SUNDAY MAGAZINE | October 30, 2009 Motherlode: Math’s Too Hard for a Parent’s Help By Lisa Belkin Many parents would rather talk to their kids about sex and drugs than math and science.

(with thanks to muriel)

Apparently, by a woman mathematician:

http://ideafoundlings.blogspot.com/

http://www.logicomix.com/en/

(with thanks to Jean-Michel Kantor).

I have always been rather partial to plane geometry; probably because it was the only branch of mathematics that was ever taught to me in such a way that I could understand it. For though I have no belief in the power of education to turn public school boys into Newtons (it being quite obvious that, whatever opportunity may be offered, it is only those rare beings desirous of learning and pssessing a certain amount of native ability who ever do learn anything), yet I must insist, in my own defence, that the system of mathematics instruction of which, at Eton, I was the unfortunate victim, was calculated not only to turn my desire to learn into stubborn passive resistance, but also to stifle whatever rudimentary aptitude in this direction I might have possessed. But let that pass. Suffice to say that, in spite of my education and my congenital ineptitude, plane geometry has always charmed me by its simplicity and elegance, its elimination of detail and the individual case, its insistence on generalities.

(From Aldous Huxley’s essay “Views of Holland”, page 98 of Thomas R. Cook’s Essays in modern thought on Google books (http://books.google.ca/books?id=k1yrBlylOigC). With thanks to Dan MacKinnon.

He had not become a mathematician, but geometry had imprinted on Huxley as an aesthetic paradigm obvious from the description of the Dutch landscape that immediately follows his childhood recollections:

My love for plane geometry prepared me to feel a special affection to Holland. For the Dutch landscape has all the qualities that make geometry so delightful. A tour of Holland is a tour trough the first books of Euclid. Over a country that is the ideal plane surface of the geometry books, the roads and the canals trace out the shortest distances between point and point.

[...] I may be free to admire the farmhouse on the opposite bank of the canal on our right. How perfectly it fits into the geometrical scheme! On a cube, cut down to about a third of its height, is placed a tall pyramid. This is the house. A plantation of trees, set in a quincunx formation, surrounds it; the limits of its rectangular garden are drawn, in water on the green plain, and beyond these neat ditches extend the interminable flat fields. There are no outhouses, no barns, no farm-yard with untidy stacks. The hay is stored under the huge pyramidal roof, and in the truncated cube below live, on the one side the farmer and his family, on the other side (during winter  only; for during the rest of the year they sleep in the fields) his black and white Cuyp cows. Every farmhouse in North Holland conforms to this type, which is traditional, and so perfectly fitted to the landscape that it it would have been impossible to devise anything more suitable. An English farm with its ranges of straggling buildings, its untidy yard, full of animals, its haystacks and pigeon-cotes, would be horribly out of place here. In the English landscape, which is all accidents, variety, detail and particular cases, it is perfect. But here, in this generalised and Euclidean North Holland, it would be a blot and a discord.Geometry calls for geometry; with a sense of the aesthetic proprieties which one cannot too highly admire, the Dutch have responded to the appeal of the landscape and have dotted the plane surface of their country with cubes and pyramids.

Delightful landscape! I know of no country that it is more mentally exhilarating to travel in. No wonder Descartes preferred the Dutch to any other scene. It is the rationalist’s paradise. One feels as one flies along in the teeth of one’s own forty-mile-an-hour wind like a Cartesian Encyclopaedist—flushed with mental intoxication, convinced that Euclid is absolute reality, that God is a mathematician, that the universe is a simple affair that can be explained in terms of physics and mechanics …

I was studying at the FeMeSha 18 in Moscow around ‘73. I recall being comfortable with the definition of derivative as a limit. On the other hand, the alternative definition that the instructor provided caused me no end of anxiety. Namely, he said the derivative is a number such that

As you correctly point out, it takes a considerable amount of mathematical training to formulate precisely what the problem was. The problem was that the definition says absolutely nothing about how one could find such a ““, or how to go about SIMULTANEOUSLY (in what sequence?) finding and ““. In retrospect, what I must have been bothered by is the non-constructive nature of this definition.

