Mathematics in Britain

From the music critic of The Times, writing in 1952 (issue of 2 May 1952, page 8, column 6, review of The Background of Music, by H. Lowery, published in 1952 by Hutchinson):

At Redbrick [University] they treat mathematics as an instrument of technology; at Cambridge they regard it as an ally of physics and an approach to philosophy; at Oxford they think of it as an art in itself having affinities with counterpoint and dancing.”

Quoted (incorrectly) by Ida Winifred Busbridge, in a 1974 history of mathematics at Oxford University, here. (Note that Busbridge writes “music” instead of “counterpoint”.)
Oxford University was a strong supporter of Catholicism in Elizabeth I’s time (eg, it was home to Thomas Campion), while Cambridge and the Fens, due to their proximity to the Netherlands, was the centre for an extreme Protestant sect, called the Family of Love, or the Familists.    Elizabeth I’s religious policy often sought to find a middle ground between these two extremes.   These religious differences persisted, so that Oxford was again, in the mid 19th-century, a centre of Catholic, and, within the Anglican Church, Anglo-Catholic (“High Church”) ideas.   The Redbrick Universities (Liverpool, Birmingham, Leeds, Victoria University of Manchester, etc), mostly founded in the North and Midlands of England in the late 19th century or early 20th century, were the result of money-raising campaigns by local business people and civic worthies, who were often of a Nonconformist or Jewish religious background.   The name Redbrick arose from novels written by a professor of Spanish at the University of Liverpool, Edgar Peers, about a fictional northern university modeled on Liverpool.

I don’t think the distinct differences between Nonconformist, Protestant and Catholic world views could be better expressed than those here between the philosophies of mathematics of Redbrick, Cambridge and Oxford:   Nonconformism as pragmatic utilitarianism; Protestantism as serious reflection on life’s higher ends; and Catholicism as enjoyment of life and its pleasures!

Gradgrinds de nos jours

Peter Hitchens reviews the latest jeremiad from Anthony Grayling in The Spectator, here:

‘Atheism is to theism,’ Anthony Grayling declares, ‘as not collecting stamps is to stamp-collecting’.  At this point, we are supposed to enjoy a little sneer, in which the religious are bracketed with bald, lonely men in thick glasses, picking over their collections of ancient stamps in attics, while unbelievers are funky people with busy social lives.

But the comparison is flatly untrue.  Non-collectors of stamps do not, for instance, write books devoted to mocking stamp-collectors, nor call for stamp-collecting’s status to be diminished, nor suggest — Richard Dawkins-like — that introducing the young to this hobby is comparable to child abuse.  They do not place advertisements on buses proclaiming that stamp-collecting is a waste of time, and suggesting that those who abandon it will enjoy their lives more.

Professor Grayling is too pleased with himself to have realised this. Intoxicated with amusement at his own dud metaphor, he asks: ‘How could someone be a militant non-stamp-collector?’  I rather think he has written the manual for anyone who might like to take up this activity.

It strikes me, once again, that the new atheists – particularly Grayling, Dawkins, Dennett, and C. Hitchens – want to show the world that they can think for themselves, that they are beholden not to churchly prince nor earthly potentate, neither to ideology nor tradition, in the formulation and articulation of their own views.   But thinking for themselves seems somehow insufficient to each of these people:  they want also to think for everyone else, to impose on the rest of us their own personal view.    Perhaps it’s that common affliction of people lacking in self-confidence:  by getting others to adopt their judgments (or beliefs, or consumer purchases, or career path), they can validate their own choices.

As I have remarked before, in my experience most people who profess religious belief and/or undertake religious practices do so because of some personal, transcendant experience they have had with non-material realms, or what they perceive to be such realms.  Grayling et al. tell all these people that they are each wrong:  Here’s what you should believe, not the evidence of your own lying eyes!   Who knows, our little Graylings may even be correct, and objective science may even – one day long in the future – show the delusion or invalidity of these many billions of subjective experiences. But, in the meantime, it surely takes industrial-strength levels of presumption to assume that your own personal subjective viewpoint takes precedence over the subjective lived experience of others.

 
Noticeable too, is the complete absence of any sense of wonder or awe at the unknown and the not-yet-explained in the writings of these folks (Sam Harris excepted).   It is remark-worthy, I think, that none of these people are poets, or artists, or musicians, or mathematicians, all people who seek regularly to wrangle the transcendent.   Gradgrinds the new atheists seem to be, as well as arrogantly presumptious.

Time, gentlemen, please

Much discussion again over at Language Log over a claim of the form “Language L has no word for concept C”.  This time, it was the claim by Wade Davis (whose strange use of past tense indicates he has forgotten or is unaware that many Australian Aboriginal languages are still in use) that:

In not one of the hundreds of Aboriginal dialects and languages was there a word for time.”

