David Hilbert (Jan 23 1862 - Feb 14 1943)

David Hilbert
Image by aldoaldoz via Flickr

Today marks the 150th anniversary of the birth of David Hilbert. Hilbert was a German mathematician, recognized as one of the most influential mathematicians of the 19th and 20th centuries. Much of his work was not of direct relevance to A-Level mathematics, but his greatness demands a mention in this blog.

Hilbert worked in many areas of mathematics and developed many fundamental theorems. He also formulated the theory of Hilbert spaces, one of the cornerstones of the emerging "functional analysis". But he is perhaps most famous for his collection of problems that set the course for much of the mathematical research of the 20th century. He presented 10 of these unsolved problems at the Paris conference of the International Congress of Mathematicians, with the complete set of 23 published in 1902 in the Bulletin of the American Mathematical Society.

Many of Hilbert's problems remain unsolved, including the famous Riemann hypothesis. Many mathematicians believe the Riemann hypothesis to be unsolvable, at least given the current confines of mathematical knowledge and methodology.

But all of the problems generated a vast amount of research and investigation throughout the 20th century. They have led to new mathematical techniques and ideas, which may never have come about were it not for the problems posed.

Hilbert's 23 problems at the turn of the 20th century inspired another list of unsolved problems one hundred years later, at the turn of the millennium. The 21st century list of seven Millennium Prize Problems was chosen in 2000 by the Clay Mathematics Institute. Unlike the Hilbert problems, which offered no rewards, the solution of each problem included a million dollar prize. One of the Millennium Prize Problems, the Poincaré conjecture, was solved relatively quickly, with the Russian Grigori Perelman presenting the solution in 2002 and 2003. He famously turned down the prize money and the prestigious Fields Medal that was offered.

Hilbert strived to establish rigor in all mathematics and developed important tools used in modern mathematical physics and he is known as one of the founders of proof theory and mathematical logic.

Modern mathematics would be in a different, poorer place if it were not for the work of David Hilbert, who would have been 150 today.

Sir Francis Galton

Sir Francis Galton lived from 16 February 1822 to 17 January 1911.

Sir Francis Galton
Image from Wikipedia

He was a cousin of Charles Darwin, and like many educated Victorian gentlemen, dabbled in many different areas of learning. Among Galton's interests were anthropology, genetics including eugenics (the idea of improving the human race by genetic selection), exploration, geography, invention, meteorology and statistics. He was knighted in 1909. He actually invented the term eugenics, and is considered the first person to use the expression "nature versus nurture".

Galton's wide range of interests led him to publication of over 340 papers and books during his lifetime. But it is his mathematical contributions, particularly in the field of statistics, that we are interested in. He created the statistical concept of correlation and widely promoted regression towards the mean. He was the first to apply statistical methods to the study of human differences and inheritance of intelligence, and introduced the use of questionnaires and surveys for collecting data on human communities, which he used in his genealogical studies and his analyses of the behaviour of people, anthropometrics.

So Galton's contributions were considerable and, in some cases, revolutionary. But some of his ideas were also highly controversial, even for the time in which he lived. He attempted to draw up a 'Beauty Map' of the British Isles, and for this he classified passing girls into three categories: attractive, indifferent and repulsive, surreptitiously making pin-pricks in paper stored in his pocket as a means to build a database.

Eugenics itself is today considered a brutal concept. The practices involved in favouring certain families deemed genetically superior was practised by some governments during the early years of the 20th century, resulting in deprivation and a loss of human rights for millions.

Galton's book, Hereditary Genius (1869), was one of the first scientific attempts to study genius and greatness. It demonstrates some of the statistical techniques that Galton would develop through his life, but it is steeped in language that is far from acceptable in today's scientific literature, such as "idiots" and "imbeciles".

He founded psychometrics (the science of measuring mental faculties) and devised a method for classifying fingerprints that was later used in forensic science. He also conducted research on the power of prayer, concluding it had none, because of the fact that those prayed for lived no longer than those not prayed for.

