Fermat’s Last Theorem

fermat coverI was a sucker for this book, having been fascinated by the history of mathematics from an early age. As Simon Singh’s book Fermat’s Last Theorem explains, the origins of this theorem came from the early days of mathematics, with Pythagoras in Ancient Greece. Everyone knows Pythagoras’ theorem that the sum of the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, e.g:

32 + 42 = 52

In fact, it was eventually demonstrated that there are an infinite number of triples of integers x,y,z for which

x2 + y2 = z2

Mathematicians puzzled for centuries as to whether a similar equation might be possible with higher powers of any integers, i.e. cubes, power of 4, 5, 6,…

xn + yn = zn

Pierre de Fermat was a supreme mathematician of the 17th century, who worked largely alone, rather than with colleagues. When his work was subsequently examined he was found to have made major advances to mathematics in a number of areas. In particular there was a famous note written in a margin that he had found ‘a truly marvellous proof’ that there could be no instance where such an equation was possible, yet there was insufficient space in the margin to explain it. This became a challenge to all the top mathematicians since then.

Simon Singh takes us through much of the history of mathematics in recounting the development of efforts to solve what had become known as Fermat’s Last Theorem. And a fascinating tale he tells, with potted histories of the involvement of many leading mathematicians over the centuries – including the story of the 21-year-old Frenchman Evariste Galois, who jotted down what proved to be key insights during the night before he was shot and killed in a duel early the next morning.

Finally there came the assault by Cambridge mathematician Andrew Wiles, working for some years in a solitary fashion similar to that adopted by Fermat himself. Finally, in June 1993 Wiles outlined to a packed meeting of leading mathematicians a proposed argument that demonstrated that Fermat’s Last Theorem was true. But this was only a prelude to drama, as a fault was discovered in the logic of his proof. It was not until October 1994 that Wiles and a colleague finally laid rest to centuries of speculation and completed their proof of Fermat’s Last Theorem.

Simon Singh makes the development of deep ideas in mathematics in some way accessible to us, even though we could never understand the detail. Few people do!

Featured image of Pierre de Fermat from Wikimedia Commons

Want of foresight

Want of foresight, unwillingness to act when action would be simple and effective, lack of clear thinking, confusion of counsel until the emergency comes, until self-preservation strikes its jarring gong – these are the features which constitute the endless repetition of history.

Winston Churchill, Speech, House of Commons, May 2, 1935

Churchill was looking at the patterns of history and making a point about his own times. He could have been commenting on modern crises such as global warming, ocean acidification, over-use of pesticides, plastic pollution, Brexit, Palestine and on and on. It seems to be in the nature of humanity that only extreme crisis forces the needed change. We are not (yet), in that sense, a collectively rational species.

Picture of dark storm clouds rolling in at the Pawnee Buttes National Grasslands, Wyoming, by MichaelKirsh via Wikimedia Commons

Senatus PopulusQue Romanus

spqr coverI guess this will be my only blog title in Latin, a conceit from an early education that included this ancient language. It means ‘The Senate and People of Rome’, often abbreviated to SPQR, which is the title of classicist Mary Beard’s book.

Rigorous but readable, Mary tells the tale of 1000 years of ancient Rome. I enjoyed it, probably more so, and with greater insight, than I did reading the abbreviated version of Edward Gibbon’s Decline and Fall of the Roman Empire many years ago.

She begins in the middle, around 63 BCE, with the story of Cicero putting down the conspiracy by Cataline and other members of the Roman Senate itself, a story that is well understood because Cicero himself was a writer, many of whose works have survived, and history is written by the winners. The true story we do not know!

Mary then takes us back to the beginnings of Rome and the legend of Romulus and Remus, the foundation story of Rome. As Rome began to expand and co-opt nearby communities there eventually emerged what is characterised as a time of absolute kings, which ended around 490 BCE when Tarquin was defeated, a Republic was declared, and kingship got a bad name. Pairs of consuls were elected and ruled on an annual basis, overseen by the Senate – or rather such a system eventually emerged. Rome expanded, co-opting subject peoples into the system, particularly the army.