The rebuttal of this claim by Mark Liberman was incisive and decisive.   Davis was using this claim to support a more general argument:  that traditional Australian Aboriginal cultures had different notions of and metaphors for time to those we mostly have in the modern Western world.
We in the contemporary educated West typically use a spatial metaphor for time, where the past is in one abstract place, the present in another non-overlapping abstract place, and the future in yet a third non-overlapping abstract place.    In this construal of time, causal influence travels in one direction only:  from the past to the present, and from the present to the future.   Nothing in either the present  or the future may influence the past, which is fixed and unchangeable.   Events in the future may perhaps be considered to influence the present, depending on how much fluidity we allow the present to have.  However, most of us would argue that it is not events in the future that influence events in the present, but our present perceptions of possible future events that influence events and actions in the present.
Modern Western Europeans typically think of the place that represents the past as being behind them, and the future ahead.   People raised in Asian cultures often think of the abstract place that is the past as being below them (or above them), and the future above (or below).   But all consider these abstract places to be non-overlapping, and even non-contiguous.
Traditional Australian Aboriginal cultures, as Davis argues, construe time very differently, and influences may flow in all directions.   A better spatial metaphor for Aboriginal notions of time would be to consider a modern city, where there are many different types of transport and communications, each viewable as a network:  rivers, canals, roads, bus-only road corridors, railways, underground rail tunnels, underground sewage or water drains, cycleways, footpaths, air-transport corridors, electricity networks, fixed-link telecommunications networks, wireless telecommunications networks, etc.    A map of each of these networks could be created (and usually are) for specific audiences.  A map of the city itself could then be formed from combining these separate maps, overlaid upon one another as layers in a stack.   Each layer describes a separate aspect of reality, but the reality of the actual entire city is complex and more than merely the sum of these parts.  Events or perceptions in one layer may influence events or perceptions in other layers, without any limitations on the directions of causality between layers.
Traditional Aboriginal notions of time are similar, with pasts, the present and futures all being construed as separate layers stacked over the same geographic space – in this case actual geographic country, not an abstract spatial representation of time.  Each generation of people who have lived, or who will live, in the specific region (“country” in modern Aboriginal English) will have created a layer in the stack.   Influence travels between the different layers in any and all directions, so events in the distant past or the distant future may influence events in the present, and events in the present may influence events in the past and the future.
Many religions – for example, Roman Catholicism, Hinduism, and African cosmologies – allow for such multi-directional causal influences via a non-material realm of saints or spirits, usually the souls of the dead, who may have power to guide the actions of the living in the light of the spirits’ better knowledge of the future.   Causal influence can thus travel, via such spirit influences, from future to present.  Similarly, the view of Quantum Mechanics of space-time as a single 4-dimensional manifold allows for influences across the dimension of time as well as those of space.
I am reminded of an experience I once witnessed where the only sensible explanation of a colleague’s passionate enthusiasm for a particular future course of action was his foreknowledge of the specific details of the outcome of that course of action.  But these details he did not know and could not have known at the time of his enthusiasm,  prior to the course of action being executed.  In other words, only a causal influence from future to present provided a sensible explanation for this enthusiasm, and this explanation only became evident as the future turned into the present, and the details of the outcome emerged.  Until that point, he could not justify or explain his passionate enthusiasm, which seemed to be a form of madness, even to him.    Contemporary Western cosmology does not provide such time-reversing explanations, but many other cultures do; and current theories of quantum entanglement also seem to.
Contemporary westerners, particularly those trained in western science, have a hard time understanding such alternative cosmologies, in my experience.  I have posted before about the difficulties most westerners have, for instance,  in understanding Taoist/Zen notions of synchronicity of events, which westerners typically mis-construe as random chance.

Reincarnation

Do we each have a soul that incarnates in different bodies over time?  Most scientists in my experience dismiss any such idea, like they do most everything they cannot yet explain.  But a true scientist would (a) keep an open mind on the question, while (b) devising a scientific test of the claim.   And here’s where things become difficult – and interesting.  Exactly how would one test the hypothesis of reincarnation?

If reincarnation occurs, then there is a connection between bodies in different historical time zones.  Yet there seems to be no way that such bodies could communicate their special connectedness to one another.  In the case that reincarnation occurs, is there some way for instance that I could communicate a message with my future self (or selves), and only that person (or people), in a way that they could recognize the message as coming from me (their own past incarnation) and from no one else?  Thus far, I have not been able to imagine such a communication channel or message.  It may be possible to design a message that is public and seen by all, yet is only understood correctly by a particular recipient, as with the signal sent by the USSR’s Strategic Missile Command to the leadership of the USA during the August 1991 coup.