Finally Galton was a pioneer of scientific meteorology. He devised an early weather map, proposed a theory for the formation of anticyclones, and was the first to establish a complete record of short-term climatic phenomena on a European scale.

Francis Galton was truly a great British eccentric, whose ideas were often controversial, but whose contributions to statistics and science in general cannot be ignored.

A Brief History of Hawking, 70 Today

Stephen Hawking

Happy birthday to Stephen Hawking, who is 70 today.

Hawking will be known to most readers of this blog, an example of an individual whose extremely challenging disabilities have not prevented a life of incredible achievement. He has motor neurone disease, a condition that has progressed over the years and has now left him completely paralysed.

Born on 8th January 1942, Hawking is a theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity. Among the many awards he has been given throughout his lifetime, he was, in 2009, awarded the Presidential Medal of Freedom, the highest civilian award in the United States.

The diagnosis of motor neurone disease came when Hawking was 21, shortly before his first marriage. Forty-nine years later, he has been almost completely paralysed, but is still producing some of his finest work.

To speak, Hawking uses a computer fitted into his wheelchair. In latter years, he has operated it using a muscle in his cheek, selecting words, and those words are spoken by a synthesised voice. The voice, familiar to all of us, is no longer available as a speech synthesiser, but Hawking continues to use it because he considers it now to be his own.

Hawking was the Lucasian Professor of Mathematics at the University of Cambridge for 30 years. The post has previously been held by Isaac Newton and Charles Babbage, to name just two of the former eminent incumbents. Hawking retired from the post in 2009.

In 2007, to celebrate his 65th birthday, Hawking took a zero-gravity flight, during which he experienced weightlessness eight times. He became the first quadriplegic to float in zero-gravity. This was the first time in forty years that he moved freely, without his wheelchair. His plan is to take a sub-orbital space flight in 2013 on Virgin Galactic's space service.

Stephen Hawking is now Director of Research at the Centre for Theoretical Cosmology in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. He is best known for his contributions in the fields of cosmology and quantum gravity, especially in the context of black holes. Hawking's key scientific works have included theorems regarding gravitational singularities in the framework of general relativity, and the theoretical prediction that black holes should emit radiation, today known as Hawking radiation.

He has also written popular science books, in which he discusses cosmology in general, going a long way to making the subject accessible to the general public; A Brief History of Time was by far his best-selling title, staying on the Sunday Times best-sellers list for a record-breaking 237 weeks.

At the time of the diagnosis of motor neurone disease in 1963, Hawking's doctors gave his life expectancy as a further two or three years. Everybody who meets the man says he is a highly entertaining person, with a great sense of humour and an ability to inspire. Although he has continued to deteriorate slowly, gradually losing the use of his arms, legs, and voice, and now completely paralysed, Stephen Hawking has lived to celebrate his 70th birthday today. The world of science, indeed the world in general, is a better place because of it.

Make a Great Resource and Win a Great Prize

MEI (Mathematics in Education and Industry) and Tarquin Books are joining forces to offer an annual prize to post-16 maths students.

Entrants must design an electronic resource - e.g. Geogebra file, Excel spreadsheet, a video, etc. - that communicates the mathematical ideas needed to solve a practical problem of some kind. These mathematical ideas used should be relevant to A-Level Mathematics or Further Mathematics A-Level, or to some equivalent qualification. Your entry could be designed by you individually, or it could be a collaboration between a group of students.

Entries to the competition could also form the basis for an Extended Project Qualification.

The winner will receive books to the value of £100 and Tarquin Books will also donate £400 worth of books or materials to the winners’ school or college.

To register your interest in the competition, follow the Prizes link on the Tarquin website. The closing date for entries is 30 April 2012 and the winner will be announced by 31 May 2012. For further information email Andrew Griffin at Tarquin.