While Cicero was still alive, ever expanding overseas adventures led to the emergence of two major figures, Pompey in the East and Julius Caesar in the West. After years of accomodation, in 49 BCE Caesar ‘crossed the Rubicon’ and engaged in civil war with Pompey. Pompey was eventually defeated and Caesar became dictator, or Emperor. Not for long, he was assassinated on the Ides of March in 44 BCE. Amazingly this led to many years of stability as Augustus Caesar dominated the scene for many years, dying in old age (possibly).

Problems of succession then dominated for the next 14 Emperors. In 212 ACE Caracalla extended citizenship to all inhabitants of the Empire, and a different game began, with ever-increasing turnover of Emperors. This is where Mary Beard leaves the subject, to be taken up by someone else!

Mary cautions against trying to draw lessons for today from these historic events. But what is clear from Rome’s history is the danger of any state becoming identified with one man (it’s always a man!). Then there is the inevitable problem of succession, competitors, warring factions, and wars. This is what democratic systems avoid, by providing for the people to elect a new leader from time to time. So beware any leader or faction which seeks to extend his/its leadership beyond the specified time, who wants to rule for life or be a one-party state, in fact any dictator, populist or otherwise, and any party that seeks to rule forever. I won’t name names.

 

 

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The Templars

the templarsThere’s a thing about the Knights Templar, something romantic lodged in the European brain. Maybe it’s the idea of monk-like knights dedicated to fighting for Christendom, or the tales of valour in the holy land and the Iberian peninsula, or their tragic ending at the hands of the king of France… Who knows why stories get lodged into the collective imagination, but this one did. Historian Dan Jones’s very readable book The Templars tells the story well.

The Templars arose in the aftermath of the first crusade, which culminated in the taking of Jerusalem in 1099. When the main Christian forces had returned home the occupied lands were always vulnerable to being recaptured by local or regional forces. The Knights Templar were established to help protect these Christian outposts and keep safe routes for pilgrimage to Jerusalem.

The Knights Templar were founded in 1119, and in 1129 rules were established for the lives of these knights, based on the rules that had been established for the recently established Cistercian order of monks. The hugely influential Bernard of Clairvaux was instrumental in this. They were effectively fighting monks, with a code that meant they would fight to the death for the cause.

How strange to the modern mind that the rules of monasticism should be applied to the battlefield.

In subsequent battles, particularly from the second crusade in 1147, preached by Bernard himself, the Templars proved to be the most effective European fighting force, often, with the similar Knights Hospitaller, formed the vanguard or rearguard of the advancing forces.

Through the many crusades and the Spanish reconquista, up to the mid-1250s, the Templars played a major role. They provided trans-national services such as banking and fighting forces to the various kings in Europe. They became very powerful, which was fine while they held the confidence of those kings.

But the various crusades were not well organised and the Templars took the brunt of failures of the leaders who came looking for glory. There were a number of massacres of Templars and eventually the crusader project seemed to be coming to nought, with all gains being cancelled out. It seems there was some blame pointed at the Templars for these failures.

Jones tells the story of the various crusades and battles in an engaging manner. The balance of power clearly changed when the Mongols arrived and sacked Baghdad in 1258, and the Mamluk state joining Egypt and Syria was established in 1260. The Christians were effectively squeezed out.

The Templars remained influential across Europe until the coming of the French king Philip IV. Philip was a new sort of French king, establishing a strong centralised state, and moving against other sources of power. His first target was to get rid of the Jews, next came the Templars. On Friday 13 October 1307 all Templars in France were seized, imprisoned and tortured.

Neither the French pope Clement V nor the other European leaders agreed with Philip’s move, but it seems that the pope was persuaded to spread the investigation of Templars throughout Europe. These were the days of inquisition to detect heresy, so it was not difficult to trump up charges. The net effect was that the order of the Temple was suppressed in 1312, and the last leader of the Templars, Jacques de Molay, was burnt to death after recanting his forced confession a couple of years later.

It had been less than 200 years. What a story!