It would seem that no such inter-carnate communication is possible between incarnations of the same soul.   Yet all the scientific tests of  the hypothesis of reincarnation I can imagine would require some form of direct communications between separate human incarnations of the same soul, in the case there was reincarnation.   Suggestions for experiments most welcome.

Bird cries from the mountaintop

When the wild bird cries its melodies from the treetops,
Its voice carries the message of the patriarch.
When the mountain flowers are in bloom,
Their full meaning comes along with their scent.

I have remarked twice before that modern westerners, even very clever ones, fail to understand the nature of synchronicity in Taoist and Zen philosophy when discussing the art of John Cage.  If you believe the universe is subject to invisible underlying forces, as Taoist and Zen adherents may do (and as Cage did), then there is no chance, no randomness, no lack of relationships between events, only a personal inability to perceive such relationships.  The I Ching is intended as a means to reveal some of these hidden connections.
In a recent essay on Silence in the TLS, Paul Griffiths ends with:

Another of Cage’s favourite maxims, this one taken from Ananda Coomaraswamy and delivered five times in Silence, was that the purpose of art is to “imitate nature in her manner of operation”, which is almost another way of stating his first catchphrase, since natural objects and phenomena have nothing to say. They are not, of course, saying it. We say it for them. And in our doing so, experiencing their voicelessness and taking it into ourselves, a great deal comes to be said. There is no message in the changing pattern of cloud shadow and reflected sunlight on the sea. It may, nevertheless, thrill us, calm us, and fix our sustained attention.”

But, of course, for a Zen adherent there are indeed messages in the changing patterns of clouds and in sunlight reflected on the sea.   Even more so are there messages in human artefacts such as musical compositions, even those (perhaps especially those!) using so-called random methods for creation.    For Cage, the particular gamuts (clusters of sounds) that he selected for any particular one of his random compositions were selected as the direct result of the spiritual forces acting on him at that particular moment of selection, through his use of the I Ching, for instance.   Similarly, under this world-view, the same forces are active in those compositions allowing apparently-random leeway to the performers or listeners.
One can criticize or reject this spiritual world-view, but first one has to understand it. Griffiths, like so many others, has failed to understand it.
 

The emotions of religion

For some time, I have been arguing the irrelevance of belief for many practitioners of religion. (See posts among those listed here.)  Author Francis Spufford has a nice essay today in The Gruaniad offering his take on this question, arguing that his religious belief is made up of emotions, and not just some emotions but all.
I was reminded of George Santayana’s assessment of Roman Catholicism:

Catholicism is the most human of religions, if taken humanly:  it is paganism spiritually transformed and made metaphysical.  It corresponds most adequately to the various exigencies of moral life, with just the needed dose of wisdom, sublimity, and illusion.” (1944, p. 98)

Reference:
George Santayana [1944]:  Persons and Places. (London, UK:  Constable.)
 

Imaginary beliefs

In a discussion of the utility of religious beliefs, Norm makes this claim:

A person can’t intelligibly say, ‘I know that p is false, but it’s useful for me to think it’s true, so I will.’ “

(Here, p is some proposition – that is, some statement about the world which may be either true or false, but not both and not neither.)
In fact, a person can indeed intelligibly say this, and pure mathematicians do it all the time.   Perhaps the example in mathematics which is easiest to grasp is the use of the square root of minus one, the number usually denoted by the symbol i.   Negative numbers cannot have square roots, since there are no numbers which when squared (multiplied by themselves) lead to a negative number.  However, it turns out that believing that these imaginary numbers do exist leads to a beautiful and subtle mathematical theory, called the theory of complex numbers. This theory has multiple practical applications, from mathematics to physics to engineering.  One area of application we have known for about a  century is the theory of alternating current in electricity;  blogging – among much else of modern life – would perhaps be impossible, or at least very different, without this belief in imaginary entities underpinning the theory of electricity.
And, as I have argued before (eg, here and here), effective business strategy development and planning under uncertainty requires holding multiple incoherent beliefs about the world simultaneously.   The scenarios created by scenario planners are examples of such mutually inconsistent beliefs about the world.   Most people – and most companies – find it difficult to maintain and act upon mutually-inconsistent beliefs.   For that reason the company that pioneered the use of scenario planning, Shell, has always tried to ensure that probabilities are never assigned to scenarios, because managers tend to give greater credence and hence attention to scenarios having higher-probabilities.  The utilitarian value of scenario planning is greatest when planners consider seriously the consequences of low-likelihood, high-impact scenarios (as Shell found after the OPEC oil price in 1973), not the scenarios they think are most probable.  To do this well, planners need to believe statements that they judge to be false, or at least act as if they believe these statements.
Here and here I discuss another example, taken from espionage history.

