Crafty Clue

From The Daily Telegraph:

 He is number two here (8,5)

The cunning distraction in this clue is the apparent nominative male pronoun. But that reading would make the solution ambiguous: are we looking for who "he" is or where "here" is? So there has to be another way of looking at it.

(By the by, such a clue would not be possible in Finnish, which has only gender-neutral pronouns and completely lacks grammatical gender.)

Mathematics for the Million?

The integer 1936 is not only a perfect square and the year of my birth, it was also the year in which a scientist called Lancelot Hogben published a popular book called Mathematics for the Million. (It is still in print. A 2017 reprint by Prelude Press is available on Amazon UK for £10.99 but the reviewers complain of the poor quality of the typesetting and layout.)  Hogben’s parents were Plymouth Brethren but he rejected his family’s religion during adolescence. After retiring from front-line ambulance service in the First World War and then being imprisoned as a conscientious objector, he went to Cambridge as a medical student. He became, in his own words, a ”scientific humanist” and subsequently followed successful careers in experimental zoology and medical statistics. The book’s didactic method was to develop mathematical ideas in the context of their history and their applications to everyday life. Readers found his approach refreshingly different from the dry textbooks used in schools at the time.  The book covers roughly the current secondary school mathematics curriculum for 18-year-old school-leavers in England. In its day, it was praised by the science fiction writer H.G. Wells (“a great book of first class importance”) and by Albert Einstein (”it brings alive the contents and elements of mathematics”).

Hogben’s writings stirred up a lively controversy about the relative values of pure and applied mathematics and the contrasting mindsets of their practitioners, and many took sides. Hogben made clear his dismissive view of pure mathematicians as a coterie of self-indulgent high priests, protected in their ivory towers from realities of everyday life. After condemning Greek mathematicians and philosophers for their fondness for abstraction and obsession with aesthetics, he wrote “The fact that mathematicians are often like this may be why they are so inclined to keep the high mysteries of their Pythagorean brotherhood to themselves.” His main sparring partner in this philosophical spat was the eminent Cambridge number theorist G.H. Hardy, perhaps best known for his collaboration with the Indian mathematical genius, Srinivasa Ramanujan (Their relationship is popularized in the film The Man Who Knew Infinity in which Hardy is played by Jeremy Irons).

Hogben viewed mathematics as the science of measurement, to be valued for its practical applications and its potential, through science, to make the world a better place. Hardy was passionate about the beauty of his discipline and dismissed applied mathematics as trivial:

“There are then two mathematics. There is the real mathematics of the real mathematicians, and there is what I will call the ‘trivial’ mathematics, for want of a better word. The trivial mathematics may be justified by arguments which would appeal to Hogben, or other writers of his school, but there is no such defence for the real mathematics, which must be justified as art if it can be justified at all. “ 

This quotation is taken from his personal credo A Mathematician’s Apology, published in 1940 towards the end of his career.  In further quotations Hardy continues:

“A mathematician, like a painter or a poet, is a maker of patterns”,  …  “Even Professor Hogben, who is out to minimize at all costs the aesthetic element in mathematics, does not venture to deny (the reality of its beauty).”  …  he proudly asserts “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating.” 

Posterity has shown him to have been dramatically wrong in claiming that pure mathematics could never serve the deadly purposes of warfare -- think of Alan Turing at Bletchley Park or Hiroshima and Nagasaki. But as the “two mathematics” have cross-fertilized and enriched each other over the years, this squabble about their relative worth has run its course. Nowadays there are two new kinds of mathematicians: those - a dying breed -- who still believe there are two kinds of mathematics and those who don’t.

This post was prompted by the recent death at 94 of John Tate, a distinguished American number theorist. Like Hardy, he relished the beauty of mathematics but realized it was not something that could be easily shared with those not in his field. 'Unfortunately it's only beautiful to the initiated, to the people who do it,' he said in an American Mathematical Society interview. 'It can't really be understood or appreciated much on a popular level the way music can. You don't have to be a composer to enjoy music, but in mathematics you do. That's a really big drawback of the profession. A non-mathematician has to make a big effort to appreciate our work; it’s almost impossible."