Actually I am currently writing a text on constructivism, and it could be that even after all these years I would still be unable to identify the source of the anxiety were it not for the fact of having understood constructivism better recently.

It should be noted that the noncompactness of the group of conformal transformations of is a nontrivial phenomenon which contradicts everybody’s geometric intuition. It is not clear at all why there exists a single conformal transformation of , which is not a rigid rotation. Similarly, one cannot see by a plain eye not equipped with mathematical machinery any non-trivial conformal transformation of (which as we know maps round spheres to round spheres) where “trivial” refers to the similarity transformation.

Even geometrically minded artists, designers of symmetric patterns, could not overcome this limitation of human imagination. If we look at the incredible number of ornaments designed through the centuries all over the world, we see all kinds of translational and rotational symmetries but never a conformal symmetry. Yet, in recent times conformal symmetries were displayed in many beautiful drawings of Escher. However, the idea of those was communicated to the artist by a mathematician, namely Coxeter.

(G. d’Ambra and M. Gromov, Lectures on transformation groups: Geometry and Dynamics, in: Surveys in Differential Geometry, 1, 1991. Quoted from Eremenko, with thanks.)

Thank you for signing this petition. The Prime Minister has written a
response. Please read below.

Prime Minister: 2009 has been a year of deep reflection – a chance for
Britain, as a nation, to commemorate the profound debts we owe to those who
came before. A unique combination of anniversaries and events have stirred
in us that sense of pride and gratitude which characterise the British
experience. Earlier this year I stood with Presidents Sarkozy and Obama to
honour the service and the sacrifice of the heroes who stormed the beaches
of Normandy 65 years ago. And just last week, we marked the 70 years which
have passed since the British government declared its willingness to take
up arms against Fascism and declared the outbreak of World War Two. So I am
both pleased and proud that, thanks to a coalition of computer scientists,
historians and LGBT activists, we have this year a chance to mark and
celebrate another contribution to Britain’s fight against the darkness of
dictatorship; that of code-breaker Alan Turing.

Turing was a quite brilliant mathematician, most famous for his work on
breaking the German Enigma codes. It is no exaggeration to say that,
without his outstanding contribution, the history of World War Two could
well have been very different. He truly was one of those individuals we can
point to whose unique contribution helped to turn the tide of war. The debt
of gratitude he is owed makes it all the more horrifying, therefore, that
he was treated so inhumanely. In 1952, he was convicted of ‘gross
indecency’ – in effect, tried for being gay. His sentence – and he
was faced with the miserable choice of this or prison – was chemical
castration by a series of injections of female hormones. He took his own
life just two years later.

Thousands of people have come together to demand justice for Alan Turing
and recognition of the appalling way he was treated. While Turing was dealt
with under the law of the time and we can’t put the clock back, his
treatment was of course utterly unfair and I am pleased to have the chance
to say how deeply sorry I and we all are for what happened to him. Alan and
the many thousands of other gay men who were convicted as he was convicted
under homophobic laws were treated terribly. Over the years millions more
lived in fear of conviction.

I am proud that those days are gone and that in the last 12 years this
government has done so much to make life fairer and more equal for our LGBT
community. This recognition of Alan’s status as one of Britain’s most
famous victims of homophobia is another step towards equality and long
overdue.

But even more than that, Alan deserves recognition for his contribution to
humankind. For those of us born after 1945, into a Europe which is united,
democratic and at peace, it is hard to imagine that our continent was once
the theatre of mankind’s darkest hour. It is difficult to believe that in
living memory, people could become so consumed by hate – by
anti-Semitism, by homophobia, by xenophobia and other murderous prejudices
– that the gas chambers and crematoria became a piece of the European
landscape as surely as the galleries and universities and concert halls
which had marked out the European civilisation for hundreds of years. It is
thanks to men and women who were totally committed to fighting fascism,
people like Alan Turing, that the horrors of the Holocaust and of total war
are part of Europe’s history and not Europe’s present.

So on behalf of the British government, and all those who live freely
thanks to Alan’s work I am very proud to say: we’re sorry, you deserved
so much better.

Gordon Brown

If you would like to help preserve Alan Turing’s memory for future
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Petition information – http://petitions.number10.gov.uk/turing/

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