. The philosophical implications of this have never been truly resolved.

My conclusion was to accept non-locality (faster-than-light connections) in order to keep objective reality, and I made specific reference to David Bohm’s unpopular interpretation, known as the ‘hidden variables theory’. Bohm believed that there was a hidden set of parameters that govern the particles which we can’t see or detect.

To quote David Deutsch (who doesn’t agree with Bohm at all): ‘A non-local hidden variable theory means, in ordinary language, a theory in which influences propagate across space and time without passing through the space in between.’

And this leads me to quantum tunneling, because that’s exactly what quantum tunneling does, only it happens over short distances, not the distances used in the EPR experiment, which could theoretically include the other side of the universe.

I’ve just read an excellent book on this subject, Zero Time Space subtitled, How Quantum Tunneling Broke the Light Speed Barrier, authored by Gunter Nimtz and Astrid Haibel. Originally published in German in 2004, it was published in English in 2008. This book could be read by people with only a rudimentary knowledge of physics, as it contains only a few simple equations, among them Planck’s equation: E = hf where E is energy, f is frequency of a ‘wave’ and h is Planck’s constant, 6.6 x 10-34 Js (Joules seconds). The authors also include Snell’s law of refraction and the universal wave equation of wavelength times frequency equals velocity (I can’t find the symbol, lambda, for wavelength, in my arsenal of fonts). One of the annoyances is that there is a type-setting error in this particular equation (in the book). If someone is going to include equations, especially for people unfamiliar with them, I wish they could at least get them checked during type-setting. The same applies to Richard Feynman’s excellent book on relativity theory, Six Not-So-Easy Pieces where I found 3 type-setting errors amongst the equations scattered throughout the book. In both cases the books are aimed at people who are not familiar with the material, which means they won’t know the errors are there.

Putting that one (some may say petty) criticism aside, it’s a very good book on quantum mechanics for people who know very little about physics. It includes a short history of physics leading up to Einstein’s theories of relativity (with particular reference to the Special Theory) as well as quantum mechanics. They do this because the whole point of the book is to highlight how quantum tunneling breaks Einstein’s special theory of relativity, and therefore reinforces non-locality, as I described in my previous post. So the authors go to some pains to give the reader an overview of both Einstein’s theory and quantum mechanics, in conjunction with the historical context. It’s very well done.

Nimtz and Haibel, by the way, make no reference to Bell’s Theorem, as it would probably confuse readers who are unfamiliar with it – I hope I haven’t put people off by referencing it here. Having said that, they do discuss ‘entanglement’ towards the end of their book, which is the state I described above concerning ‘twin’ particles interacting at a distance. In particular, they raise this phenomenon in their lengthy discussion on causality, as they are at pains to explain that ‘tunneling’ does not affect causality as some people might be led to believe. Even so, it still manages to confound common sense, as I explicate below.

In the forward to the book, they briefly discuss the ‘myth… about the half-life of knowledge… It suggests that our knowledge is being declared invalid every five years by new knowledge.’ They then go on to dispel the most common representation of that myth: Newton’s theory of gravitation is still valid, even in the light of the theory of relativity… Einstein’s theory has extended theory rather than disproved Newton’s theory.’

I made the same point in my essay on The Laws of Nature (March 08), explaining that Einstein’s equations reduce to Newton’s when certain parameters become negligible. The authors raise this point, because, whilst quantum tunneling appears to contradict Einstein’s special theory of relativity, in their own words: ‘Einstein deals with free space, whereas tunneling is not free space.’ In other words there are constraints on relativity theory in the same way that there are constraints on Newton’s theory, but there are various aspects of nature where one is more significant than the other. It’s one of the reasons that I’m a bit sceptical about a grand unified theory (GUT), a meta-theory of everything. Many people would love to prove me wrong, and a part of me would like to see that, but another part wouldn’t because I don’t believe there will ever be an end to physics.