Perhaps my title for this post should have been “The Loneliness of the Long-Distance Mathematician”.

Rest in peace

Two cryptic crossword clues I enjoyed this week. The first is from one of Tim Morey's regular puzzles in The Week  and the second  from The Times:

In which shameful operations may be seen (6, 2, 10)

A sentiment for those who died on Scott's last journey (4)

The surprising anagram in Tim Morey's clue makes a wry comment on the current state of our politics. A hint for the second clue is that "Scott's last" is the letter 't'.

Two creative men of letters from the world of arts and entertainment died this week. RIP Clive James and Jonathan Miller.

The Garden of Eden and Mitochondrial Eve

Satan Watching the Endearments 
of Adam and Eve

William Blake’s etching of Adam and Eve affectionately canoodling while the Devil looks on is one of over 300 works by the artist and poet currently on show at Tate Britain. So deep-rooted is our religious culture that this scene of innocence comes loaded with ideas of guilt from the creation story: disobedience to God, original sin, and so forth.

If you believe in the literal truth of the bible, it is natural to wonder where the Garden of Eden was located, and even if you don’t, it is interesting to ask where on earth Homo Sapiens first evolved. Now Vanessa Hayes. a geneticist at the Garvin Institute of Medical Research in Sydney, and her colleagues believe they have found the answer, with the help of mitochondrial DNA and some recent paleo-geographical research.

Mitochondria are organelles (semi-independent sub-units) in human cells that have many useful functions; most importantly they produce about 90% of the chemical energy the cell needs to stay alive and do its work. They are thought to have once been -- long long ago -- invading bacteria that decided to stick around and seal a symbiotic pact with their host. These freeloading mitochondria have their own modest store of DNA (called mtDNA) that is separate from the main human DNA in the nucleus There are roughly 16,500 letters (nucleotides A, C, G and T) in a mitochondrion’s DNA compared with 3.2 billion in the human genome. However, unlike human DNA, most of which gets scrambled through sexual reproduction in every generation, mtDNA is passed on unchanged from mother to child through her egg. This fact means that it can be traced back indefinitely along the maternal line to form a mitochondrial tree.

At the top of this tree there is a “matrilineal most recent common ancestor” (MRCA) of all living humans; observe that, of necessity, she would have had at least two daughters. because had she had only one, that daughter would be a more recent common ancestor. Journalistic accounts have dubbed her “Mitochondrial Eve” after Adam’s mate in the biblical story of creation that caught Blake’s artistic imagination. Unfortunately this description has led to some misunderstandings. Unlike the Eve in the Book of Genesis, she was in no sense the “first woman”. This is because she is believed to have lived sometime between 100,000 and 200,000 years ago in southern Africa and there were plenty of other women around at the time. Secondly, as matrilineal lines die out, the identity of the MRCA will change and move forward in time. Here is a simple example to illustrate why: imagine that your maternal grandmother had just one sister and that this sister had a just one (now deceased) daughter and via her just one grand-daughter.  While that granddaughter is still alive, your great-grandmother is the matrilineal most recent common ancestor of your family, but if that granddaughter were to die daughterless, then the MRCA role would pass to your maternal grandmother. (There is also a Y-chromosomal Adam. but that is another story.)

The Okavango River Delta

The Makgadikgadi desert and the lush inland delta of the Okavango river now in Northern Botswana were once covered by the largest lake in Africa, 200,000 years ago the fertile wetlands surrounding this lake were teeming with wildlife and offered the perfect nursery for the early development of Homo Sapiens. For the following 70,000 years it is conjectured that these early humans evolved in idyllic isolation, hemmed in by the inhospitable desert-like lands encompassing them. Then, in two bursts of local climate change, fertile corridors opened up first to the south around 130,000 years ago and later to the north 110,000 years ago allowing these people to leave their Garden of Eden and migrate further afield to settle in southern and central Africa.