Footnote. Since the fall of the Turkish (Ottoman) Empire Western forces have been engaged in the Middle East in many ways, including establishing the Israeli outpost safeguarding Jerusalem, several encounters in Iraq, Syria, catastrophe in Egypt (Suez),… Le plus ça change…

Featured image from the website Knight Templar International.

 

 

Interview with René Descartes

I recently came across this interview, dated 2001/1649.

Interviewer: Bonjour, Monsieur Descartes. Can I call you René?

Descartes: Allô, allô. Mais of course!

Interviewer: I have come back from the twenty first century to ask you a few questions. People there are very interested in your ideas, but you have been getting some bad press lately. Are you happy to take part?

Descartes: I think so.

Read More »

Travellers in the Third Reich

The Rise of Fascism Through the Eyes of Everyday People

As one of a generation haunted by discovering the then-recent calamity of WW2, now disturbed by the rise in populism across the world, I found this a timely book by Julia Boyd.

It tells the story of the Third Reich through the eyes of people who visited or lived in Germany through the days of Adolf Hitler’s ascent to power, its consolidation, the increasing drumbeats towards war, and the war itself.

What is remarkable is how many people gave the Nazi regime the benefit of the doubt, despite the clear signs, such as the centralisation of all power, rescinding of civil liberties and press freedom, the early concentration camps, the persecution of Jews, the burning of books (all in 1933) through to Kristallnacht (1938) and the subsequent descent into war.

Of course, the desire to avoid another war was a major part of this, and there is the interesting story of Neville Chamberlain’s vain attempt to make peace in Munich in 1938, and Hitler’s dismissive attitude to the whole affair.

The book presents an interesting story, perhaps a bit long-winded at times. It certainly opened my eyes to some things, such as the fact that Germany welcomed English and American tourists throughout the 1930s, and many found the country very efficient and friendly, except where they came face to face with the persecution of Jews and supposed non-aryans.

The stories from the 1920s and early 1930s show that, after making a fair recovery from WW1, Germany was not in a good place after the shock of the great depression. The arduous reparation terms imposed by the Allies at the end of WW1 were a major cause of German suffering and dissatisfaction. It seems that these were major factors in the rise to power of Hitler.

The evident parallel today is the rise of populism following the 2008 financial crash, and the subsequent failure to make due reckoning with its causes. The missing factor today is there is no sense of national persecution similar to that caused in Germany by the WW1 armistice terms.  

In the case of Donald Trump and the US, it is maybe too early to say how far the parallels go – but he clearly came to the presidency by exploiting white male dissatisfaction with the status quo that had come about – economic, racial and misogynistic. On the positive side, the US constitution appears to be much more robust in resisting over-centralization of power than was Germany in the 1930s.


Cistercian Simplicity

I’ve long found inspiration and sustenance from the beauty and simplicity of the Cistercian abbeys, still found in various states of repair across Europe. For me their simplicity of form is unfailingly beautiful.

In this context I’ve also been aware of the towering spiritual figure of St Bernard of Clairvaux, one of the main instigators of the Cistercian movement, and wondered what sort of person he might have been.

spirit of simplicitySo I couldn’t resist the book ‘The Spirit of Simplicity’, being translations of classical French texts by that modern spiritual seeker Thomas Merton. The book is in two parts. The first part is a text with the book’s title, attributed to Jean-Baptiste Chautard in the mid 1920s. The second part contains selected texts by St Bernard himself on Inner Simplicity. Could this explain what lay behind the beauty of those old Abbeys?

The original Cistercian movement was one of renewal, aiming to return to the Rule of the monastic life originally established by St Benedict (c. 480-550 AD). Inner simplicity was a founding principle, and from this flowed the external simplicity of the forms created. The fathers of the first Abbey at Citeaux in the early 1100s were dedicated to this.

