Sixty-three year-old Avraham Trakhtman has solved one of the current generation's toughest mathematical problems — the 38 year-old road colouring problem. The solution will shortly be published in the Israel Journal of Mathematics.
Born: 12 March 1685 in Kilkenny, County Kilkenny, Ireland
Died: 14 Jan 1753 in Oxford, England
George Berkeley was an Irish bishop and philosopher who is best known for his attacks on the logical foundations of the calculus as developed by Newton.
Berkeley started at Trinity College, Dublin at the tender age of 14 and graduated with a masters when only 18 years old. His attacks on the foundations of calculus were first aired in 1734 when he published The analyst: A discourse addressed to an infidel mathematician. The infidel mathematician is believed to have been either Edmond Halley or Isaac
Newton. He argued that although the calculus led to true results, its foundations were no more secure than those that underpin religion. He stated that the calculus involved a logical fallacy and described derivatives thus:
"And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?"
In modern language, this could be read as:
"What are these 'instantaneous' rates of change? The ratios of vanishing increments? And what are these 'vanishing' increments? They are neither finite quantities nor 'infinitesimal' quantities, nor yet nothing. May we not call them the ghosts of departed quantities?"
His interesting theory as to why the calculus actually worked was that it was the result of two compensating errors.
As a consequence of the controversy surround Berkeley's publication, the foundations of calculus were rewritten in a much more formal and rigorous manner using limits. It was not until 1966, with the publication of Abraham Robinson's book Non-standard Analysis, that the concept of the infinitesimal was made rigorous. This gave an alternative way of
overcoming the difficulties that Berkeley found in Newton's approach.
Berkeley's influence in mathematics is reflected by the fact that the University of California, Berkeley, and the city of Berkeley that grew up around it, are named after him. Berkeley died of a heart attack on 14 January 1753, sitting with his family listening to his wife reading. He died so peacefully that the event went unnoticed, his family thinking he had fallen asleep. He left
instructions that he was not to be buried for at least five days and was buried at Christ Church, Oxford on 20 January.
I would do anything for maths, but I won't do that
One of the latest group emails that I have received concerns maths and music. Good to see that mathematics has embarked on viral marketing. See if you can pick the songs referenced in the pictures below.
Profesor Rapley's talk was entitled "Great while it lasted; now what?" referring to how humans have exploited carbon-based energy sources for the last century, but now must confront the fact that we are exceeding the safe limits of such use. Professor Rapley is a systems scientist and has studied the Earth mathematically as a complex system through roles as Director of the British Antarctic Survey, Executive Director of the International Geosphere-Biosphere Programme, and Professor of Remote Sensing Science and Associate Director of University College London's Mullard Space Science Laboratory.
Professor Rapley is now Director of the Science Museum in London, and considers that the best way to study the Earth is to bring together "many -ologists — geologists, biologists, sociologists etc." As a complex system, the Earth has many inbuilt checks and balances — the oceans and the biosphere can act as carbon sinks, reducing the amount of
carbon dioxide in the atmosphere. However, Rapley thinks that the Earth will also have a "tipping point" whereby the ocean and biosphere can no longer take-up carbon dioxide. Using recent ice melts in the Antarctic and Greenland as examples, he highlighted the incredibly difficult task in modelling the Earth as a complex system. Describing the Earth as "the most complex system in the Universe",
he said that the Earth system "has many surprises and can't be modelled too much." He added, soberly, that whilst the greenhouse effect was understood in the 1800s by mathematician Joseph Fourier , the Earth now:
"...has no user's manual, and no spares... . Many senior economists act like they don't understand this."
He likened our current situation to that of the next day after a great party. Burning immense amounts of carbon has given humans a much higher quality of life than before the use of carbon fuels, however the carbon dioxide is the "hangover." He attacked sceptics of climate change by saying that the weight of scientific opinion supports the assertion that climate change is real and man-made,
highlighting that 90% of climate change is anthropogenic.
Another interesting, and perhaps unfortunate, point (well, for me anyway) was that meat-eaters contribute far more than vegetarians to climate change. This is because much more energy goes into meat production than vegetarian foods.
Professor Rapley's talk was preceded by an oration by Dr James Lovelock. Dr Lovelock is a systems modeller and is well-known for proposing the Gaia hypothesis — that living and non-living parts of the Earth are a complex interacting system that can be thought of as a single
organism, and that all living things have a regulatory effect on the Earth's environment that promotes life overall. Lovelock and Rapley have recently proposed a radical idea to "Help the Earth heal herself" in a letter to Nature Magazine. The proposal is to use huge pipes to pump lower ocean water, rich in nutrients,
to the surface, stimulating the growth of algae and plankton in the upper ocean. The surface layer of the ocean is increasingly starved of nutrients such as phosphates and nitrates. These nutrients are needed by the algae to grow, but as the surface warms, there is less mixing with the nutrient-rich water layers beneath. The mixing of ocean layers would encourage algal blooms — the algae take up
more carbon dioxide, which is eventually locked up in the tiny shells of plankton when they die and fall to the bottom of the ocean. This proposal essentially assumes that it's too late for carbon emission controls and other alternatives to have any effect on climate change — the damage is already done.