All genetic information is subject to random mutations and the mitogenome is no exception. It has a certain region of higher variation with roughly 1000 nucleotide letters, and the mutations occurring in this region. together with knowledge of the average rates of mutation, can be used to establish the order in which mutations occurred and thereby to deduce the structure of the mitochondrial tree. As the branches of the tree move back through time, they converge on a group of mitogenomes known as L0, which are characteristic of a group of Khoisan people found in southern Africa who long predate the arrival of the Bantu peoples and Europeans in the area. Professor Hayes and her colleagues assembled a total of 1,217 variants of the LO genome, and from this mtDNA data and knowledge of where it was collected, they were able to construct a more refined mitogenomic tree. This enhanced tree was found to converge to an approximate time of 200,000 years ago and to a location in Northern Botswana. The geographical hypothesis about the isolated fertile region of prehistoric Northern Botswana is supported by research of Axel Timmermann at the Institute for Basic Science in Busan, South Korea. He is a climatologist who has used paleo-geographic and astronomical evidence to deduce the existence of the two periods when the opening of green corridors made migration away from Botswana possible. Thus we have a new creation story, now supported by science, about a Garden of Eden in Botswana and man’s escape -- but not expulsion --  to go forth and multiply and populate the earth.

Other professionals in the field have challenged Professor Hayes’s conclusions as highly speculative, raising doubts about the validity of extrapolating from living mtDNA data into the distant past and about our imperfect knowledge of the migration of peoples over a such a long period as 200,000 years. It has recently been found possible, in rare cases, to retrieve usable genetic information from ancient human remains, and so perhaps older corroborative evidence will be discovered to give stronger credence to this story.

More thoughts of Norway

By the end of the week our walking group had become more like an extended family. We were a friendly and talkative crowd, mostly ‘getting on’ in years but willing to share personal stories between gasps for breath on the guided walks. These were circular rambles and scrambles on various islands typically between 7 -- 10 miles long and involving a lot of up and down, especially for those who chose the harder walk instead of the easier one on offer each day. Our two leaders were experienced guides and very concerned for our safety. Many of us were retired but had led interesting lives in a variety of jobs and personal roles. There was usually some tempting activity laid on after dinner each day - a talk, a museum visit, a boat trip - which heightened the feeling of full-on engagement during our waking hours.

The landscape had a marked individual character and the walks offered some spectacular views of the spiky mountains of the islands and complex patterns of waterways between them.

One interesting feature of the rural Norwegian landscape is the limited choice of colours for their painted wooden houses. There are just three acceptable hues: red (verging on maroon), yellow or white. The red paint is the cheapest, and was made traditionally from a mixture of blood and oil from fish or whales; the more expensive yellow paint is from powdered ochre, again bound with fish oil; and the white is made from zinc oxide, which is costly enough to signal your richer status in the community. It is not widely known that Norway still has a small whaling fleet and in 2018 it caught and killed 434 minke whales. Back in the 1950s Norway had around 350 whaling vessels, but the fleet has now declined to just 11, which may explain why its catch fell well short of the quota of over 1200 controversially set by the Norwegian Fisheries Department. It may also explain why whales’ blood is no longer used to make the red paint for the houses and farm buildings.

