Mathematics and proof

One of the great myths of mathematicians is that mathematical knowledge, once proven, is solid, and not subject to later contestation.   Thus, Oxford mathematician Marcus du Sautoy, writing in the New Scientist (2006-08-26), says:

Proof is supposed to be what sets mathematics apart from the other sciences. Traditionally, the subject has not been an evolutionary one in which the fittest theory survives. New insights don’t suddenly overturn the theorems of the previous generation. The subject is like a huge pyramid, with each generation building on the secure foundations of the past. The nature of proof means that mathematicians, to use Newton’s words, really do stand on the shoulders of giants.
In the past, those shoulders have been extremely steady. After all, in no other science are the discoveries of the Ancient Greeks still as valid today as they were at the time. Euclid’s 2300-year-old proof that there are infinitely many primes is perhaps the first great example of a watertight proof.

The reason for this widespread view is that mathematics uses deduction to reach its conclusions.  At least, that is true of pure mathematics, or was so until computers began to be used in proofs (a topic which du Sautoy discusses in that article).  But all deduction does is to show that, given some assumptions and given some rules of inference, a certain conclusion follows from those assumptions by applying those rules of inference.  If either the assumptions are false or the rules of inference not acceptable, then the stated conclusions will not, in fact, follow.
Du Sautoy is quite wrong to claim that new insights do not overturn the theorems of the previous generation.  The history of pure mathematics is replete with examples where proven conclusions were later revealed to depend on assumptions not made explicit, or on assumptions previously thought to be obvious but which were later shown to be false, or on rules of inference later considered invalid.   For over a century, mathematicians thought that everywhere-continuous functions were also everywhere-differentiable, until shown a counter-example.  For a similar period, they thought that the convergent limit of an infinite sequence of continuous functions was itself also continuous, until shown a counter-example.  They thought that there could not exist a one-to-one and onto mapping between the real unit interval and the real unit square, until shown such a mapping (a so-called space-filling curve).  In fact, there are infinitely-many such mappings; indeed, an uncountable infinity of them.  In all these case, “proofs” of the erroneous conclusions existed, which is why the earlier mathematicians believed those conclusions.  The proofs were later shown to be flawed, because they depended on (usually-implicit) assumptions which were false.   For the differential calculus, the fixing effort was begun by Cauchy and Weierstrauss, using epsilon-delta arguments which were more rigorous than the proofs of the earlier generation of analysts.
Not only does Du Sautoy have his history wrong, but there is shurely shome mishtake in his mentioning Euclid here.  The 19th century was consumed by a controversy over the truth-status of Euclidean geometry, and the discovery of apparently-logical alternatives to it.   As clever a man as the logician and philosopher Gottlob Frege (an intellectual hero of Wittgenstein) could not get his head around the idea that these different versions of geometry could all simultaneously be true.   Yet that is the conclusion mathematicians came to: that, depending on the assumptions you made about the surface on which you doing geometry, there were in fact valid alternatives to the discoveries of the Greeks:  draw your triangles on the surface of a sphere, instead of on a flat plane, for example, and you could readily draw triangles whose three angles did not sum to 180 degrees.  You choose your assumptions, you gets your geometry!  This is not a secure pyramid of knowledge, but many pyramids, post-modernist style.
And in the first part of the 20th century, pure mathematics was consumed with a bitter argument over whether a particular rule of inference – reductio ad absurdem (RAA), or reasoning from an assumption thought to be false – was valid in deductive proofs of the existence of mathematical objects.   The dissidents created their own school of pure mathematics, constructivism, which is still being studied.  Indeed, it turns out that a closely-related logic, intuitionistic logic, appears naturally elsewhere in mathematics (as part of the internal structure of a topos). Once again, you choose your rules of inference, you gets your mathematical theorems.
There is no single, massive pyramid of knowledge here, as du Sautoy claims, but lots of smaller pyramids.  Every so often, a great mathematician is able to devise a new conceptual framework which allows some or all of these baby pyramids to appear to be part of some larger pyramid, as Pieri and Hilbert did with geometry in the 1890s, or as Lawvere and others did with category theory as a foundation for mathematics in the 1960s.   But, based on past experience, new baby pyramids will continue to be created by mathematicians arguing about the assumptions or rules of inference used in earlier proofs.    To consider this process of contestation, splitting, and attempted re-unification to be somehow different to what happens in other domains of human knowledge may be comforting to mathematicians, but is myth nonetheless.

