Paul P. Mealing

Check out my book, ELVENE. Available as e-book and as paperback (print on demand, POD). 2 Reviews: here. Also this promotional Q&A on-line.

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).