I know of one counter-example to the limited colour palette for Norwegian country homesteads: In the summer of 1958 I worked with 3 student friends in the virgin forests high above the village of Frønningen on the Sognefjord. Our main job was to encourage natural regeneration by chopping and spreading the abandoned tops of the winter-harvested pine trees. The vast estate of 30 square miles was then owned by Johan Rumohr, who played the part of a benign but almost feudal laird and lived in a grand and ancient house called Frønningen-Godset (trans. Manor):

Logging provided the main livelihood of the families that lived and worked there. They cut down trees in the autumn and winter, transported them in spring down the rivers in spate from the melting snows, and cultivated their small-holdings and worked in the saw mill in the summer. Johan’s younger son, Knut Rumohr, who had become an established artist in Oslo, returned to Frønningen in that summer of 1958 with his newly-wed wife Aagot. He was closely involved with the local community and not only cut everyone’s hair, including ours, during his visit, but had previously supplied a range of outdoor paints in harmonizing pastel shades with which the villagers were encouraged to paint their fjord-side houses. The village of Frønningen lining the edge of the fjord presented an impressively artistic and colourful aspect as we four students approached by boat, which was then, and in fact still is, the only way to get there.  

We returned to Frønningen as a family 50 years later -- part of my wife’s ambitious project to take me back to the haunts of my youth -- and I was moved to find that the areas of forest in which we had worked had regrown to full maturity; sadly however, the sustainable logging activities had long since ended, the saw mill was derelict and the villagers had all left. Knut’s son Vilhelm, who now lived alone in Frønningen-Godset, invited us to stay in one of the empty houses and gave us a warm welcome. His father Knut had died by then, but we met his mother Aagot and his sister Liv, who came for a visit during our stay. Since foreign competition put the timber industry out of business, Vilhelm has worked hard to restore Frønningen’s fortunes with hydroelectric projects and by providing facilities for visitors’ to stay for hunting, fishing and artists’ retreats.

Back home from our holiday with memories fading, overtaken by the daily round, my thoughts were returned to Lofoten in an unexpected way. I became a Robert Macfarlane fan when my son Lachlan presented me with a copy of The Old Ways -- A Journey on Foot several years ago. A Good Reads review extols its virtues:

“In this exquisitely written book, Robert Macfarlane sets off from his Cambridge, England home to follow the ancient tracks, holloways, drove roads, and sea paths that crisscross both the British landscape and its waters and territories beyond. The result is an immersive, enthralling exploration of the ghosts and voices that haunt old paths, of the stories our tracks keep and tell, and of pilgrimage and ritual. Told in Macfarlane’s distinctive voice, The Old Ways folds together natural history cartography, geology, archaeology and literature. His walks take him from the chalk downs of England to the bird islands of the Scottish northwest, from Palestine to the sacred landscapes of Spain and the Himalayas.” 

Five years on and Macfarlane has done it again with Underland, recommended once more by Lachlan, this time as an audiobook beautifully read by Roy McMillan. This exploration of mankind’s relationship with a darker world of caves, potholes, catacombs and mines is a breath-taking story of hidden beauty, danger, mystery and unpredictable forces - but definitely not for the claustrophobic. It embraces a broad sweep of human experience from our atavistic past to the modern science of our anthropocene era, and speaks with passion, precision, human warmth and poetry. One of the chapters describes Macfarlane’s treacherous overland journey from the town of Å on the southern tip of Lofoten to the Refsvikhula Cave, an inaccessible prehistoric site on the western coast protected from a seaward approach by a dangerous maelstrom. Over 3000 years ago our ancestors braved the journey and left their mark with scores of red stick figures, both males spread widely over the cave walls and females cleaving together.

I have recommended Underland to anyone who might be interested, including Peter Gill, a fellow walker on our holiday. He sums up his response to the book better than I could my own:

“Thank you for pointing me in the direction of what I can only describe as Robert Macfarlane's masterpiece. I have been enthralled by his words as he weaves his way through landscapes, mountains and, of course, caverns. Truly poetic prose and so sympathetically read. However, as one progresses through the depths it becomes increasingly clear how much planning has gone into each chapter and the book as a whole coming to a final dark but rather inspiring crescendo. I am constantly amazed by his breadth of knowledge in so many fields whilst presenting it with such lyricism. Having reached the end I want to start it all over again.”