Animal Farm: The Limerick

The superb winning entry of a competition run by New Statesman magazine (2009-12-14) to summarize a work of literature with a limerick, due to performance poet and photographer Anneliese Emmans Dean:

From the farm they banished the people.
“Hurrah!” cried the beasts. “We’re all equal!”
But superior plotters,
With trotters, the rotters,
Took over. The End. (There’s no sequel.)

The second time as farce

Rory Stewart, in his book about walking across Afghanistan, has this to say about the post-colonial cadres working for the UN and other international agencies in developing countries:

Critics have accused this new breed of administrators of neo-colonialism.   But in fact their approach is not that of a nineteenth-century colonial officer.  Colonial administrations may have been racist and exploitative but they did at least work seriously at the business of understanding the people they were governing.  They recruited people prepared to spend their entire careers in dangerous provinces of a single alien nation. They invested in teaching administrators and military officers the local language.  They established effective departments of state, trained a local elite and continued the countless academic studies of their subjects through institutes and museums, royal geographical societies and royal botanical gardens.  They balanced the local budget and generated fiscal revenue because if they didn’t their home government would rarely bail them out.  If they failed to govern fairly, the population would mutiny.
Post-conflict experts have got the prestige without the effort or stigma of imperialism.  Their implicit denial of the difference between cultures is the new mass brand of international intervention.  Their policy fails but no one notices.  There are no credible monitoring bodies and there is no one to take formal responsibility.  Individual offices are never in any one place and rarely in one organization long enough to be adequately assessed.  The colonial enterprise could be judged by the security or revenue it delivered, but neo-colonialists have no such performance criteria.  In fact their very uselessness benefits them.  By avoiding any serious action or judgement they, unlike their colonial predecessors, are able to escape accusations of racism, exploitation or oppression.

Rory Stewart [2004]: The Places in Between. London, UK:  Picador, p.272, footnote #59.

Poem: Joseph's Amazement

Following Michael Dransfield’s poem about conflicted love, I remembered a seasonally-appropriate poem written four centuries before:  Robert Southwell’s Joseph’s Amazement, which imagines the torment and self-questioning Mary’s husband would have felt to discover that Mary was pregnant.  Southwell moves between first and third persons to describe Joseph’s anguish, which he does not resolve, instead ending in a similar place of uncertain quandary to Dransfield.  Perhaps this lack of resolution is another reason Southwell’s poetry sounds so modern, and so fresh.