Chautard suggests that there was a golden age of 150 years for the Cistercian movement, when this simplicity was effectively maintained. This was followed by a silver age of another 100 years when it was not so effectively maintained and embellishments crept in. After the middle of the 14th century decline set in – with several causes: the Black Death, religious wars, and then the Reformation. (Paradoxically, Protestantism saw a return to simplicity in the form of religious buildings. Many of the older decorated Gothic buildings now show an almost Cistercian simplicity.) Another renewal movement at the end of the 19th century ensured that there are still some Cistercian Abbeys operating today.

St Bernard himself is regarded as the finest exemplar of the movement. The second part of the book contains his reflections on that simplicity, the need for humility, and obedience in the context of the monk’s life, the importance of the monk knowing himself – so actually quite modern psychologically – the overcoming of pride and dedication to the love of God.

I was quite struck by one particular quote:

And what greater pride is there than that one man should try to impose his own opinion upon the whole community, as if he alone had the spirit of God?

Modern dictators and populists please note. Pride always comes before a fall.

So the outer simplicity of the Cistercian abbey is a reflection of the inner simplicity of the monks. The evident beauty is a reflection of the inner beauty of their souls.

I would not suggest that the life of a monk is right for everyone, but it is clear that this dedication to inner simplicity produces this wonderful contribution to the beauty in the world. Go see some of these superb buildings for yourself – Fountains Abbey in UK, Fontenay, Senanques, Silvacane, Fontfroide, Pontigny and many others in France, Orval in Belgium. There are far too many to list them all. Here are just a few random selected photos.

For most, you must travel to less frequented parts of the country. The communities were built to be self sufficient, away from centres of population. These journeys provide a scenic mini pilgrimage in themselves. Even the less well preserved abbeys, such as Abbeycwmhir in an isolated valley in mid-Wales, once one of the largest abbeys in the UK, have a special atmosphere about them.

abbeycwmhir
Abbeycwmhir

And the book is certainly very readable if it aligns with your interest. Merton knew his stuff.

 

An unfortunate legacy of Justus von Liebig

Every good idea that takes off in human thinking seems to have its downside that eventually requires correction, as it is taken to extremes. I think this is what Hegel’s dialectic of thesis –> antithesis –> synthesis was about. The German chemist Justus von Liebig provides an example very relevant today, in a story told by Beata Bishop in the recent SciMed newsletter.

Von Liebig (1802-1873) is variously regarded as the founder of organic chemistry and the father of the fertilizer industry. He notably discovered that nitrogen, phosphorus and potassium are key minerals that plants need to grow and thrive. Thus came about the modern fertilizer industry, which gradually supplanted traditional farming techniques based on manure, compost, crop rotation, leaving fields fallow, etc.

Of course, initially this approach appeared successful and crops thrived. With the development of modern pesticides the industrial approach to agriculture seemed sensible and was commercially successful. But what has only become apparent after many decades is that this approach is over-simple and other vital minerals and organic matter are being gradually lost from the now-depleted soils. The organic movement arose to try to counteract this, but still only has a foothold where people can afford it. And the agrochemical industry has become so powerful that it is difficult to change towards the organic antithesis, or indeed any new synthesis.

Of course the pendulum will swing back, they always do. Unfortunately, this is also a critical time of climate change, caused by the related explosion of fossil fuel exploitation over the same period.

Historically, civilisations come to an end when changes of climate and crop yields eventually make them unsupportable. We really now are in a critical period of human history, partly thanks to the worthy efforts of Justus von Liebig. But never say die, necessity is the mother of invention, and humanity is a very inventive and adaptable species.

 

Northwich’s Swing Bridges

Northwich in Cheshire is notable as being an old salt town, so much so that during its history there have been frequent occurrences of subsidence as the land has subsided into old salt workings below. This proved a big problem for bridges over the River Weaver, which provided Northwich’s water link with the River Dee. The bridge pillars gradually subsided.

This was solved in 1898 with the building of the Hayhurst Bridge, followed the Town Bridge in 1899, the first electrically powered swing bridges in Great Britain. Two bridges meant that one was always open to traffic.

To avoid the subsidence problem, these bridges were built on floating pontoons, said also to be also the first of their kind in the country.