Another interesting session that
Plus attended was on a computer that had been taught using hidden markov models to harmonise melodies in the style of Johann Sebastian Bach. A hidden Markov model is a statistical model in which the system being modelled is assumed to be a Markov process with
unknown parameters, and the hidden parameters need to be determined from the observable parameters. Hidden Markov models are known for their application in pattern recognition such as speech, handwriting, musical score following and bioinformatics.
This week, Plus is attending the Edinburgh International Science Festival — an educational charity aiming to engage society in the value of science and technology, placing a particularly strong emphasis on providing experiences of science that are inspiring and confidence building. The Festival is internationally well-regarded and one of the
biggest science events in the UK. Its timing suggests we are now well into science festival season, with roughly another 10 city-wide events to be held around the country, in addition to the recently run Cambridge and Brighton Science Festivals. In 2007, Plus attended the BA York Science Festival — you can read more on the Plus Blog.
There is a comprehensive range of talks from some of the world's leading scientists, and last evening Plus attended a talk by Chris Bishop, Deputy Director of Microsoft Research in Cambridge, on Machine Learning. Compared to humans,
machines are very bad at recognising some patterns — for instance, recognising that a photo of a cow is actually a cow and not a dog or a shadow or even a spoon — they simply do not innately know these things and must be taught. Whilst humans are very good at understanding messy handwriting — a subtle task — computers must learn. Bishop's work involves developing computers that can learn and so
become smarter in completing their tasks. To read more about the mathematics involved, see the Cambridge team's introduction to applied mathematics for machine learning.
Central to machine learning is the Turing Test. English mathematician Alan Turing — instrumental in breaking the Enigma code during World War II — is widely regarded as one of the founders of modern computer science — see the Plus article What computers can't do. In a 1950 paper, he described what he called the
Imitation game, now known as the Turing Test, in which a person (in a separate room) tries to distinguish between human and computer test subjects by asking them each a series of questions. If the person can't distinguish between the computer and the human, the computer is deemed to be intelligent. There is a (currently unclaimed) prize
of $100,000 for a computer that can pass the Turing Test. Bishop talked about the Forza Motorsport X-box game, in which you race against other players or against the computer. The computer car can be taught to drive like a human — you can create a "Drivatar" which will learn from the way you drive and then drive like you. The word Drivatar comes from
avatar, which in gamer-speak is the player's representation within the game. Participants in trials have said that they cannot tell the difference between a human playing and a computer, and so within a very limited context, the Drivatar passes the Turing Test.
Bishop also talked about other developments in machine learning, from Spam email detection — see the Plus article CAPTCHA if they can — to websearches and modern developments in HIV detection. New software is being developed to look for patterns in the HIV protein code. This code evolves over time, and so searching for
the same string of amino acids each time is likely not to work. He also discussed developments in the way computers play the games Chess and Go. He stated that whilst the IBM Deep Blue chess-playing computer was able to beat world chess champion Garry Kasparov in 1997, it was not because of any particular
advances in machine "intelligence" — the success was largely due to advancements in computer memory and speed that allowed the computer to analyse the game very thoroughly. If at each turn you can make around 30 possible moves, and then there are around 30 possible replies by your opponent, and then in turn another 30 possible moves you can make, and so on, then a human could never hope to
analyse each and every scenario. However, Deep Blue was able to look at the full set of possible moves and then make the best one (although it did have the intelligence to look within a set of smart moves). With the game Go however, at each turn there are around 200 possible moves, meaning that it is impossible currently to completely analyse each scenario before playing a move. Bishop is working
with computers that are learning by playing against humans.
I have also attended Wonderama and the mathematical puzzles at the Museum of Scotland. Wonderama is designed for the whole family and hosts many free activities and events for budding scientists, aspiring surgeons,
wannabe adventurers and trainee archaeologists. Thankfully, my time around science festivals means that I wasn't outdone by 8-year olds in the mathematical puzzle department. Actually, that is not completely true...
Tonight, I am eagerly awaiting the Edinburgh Medal Address by Professor Chris Rapley, Director of the Science Museum, on the dangers of climate change. The Edinburgh Medal is awarded each year to those within science whose professional achievements are judged to have made a significant contribution to the
understanding and well-being of humanity.
For more information and a comprehensive program, see the Festival website. The event runs until April 6.
One of the most important international prizes for mathematics has this year been awarded jointly to two outstanding mathematicians — even though one of them was originally unable to find a publisher for his groundbreaking work.