Visitations to a writer

Recently-deceased Italian writer Antonio Tabucchi, author of that subtle novel of political intrigue under totalitarianism, Pereira Maintains, writes about the visitation he received which inspired the novel, here.  How sad that the name of the brave Portuguese journalist whose death inspired the novel should be unmentioned by Tabucchi.

Dr Pereira visited me for the first time one September evening in 1992. In those days his name wasn’t yet Pereira. He still didn’t have distinct traits, he was rather vague, elusive, hazy, but he already nurtured the wish to be a protagonist in a book. He was only a character in search of an author. I don’t know why he chose me to tell his story. One possible hypothesis is that the month before, on a torrid August day in Lisbon, I too had made a visit.
I vividly remember that day. In the morning I bought the city’s daily newspaper and read an article about an old journalist who had died at the Santa Maria Hospital and whose remains lay in state at the hospital chapel. I shall discreetly avoid any mention of the deceased’s name. I shall say only that he was someone with whom I had a passing acquaintance in Paris, in the late 1960s, when he, a Portuguese exile, was writing for a Parisian newspaper. He was a man who had plied his journalistic trade in Portugal during the 1940s and 50s under Salazar’s dictatorship. And he had managed to ridicule the regime by publishing a savage article in a Portuguese newspaper. He naturally encountered serious problems with the police and was subsequently forced to choose exile.
I knew that after 1974, when Portugal returned to democracy, he went back to his country, but I didn’t meet him again. He wasn’t writing any more, he had retired, and I didn’t know what he was doing for a living. Sad to say, he had been forgotten. In that period Portugal lived the restless, convulsive life of a country that had rediscovered democracy after 50 years of dictatorship. It was a young country, led by young people. No one remembered an old journalist who had resolutely opposed Salazar’s dictatorship in the late 40s.
I went to view the remains at two in the afternoon. The chapel was deserted. The coffin was uncovered. The gentleman was Catholic, and they had placed a wooden crucifix on his chest. I stood beside him for nearly 10 minutes. He was robust or, rather, fat. When I knew him in Paris, he was about 50, svelte and agile. Old age, perhaps a hard life, had turned him into a fat, flabby old man.
At the foot of the coffin, on a small lectern, lay a register open to receive the signatures of visitors. A few names had been written there, but none I recognised. Perhaps they were old colleagues, people who lived through the same battles, retired journalists.
A month later Pereira paid his visit to me. I didn’t know what to say to him then and there. And yet I dimly understood that his vague self-presentation as a literary character was symbolic, metaphoric: somehow he was the ghostly transposition of the old journalist to whom I bid my last farewell. I felt embarrassed, but I warmly welcomed him.
That September evening I divined that a spirit drifting in the ether needed me to tell his story, to describe a choice, a torment, a life. In that privileged space which precedes the moment of falling asleep – and which I find most suitable for receiving visits from my characters – I told him to come back, to confide in me, to tell me his story.
He came back, and I immediately found a name for him: Pereira. In Portuguese “Pereira” means “pear tree”, and like all the names for fruit trees, it is a surname of Hebrew origin, just as in Italy the surnames of Hebrew origin are the names of cities. With this name I wanted to pay homage to a people who had left a great imprint on Portuguese culture and suffered the injustices of history. But there was another reason, literary in origin, which led me to this name: a brief interlude by TS Eliot entitled “What About Pereira?” in which a fragmentary conversation between two friends evokes a mysterious Portuguese man named Pereira, about whom nothing can ever be known.
About my Pereira, however, I began to know many things. In his nocturnal visits he told me that he was a widower who suffered from heart disease and unhappiness. He loved French literature, especially Catholic writers between the wars, such as François Mauriac and Georges Bernanos. He was obsessed with the idea of death. His closest confidant was a Franciscan named Father Antonio, to whom he shuddered to confess his heresy: he didn’t believe in the resurrection of the body.
Later Pereira’s confessions, joined to my writerly imagination, produced the rest. Through Pereira I located a crucial month in his life, a torrid month, August of 1938. I recalled Europe on the brink of disaster, the second world war, the Spanish civil war, the tragedies of our recent past. And in the summer of 1993, when Pereira – who had now become my old friend – told me his story, I was able to write it. I wrote it at Vecchiano, in two equally torrid months of furiously intense work.
By a lucky coincidence, I finished writing the last page on the 25 August. I wanted to record that date on the page because it is an important day for me: my daughter’s birthday. I felt it was a sign, an omen. The happy day of my child’s birth also gave birth – thanks to the effort of writing – to the story of a man’s life. Perhaps, in the inscrutable weave of events that the gods bestow on us, everything has its meaning.”
• Antonio Tabucchi died on 25 March 2012. This article about the writing of Pereira Maintains (Canongate) was translated by Lawrence Venuti.