Joseph’s Amazement
When Christ, by growth, disclosed his descent
Into the pure receipt of Mary’s breast
Poor Joseph, stranger yet to God’s intent,
With doubts of jealous thoughts was sore oppressed
And, wrought with diverse fits of fear and love,
He neither can her free nor faulty prove.
Now sense, the wakeful spy of jealous mind,
By strong conjectures deemeth her defiled,
But love, in doom of things best loved blind,
Thinks rather sense deceived than her with child
Yet proofs so pregnant were that no pretence
Could cloak a thing so dear and plain to sense.
Then Joseph, daunted with a deadly wound,
Let loose the reins to undeserved grief.
His heart did throb, his eyes in tears were drowned,
His life a loss, death seemed his best relief.
The pleasing relish of his former love
In gallish thoughts to bitter taste doth prove.
One foot he often setteth forth of door
But t’other’s loath uncertain ways to tread.
He takes his fardel for his needful store,
He casts his Inn where first he means to bed.
But still ere he can frame his feet to go,
Love winneth time till all conclude in no.
Sometime, grief adding force, he doth depart.
He will, against his will, keep on his pace.
But straight remorse so racks his ruing heart,
That hasting thoughts yield to a pausing space;
Then mighty reasons press him to remain.
She whom he flies doth win him home again.
But when his thought, by sight of his abode,
Presents the sign of mis-esteemed shame,
Repenting every step that back he trod,
Tears drown the guides; the tongue, the feet doth blame.
Thus warring with himself a field he fights,
Where every wound upon the giver lights.
“And was my love,” quoth he, “so lightly prized?
Or was our sacred league so soon forgot?
Could vows be void, could virtues be despised?
Could such a spouse be stained with such a spot?”
O wretched Joseph that hast lived so long,
Of faithful love to reap so grievous wrong.
Could such a worm breed in so sweet a wood?
Could in so chaste demeanour lurk untruth?
Could vice lie hid where virtue’s image stood?
Where hoary sageness graced tender youth?
Where can affiance rest to rest secure?
In virtue’s fairest seat faith is not sure.
All proofs did promise hope, a pledge of grace,
Whose good might have repaid the deepest ill.
Sweet signs of purest thoughts in saintly face
Assured the eye of her unstained will.
Yet in this seeming lustre seem to lie
Such crimes for which the law condemns to die.
But Joseph’s word shall never work her woe:
“I wish her leave to live, not doom to die.
Though fortune mine, yet am I not her foe,
She to herself less loving is than I.
The most I will, the lest I can, is this,
Sith none may salve, to shun that is amiss.
Exile my home, the wilds shall be my walk,
Complaints my joy, my music mourning lays,
With pensive griefs in silence will I talk;
Sad thoughts shall be my guides in sorrow’s ways.
This course best suits the care of cureless mind,
That seeks to lose what most it joyed to find.
Like stocked tree whose branches all do fade,
Whose leaves do fall, and perished fruit decay,
Like herb that grows in cold and barren shade,
Where darkness drives all quick’ning heat away,
So must I die, cut from my root of joy,
And thrown in darkest shades of deep annoy.
But who can fly from that his heart doth feel?
What change of place can change implanted pain?
Removing moves no hardness from the steel.
Sick hearts that shift no fits, shift rooms in vain.
Where thought can see, what helps the closed eye?
Where heart pursues, what gains the foot to fly?
Yet still I tread a maze of doubtful end.
I go, I come, she draws, she drives away,
She wounds, she heals, she doth both mar and mend,
She makes me seek and shun, depart and stay.
She is a friend to love, a foe to loathe,
And in suspense I hang between them both.”

Notes and Reference:
A fardel is a package.  Affiance is a binding marriage pledge.  I have modernized the spelling and added punctuation.   Previous poems by Robert Southwell are here and here.
Robert Southwell [2007]: Collected Poems. Edited by Peter Davidson and Anne Sweeney. Manchester, UK: Fyfield Books, pp. 19-21.

Film: The New World

I am a great fan of the films of Terence Malick, and so I was delighted to read John Patterson’s recent article proclaiming Malick’s The New World as the single film masterpiece of the decade just ending.

It may seem like an exaggeration, but with The New World cinema has reached its culmination, its apotheosis. It is both ancient and modern, cinema at its purest and most organic, its simplest and most refined, made with much the same tools as were available in the infancy of the form a century ago to the Lumières, to Griffith and Murnau. Barring a few adjustments for modernity – colour, sound, developments in editing, a hyper-cine-literate audience – it could conceivably have been made 80 years ago (like Murnau and Flaherty’s Tabu). This is why, I believe, when all the middlebrow Oscar-dross of our time has eroded away to its constituent molecules of celluloid, The New World will stand tall, isolated and magnificent, like Kubrick’s black monolith. Anything else that survives from now till then will by comparison probably resemble 2001’s grunting apes. To quote, simultaneously, Godard’s Pierrot le Fou and primitivist auteur Sam Fuller – whose 1957 western Run of the Arrow is a sort of thematic inbred bastard cousin of The New World – Malick is seeking “in a word: emotion!”
Continue reading ‘Film: The New World’

Bridget Riley on drawing as thinking

Ealier, I quoted Marion Milner on the zen of sunday-painting.   The British artist, Bridget Riley, writing for a catalog that accompanied a retrospective of her work presented recently at the Walker Gallery in Liverpool, UK, talks about the non-propositional thinking involved in drawing and painting, particularly during the exploration that she undertakes as she begins each new work.

For me, drawing is an inquiry, a way of finding out – the first thing that I discover is that I do not know. This is alarming even to the point of momentary panic. Only experience reassures me that this encounter with my own ignorance – with the unknown – is my chosen and particular task, and provided I can make the required effort the rewards may reach the unimaginable. It is as though there is an eye at the end of my pencil, which tries, independently of my personal general-purpose eye, to penetrate a kind of obscuring veil or thickness. To break down this thickness, this deadening opacity, to elicit some particle of clarity or insight, is what I want to do.