I’ve included monochrome images, trying out the capabilities of my new phone’s camera. I thought the black-white bridges could look better this way, but the colour versions actually look better to me.

What Newton really thought

Alert readers of this blog may have realised that I am reading Henri Bortoft’s book Taking Appearance Seriously: The Dynamic Way of Seeing in Goethe and European Thought. Bortoft throws interesting insight into the role of Isaac Newton in creating the modern scientific world, confirming Edi Bilimoria’s article mentioned in an earlier post.

Isaac Newton basically invented modern mathematical physics in his masterwork, Principia Mathematica (1687). To the theory of atomism and mechanical philosophy he added the notion of forces which act between bodies that are not in contact.

Bortoft suggests that from the eighteenth century onwards,  gravity began to be thought of as a ‘property of matter’, as if it were an attractive force inherent to matter. This is not what Newton thought. He did not believe in attraction as a real, physical, force.

For example, in a letter Newton said:

Pray do not ascribe that notion to me, for the cause of gravity is what I do not pretend to know and therefore would take more time to consider of it… Gravity must be caused by an agent acting constantly according to certain laws, but whether this agent be material or immaterial I have left to the consideration of my readers.

So Newton’s major discovery was to the effect that we could create mathematical models of the real world, what we now call ‘physics’. Subsequent founders of modern science were dedicated to the mathematical approach to nature, but ultimately the ascendancy of the mathematical was accompanied by the downgrading of the sensory and increasingly seeing the world as a mathematical abstraction. To many scientists the world became de-spiritualised and dead.

This was not Newton’s intention, although his name is often invoked as the originator of such a viewpoint.

The four little girls

Birmingham (Burr-ming-HAM) Alabama is renowned for its role in the civil rights campaigns of the 1960s, that were spearheaded by Martin Luther King Jr. In 1963 there was the bombing by the Ku Klux Klan of the 16th Street Baptist church that was at the heart of the movement, 4 little girls were killed. Birmingham police with dogs and water cannon attacked defenceless crowds, including children, in the nearby park. All this was orchestrated by the renowned mayor Bull O’Connor. I remember it all so well from the UK media of that time.

That park (Kelly Ingram Park) is now a moving memorial to these events, with a number of evocative statues. Near the entrance are statues to the four little girls, and to King himself.Read More »

Geography and Stupidity

The breakout of WW1 is a haunting occurrence for those of us born in the dying days of WW2, which finally brought an end to the European conflict begun in 1914, leading to the peace of the European Union since then.

How did that prosperous and confident Europe of the late 19C descend to such a self-defeating process?

It seems the answer lies in geography and stupidity. Read More »

Vanity and Happenstance

Archduke Franz Ferdinand was heir to the Habsburg Monarchy, Emperor of Austria and King of Hungary. On 28 June 1900 he married Countess Sophie Chotek. The Countess was too lowly placed for an imperial Habsburg marriage, so did not become her imperial highness, and their children did not have the right of succession. She was not allowed to sit by the Archduke’s side on public occasions.

Franz Ferdinand was irked, but there was one loophole – his wife could be by his side when he was acting in a military capacity as Inspector General of the army. Thus it was that, in 1914 on their 18th wedding anniversary, he inspected the Bosnian army in Sarajevo, in an open carriage with his wife by his side.

With hindsight this was not a good plan. Bosnia was recently acquired by the Habsburgs and there was unrest from young men who wanted it to join Serbia instead. Several conspired, aiming to assassinate the archduke. They were young and inexperienced and there were several blunders.

By accident the archduke’s chauffeur took a wrong turning and had to turn round. One of the conspirators just happened to be there, saw them and shot the couple. Thus began a World War that was only fully resolved 31 years later.

Did it all really begin by chance? Some would say otherwise.

This story is told at the beginning of AJP Taylor’s book ‘The First World War’, an engaging read first published in 1963. I well remember Taylor’s articles in the Daily Express around that time – he was one of the great popularisers of history and then very controversial.