Caleb Gattegno (1911 - 1988)

Image via Wikipedia

Today marks the 100th anniversary of the birth of Caleb Gattegno. He was an educator, particularly in mathematics, but also in linguistics, publishing many seminal works on the theory of education. But none of this theory was detached from the real world. His ideas were based on practical, real-life situations, and what, according to his observations, helped his students the most. He was also an inventor, creating tools for the classroom that are still used and found invaluable by today's teachers.

Gattegno grew up in Alexandria, then lived in Cairo, London and New York. He worked all over the world and devoted his life to a study of learning, not only creating a number of important innovative techniques for the teaching of languages and of mathematics, but also made a remarkable, seminal contribution to the understanding of the learning process at all ages. In 1952 he founded the organisation that would later become the Association of Teachers of Mathematics, and its magazine Maths Teaching. This year, the ATM is making Gattegno and his work the theme of their annual conference.

Caleb Gattegno made a significant impact on teaching and thinking about education not only in the UK but also in many countries around the world. Within mathematics, apart from the creation of ATM, his work included the promotion and use of Cuisenaire Rods, the creation of geoboards, developments of the animated geometry films of Nicolet, and the Gattegno ‘tens’ chart for number. Cuisenaire Rods were so effective, he thought, that he founded the Cuisenaire company in the UK.

In linguistics, Gattegno pioneered the learning of reading and foreign languages with ‘infused reading’ and ‘the silent way’. He was the first English translator of Piaget, he was influential in spreading awareness of developmental psychology.

In addition he challenged many in the educational world to consider what is involved in learning, encouraging them to allow an explorative approach. The teacher's actions should be subordinated to the way in which the child learns through exploration and investigation. His idea that only awareness is educable breaks the learning process down into four stages. The first, the most important, is a single act of awareness, for example that something is there to be learned or can be explored. Without this realisation, or any notion that there is a challenge, problem or issue to be explored, the pupil will not be able to proceed to the next three stages. These stages can be described as exploration, transition (which begins with the new skill being something that can be achieved with a lot of concentration, and ends with it being automatic) and transfer, in which the new skill can be used and applied to the learning of yet other skills and ideas.

Use of his teaching aids was developed in classrooms, and to demonstrate their power he was prepared to teach children of any age or ability, at any time, in front of other teachers. He ran many seminars through his lifetime and his personal influence was felt profoundly by those who saw him in action.

A-Level Exams Earlier, University Applications Later?

Image by Getty Images via @daylife

UCAS, the body that administers UK university admissions, has put forward proposals for changes to the admissions system. These recommendations include bringing the A-Level examinations forward and completing most of the university application process when the exam results have been released. This system, UCAS argues, would be fairer and less complex.

Currently, pupils in their final year of school must make their UCAS applications by mid-January. Universities judge each application based on predicted A-Level grades, references from teachers, personal statements and possibly an interview. The universities then award conditional offers, dependent on certain A-Level grades being gained.

Many schools, particularly private schools, give university admissions advice that can maximise the chances of successful entry. UCAS argues that this system makes it unfair for pupils who do not have such a system of support available to them. In short, as the Guardian puts it, the current system favours the rich.

An overhaul would lead to a fairer and more transparent applications process, with the actual grades gained being central to a university’s decision.

An application later in the year would also give pupils more time to discover their real interests, which subjects they are excelling in, and would like to spend further time studying. The downside to such a plan would be the timing. A-Level examinations, the marking, awarding of grades, university applications and decision-making would all need to take place in the summer term. In Northern Ireland and Scotland, this problem would be exacerbated because schools break up for their summer holiday earlier than in England and Wales.

The last Labour government attempted to bring in similar reforms of the universities applications process, without success, largely because of opposition from teaching unions. Although many teachers see some benefits to such a scheme, the amount of teaching time for the A-Level examinations would be shortened. There is also a feeling that there would be simply too much to achieve during the summer term.