The strange thing is that the information I am looking for is, of course, there all the time and as present to one’s naked eye, so to speak, as it ever will be. But to get the essentials down there on my sheet of paper so that I can recover and see again what I have just seen, that is what I have to push towards. What it amounts to is that while drawing I am watching and simultaneously recording myself looking, discovering things that on the one hand are staring me in the face and on the other I have not yet really seen. It is this effort ‘to clarify’ that makes drawing particularly useful and it is in this way that I assimilate experience and find new ground. (p. 15)

. . .
You cannot deal with thought directly outside practice as a painter: ‘doing’ is essential in order to find out what form your thought takes. The ‘new curves’ that I started in 1998 grew directly out of paintings such as Shimmered Shade. The latent visual arcs and sweeping movements came to the fore in Painting with Verticals 1 (2006) and Red with Red 1 (2007). Retaining the diagonals and verticals of the earlier group of paintings, I introduced a curve that connected to the existing structure. This is the underpinning of my new curvilinear work. The vertical is still there, acting like a break in the movement across the canvas. The cut collage pieces define the various contours that arise from combining and recombining the slender curve with its diagonal accents. This has developed into a layering technique that allows me to weave forms and colours together in a supple plastic space. I have reduced the number of colours and increased the scale of the imagery. Would it be possible to once again build up a repertoire of these invented forms, a repertoire that might gradually acquire sufficient momentum to put itself at risk, to precipitate its own kind of hazard? It is only through the experience of working that answers may be discovered within the inner logic of an invented reality such as the art of painting.” (p. 18)

The image is Red with Red by Bridget Riley, 2007.
Bridget Riley [2009]: Work.  pp. 15-18 of:   Michael Bracewell and Bridget Riley [2009]: Bridget Riley Flashback.  London, UK:  Hayward Publishing.
This essay was republished in The London Review of Books (31 (19): 20-21, 8 October 2009) and is online here.

Political activists of renown

Recently, I have listed the teachers and writers who have influenced me, along with the managers whom I admire.  I now list the politicians and political activists whom I admire.  Some of these led conventional political careers, others were community organizers or single-issue advocates, and yet others were spies, or were accused of being such.

Edmund Campion, Robert Persons, Robert Southwell, Thomas Aikenhead, Tom Paine, Abe Lincoln, Teddy Roosevelt, Solomon Plaatje, Franklin Roosevelt, Ted Theodore, John Curtin, Doc Evatt, Richard Sorge, Imre Nagy, Zhou Enlai, Milada Horakova, Bram Fischer, Salvador Allende Gossens, Lyndon Johnson, Donal Lamont, Rudolf Margolius, Gough Whitlam, Helen Suzman, Andrei Sakharov, Alexander Dubcek, Nelson Mandela, Zhao Ziyang, Martin Luther King Jr, Zdenek Mlynar, Mikhail Gorbachev, Vaclav Havel, Michael Schneider, Bella Subbotovskaya, Paul Keating, Vadim Delone, Jes Albert Möller, Barack Obama and Rory Stewart.

Australia (5), Czechoslovakia (5), and South Africa (5) have produced more than their per capita share of political heroes, it would seem, but the distribution no doubt reflects my reading and interests.  Of course, it hardly needs to be said that I do not necessarily agree with any or all the views these people have expressed or hold, nor necessarily support all their actions.

Poem: Pas de deux for Lovers

While quoting poetry by Michael Dransfield, here is another of his fine poems, along with Caspar David Friedrich’s painting, Cape Arkona at Sunrise (1803):

“Pas de deux for Lovers”
To wake
and go
would be so simple.
Morning ought not
to be complex.
The sun is a seed
cast at dawn into the long
furrow of history.
how the
first light
makes gold her hair
upon my arm.
How then
shall I leave,
and where away to go. Day
is so deep already with involvement.

Thomas W. Shapcott (Editor) [1970]:  Australian Poetry Now. Melbourne, Australia:  Sun Books.