Parkgate

One of the delights of living is Cheshire is the occasional visit to Parkgate, a pretty village facing onto the marshes of the estuary of the River Dee. I remember visiting in the 1960s and seeing the most spectacular of sunsets.

parkgate

Parkgate has an interesting history, and provides a pleasant walking promenade along by the marshes, which are an RSPB reserve. There are plenty of birds to be seen, albeit usually at some distance. The biggest high tides are always popular, as the birds are driven closer to the land, and the occasional rodent emerges from the marshes to escape the rising water.Read More »

How the Renaissance Began?

the swerve coverHow did the Renaissance begin? If we knew that, it would surely be useful in understanding what is needed for a New Renaissance. Well here’s a book that claims to give an answer: The Swerve: How the Renaissance Began by Stephen Greenblatt.

In a way, it does, although I suspect this is a gross over-simplification. Roughly, the story is that a very clever man Poggio Bracciolini, one time right hand man of a disgraced pope, discovered and had copied key texts that had been preserved over the centuries by monks regularly copying manuscripts.

The key text, De Rerum Natura (On The Nature of Things), by Roman philosopher/poet Lucretius contained explosive ideas that, once they began to circulate, overcame the stranglehold of the church on European ideas and led to the explosion of creativity that was the Renaissance.

In particular they directly influenced men such as Marsilio Ficino, Botticelli, Raphael, Giordano Bruno, Galileo, Thomas More, Machiavelli, Montaigne, Shakespeare, Francis Bacon, Dryden,  Isaac Newton, Darwin, Thomas Jefferson, and on and on…Read More »

Conques

The village of Conques in Aveyron, France, has been a target of pilgrimage since medieval times, lying as it does on the route from Le Puy en Velay to Santiago de Compostela in Northern Spain. The isolated position of Conques in hilly terrain means that it has never been subject to much modern development, so the medieval streets are essentially as they were.

conques rainbow

This view is the first the pilgrim coming from Estaing sees of Conques, nestling in the treed valley. We were lucky on our recent visit when, after a day of rain, the sun came out as we reached Conques. The dramatic welcome became spectacular when this rainbow appeared over the village.Read More »

Le Puy en Velay

I first visited Le Puy en Velay nearly 30 years ago, with Alf, as we traced the steps of pilgrims on the route from this starting point to Santiago de Compostela in northern Spain – one of Europe’s most popular pilgrimages since the Middle Ages. What a special place to begin a pilgrimage.

st michel aiguilhe

This area of the former county of Velay, in the south western Massif Central, is volcanic. As well as numerous dead volcanoes, it contains various strange landscape features, notably several isolated plugs of rock pointing skywards. On one of these is the chapel of St Michel d’Aiguilhe. The well-formed stairway up this rock brings you to wonderful views over Le Puy and the tiny Romanesque chapel at the top. Linger a while in here and it provides an experience of perfect peace before the start of the journey.

Read More »

Ghent

The Friday SquareIn the square of the Vrijdagmarkt in the delightful medieval centre of Ghent in Belgium, there is a statue of a local luminary – Jacob van Artevelde. He is so honoured because Ghent survived the 100-year war between England and France due to his efforts at ensuring neutrality and maintaining links with England. The statue points towards England.

Of course England has always had close ties with Europe, and periods of major influx of Europeans, such as the Romans, Angles, Saxons, Vikings, Normans. And we’ve been involved in so many continental wars over the centuries, culminating in the World Wars of the 20th century, when we took in many more as refugees.

The wise men of the immediate post-war world decided to link the countries of Europe together to avoid the possibility of future pointless wars, encouraged by the British, and Winston Churchill in particular. Of course this led to today’s European Union.

Strange then, that unscrupulous UK politicians have fostered the very nationalism that potentially leads to wars to engineer the UK’s exit from that Union. Something about ‘taking back control’ and ‘making Britain great again’ – the freedom to go off on our own, ‘do deals’, form alliances, and repeat unlearned lessons of history.

They should be ashamed of themselves.

Hopefully the duped UK populace will support a change of course in time.

Inline image by Sergey Ashmarin, via Wikimedia Commons