What do you think? Would you be happy to delay your university application until after you have received your A-Level results? How do other countries manage university admissions? Would a compulsory gap year be one radical solution to the problem (which they once called National Service), giving pupils further time to think about their futures and time to do something useful in the workplace, while ensuring our students are more mature when entering university? Let us know your thoughts.

Evariste Galois (25 Oct 1811 – 31 May 1832)

A sketch of Galois, image via Wikipedia

Of all the mathematicians who ever lived, Évariste Galois, born 200 years ago today, is the one who has given me the most inspiration for the subject.

He was a hot-headed genius, living in a period of political instability in post-revolutionary France. The ideas he formulated, during his incredibly brief life, were not only ground-breaking, they were so ahead of his time they could be described as alien to the 18th century world he lived in. Only now are Galois's ideas becoming of dramatic importance. If you are interested in quantum computing, or particle physics, to name just two of today's hottest scientific topics, you should know about Galois.

At the age of 16, with a head full of novel mathematical ideas, he sat the entrance exam for the École Polytechnique, failing to get in. The following year, he tried again. This time, the application ended in frustration and acrimony. The examiner reported the student ‘Knows nothing’ and was of ‘Little intelligence’. More likely, Galois’s approach took too many logical leaps, confusing his supposedly superior examiner. He entered the inferior École Normale, despite being described by the authorities there as ‘Obscure in expressing his ideas’.

Aged 19, Galois sent two papers to the Academy of Sciences as an entry to the Academy’s prestigious Grand Prix. This first attempt at recognition failed for mysterious reasons, although it seems that Cauchy, the eminent mathematician refereeing the works, simply wanted the two papers combined. Galois’s second submission, however, failed in farcical circumstances. The revised paper was sent directly to Fourier, who was secretary to the Academy and appeared sympathetic to Galois’s ideas. But Fourier died shortly afterwards and the paper was lost.

One year later, political tension in France was running high. Galois had friends in revolutionary circles and attended a banquet with them. He was overheard, perhaps by spies, uttering the name of the king, Louis Phillipe, while brandishing a dagger. This being construed as a threat on the king’s life, Galois was arrested. This time, he was released, but he was later re-arrested and this time sentenced to 9 months in prison.
He was not well treated in prison, being forced into drinking matches with his fellow inmates. When Galois complained about his ill-treatment, he was put into solitary confinement.

When out of prison, Galois returned to his mathematics, but his political interests continued to divert his attention. He finally received a response from the Academy of Sciences. His submission was declared “neither sufficiently clear nor sufficiently developed to allow us to judge its rigor”.

The exact circumstances surrounding the final days of Galois’s life and the fateful early morning duel, on 30 May 1832, are unclear. It has been rumoured that he was involved with a young lady, and perhaps the duel was instigated by Galois for her honour. The night before the duel, he set out to document all he could of his life’s mathematical work. He finished his rushed document with “… j'espere, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.” (I hope some people will find it to their advantage to decipher all this mess.)

Galois lost the duel and his short life was brought to a dramatic and violent end, the lack of sleep perhaps contributing to his slower reactions. His final words were “Do not cry, it takes all my courage to die at the age of 20.”

The work of Galois that survives amounts to no more than 60 pages. In its modern form, it bears little relation to the scrawl of his last document, but the ideas remain. He analysed the symmetry in the solutions of equations in ways never previously imagined. These new approaches, combined with his chaotic presentation, were what had confused his superiors at the various academic institutions.

Today, Galois Theory is of fundamental importance in the field of Quantum Theory. It is helping in answering questions about the structure and origin of the universe. It has been applied not only to describing the sub-atomic particles that have already been discovered, but to predicting the existence of new ones. The workings of the Large Hadron Collider, CERN’s giant particle accelerator, are dependent on the strength of Galois Theory.

Who knows how differently mathematics - even human civilisation - may have evolved, had Galois's life not been brought to such a sudden and tragic end.

Related articles

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• The mystery of mass: What makes one particle light and another heavy? | Ian Sample (guardian.co.uk)