During this discussion they make another statement, relevant to the stability of scientific knowledge: ‘Mathematical proof has been regarded since Pythagoras and Plato as eternal, metaphysical truth.’ A statement I would agree with. For example, Reimann geometry hasn’t displaced Euclidean geometry, it has just extended our knowledge, both of the mathematical world and the physical world (through Einstein’s theory of General Relativity).

I’ve discussed on other posts, the relationship between mathematics and the natural world (refer The unreasonable effectiveness of mathematics, March 09), but no where is that more significant than in quantum mechanics. QED (Quantum Electrodynamics), for which Richard Feynman, Julian Schwinger and Sin-Itoro Tomonaga jointly won the Nobel Prize, is the most successful theory of all time. Without mathematics, quantum mechanics would be indecipherable, quite literally. Intriguingly, there are imaginary numbers in quantum theory that are completely relevant to quantum tunneling. Without imaginary numbers (created by the square route of -1, called i) quantum mechanics would never have been articulated as a meaningful theory at all.

As Nimtz and Haibel point out, it is the imaginary component of the equation that does the tunneling. When this was first derived, people just assumed that these imaginary components were unnecessary remnants of the mathematics, but that’s not the case. When tunneling occurs there is an interface where part of the signal is reflected and part is transmitted through ‘the tunnel’. The part that is reflected is mathematically ‘real’ and the part that is transmitted is mathematically ‘imaginary’. (I've since been informed this is not correct - refer Addendum 2 below.) A tunnel, by the way, is a barrier, where the particle or wave theoretically can’t travel, because it doesn’t have enough energy. The authors point out that it even occurs in the sun, otherwise the fusion, which gives us sunlight, would never occur. I should add that quantum tunneling is a feature of all transistor devices. In fact, it's the very feature that makes transistors work (called 'tunnel diode' by Nimtz and Haibel).

Both of the authors have performed experiments, to not only detect quantum tunneling, but to also measure the time elapsed. As predicted by Thomas Hartman in 1962, there is a time elapse at the ‘entrance’ to the tunnel, or the ‘interface’, between the medium and ‘the tunnel’, but the actual time spent in the tunnel is zero. This is called the Hartman effect. To quote the authors: ‘So the wave packet spreads in the tunnel in zero time and is everywhere from the entrance to the exit. This non-local phenomenon makes one feel eery.’ An understatement, if I’ve ever read one.

One of the authors, Gunter Nimtz, participated in an experiment that tunneled Mozart’s symphony in g-minor through a waveguide at superluminal speed: 4.7 times the speed of light. The elapsed time occurred at the entrance to the tunnel, as predicted by Hartman, not in the tunnel itself. In an exposition, that I will not try to repeat here, the authors explain how this quirk of nature (the elapsed time at the entrance to the tunnel) allows superluminal communication without impacting causality. The speed in the tunnel is infinite – as the Americans like to say: go figure. The title of the book, Zero Time Space, is therefore entirely appropriate.

They end the book with a brief description of wormholes and hypothetical warp drives, beloved of Sci-Fi writers, like me, that require exotic negative gravity amongst other improbabilities.

Of all the incredible manifestations of the universe, only consciousness is arguably more inexplicable or more mysterious (but no more weird) than quantum phenomena. If we didn’t observe it, no one would believe it. And if we didn’t have the mathematics to describe it, no one would be able to fathom it, even remotely.


Addendum 1: I came across this - it's very entertaining as well as informative.

Addendum 2: I would like to acknowledge Timmo (refer comments thread below) who has valiantly tried to correct all my mistakes. In particular, that the imaginary component of Schrodinger's equation plays no greater role in tunneling than the real component, if I understand Timmo correctly. Also he points out that tunneling and non-locality are independent phenomena, and possibly I misled people on that point.

He also corrects some faux pas I made concerning the Lorenz transformation and Godel's Incompleteness Theorem in response to comments I've made since the post was posted.


I confess I don't know as much as I appear to, and I wish I understood more than I actually do.

And I would like to thank Timmo for reminding me of how much I don't know.