Research funding myopia

The British Government, through its higher education funding council, is currently considering the use of socio-economic impact factors when deciding the relative rankings of university departments in terms of their research quality, the Research Assessment Exercise (RAE), held about every five years.   These impact factors are intended to measure the social or economic impact of research activities in the period of the RAE (ie, within 5 years). Since the RAE is used to allocate funds for research infrastructure to British universities these impact factors, if implemented, will thus indirectly decide which research groups and which research will be funded.    Some academic reactions to these proposals are here and here.
From the perspective of the national economy and technological progress, these proposals are extremely misguided, and should be opposed by us all.    They demonstrate a profound ignorance of where important ideas come from, of when and where and how they are applied, and of where they end up.  In particular, they demonstrate great ignorance of the multi-disciplinary nature of most socio-economically-impactful research.
One example will demonstrate this vividly.  As more human activities move online, more tasks can be automated or semi-automated.    To enable this, autonomous computers and other machines need to be able to communicate with one using shared languages and protocols, and thus much research effort in Computer Science and Artificial Intelligence these last three decades has focused on designing languages and protocols for computer-to-computer communications.  These protocols are used in various computer systems already and are likely to be used in future-generation mobile communications and e-commerce systems.
Despite its deep technological nature, research in this area draws fundamentally on past research and ideas from the Humanities, including:

  • Speech Act Theory in the Philosophy of Language (ideas due originally to Adolf Reinach 1913, John Austin 1955, John Searle 1969 and Jurgen Habermas 1981, among others)
  • Formal Logic (George Boole 1854, Clarence Lewis 1910, Ludwig Wittgenstein 1922, Alfred Tarski 1933, Saul Kripke 1959, Jaakko Hintikka 1962, etc), and
  • Argumentation Theory (Aristotle c. 350 BC, Stephen Toulmin 1958, Charles Hamblin 1970, etc).

Assessment of the impacts of research over five years is laughable when Aristotle’s work on rhetoric has taken 2300 years to find technological application.   Even Boole’s algebra took 84 years from its creation to its application in the design of electronic circuits (by Claude Shannon in 1938).  None of the humanities scholars responsible were doing their research to promote technologies for computer interaction or to support e-commerce, and most would not have even understood what these terms mean.  Of the people I have listed, only John Searle (who contributed to the theory of AI), and Charles Hamblin (who created one of the first computer languages, GEORGE, and who made major contributions to the architecture of early computers, including invention of the memory stack), had any direct connection to computing.   Only Hamblin was afforded an obituary by a computer journal (Allen 1985).
None of the applications of these ideas to computer science were predicted, or even predictable.  If we do not fund pure research across all academic disciplines without regard to its potential socio-economic impacts, we risk destroying the very source of the ideas upon which our modern society and our technological progress depend.
M. W. Allen [1985]: “Charles Hamblin (1922-1985)”. The Australian Computer Journal, 17(4): 194-195.

Heroes: the underground railroad in Rhodesia

Talking about Zimbabwean history reminded me that there are some unsung heroes of Zimbabwe’s struggle for majority rule whom I wish to salute.   These are the people who, rejecting the racist policies of the Rhodesian Front government, organized an illegal underground railroad to secretly transport black and white resisters across the border, mostly to Botswana and Zambia.   The whites transported were usually resisting military conscription to fight in a war they disagreed with, a war in support of a cause they believed immoral.  I knew a couple of these railwaymen:  AP (“Knotty”) Knottenbelt, who had been headmaster of Fletcher High School, a state boarding school for black boys, from where he resigned in 1969 rather than raise a Rhodesian flag; he is said to have tied the flag to the back of his car and driven it through the dust of the schoolyard in front of the assembled students before hoisting it; he later tutored at the University of Zimbabwe, and the Mugabe Government appointed him to the board of the Posts and Telecommunications Corporation after Independence.     Another railwayman was his bridge partner, Nick Holman (1919-2002), father of the (now former) Financial Times Africa Editor, Michael Holman.   These men and their collaborators deserve praise and admiration for their great personal courage in support of a non-racial society.

One of those transported by this railroad was the late Christopher Lewis, son, grandson, and great-grandson of Rhodes Scholars.  His father, Charles Patrick Jameson (“Pat”) Lewis (d. 1975) was a lawyer in partnership with Hardwicke Holderness MP (1915-2007), and Chairman from 1961-1969 of the Constitutional Council established under the 1961 Rhodesian constitution; Christopher Lewis’s paternal great-grandfather Vernon Lewis CMG (d. 1950), was a Rhodes Scholar later appointed in 1950 Chief Justice of Southern Rhodesia (being succeeded on his death the same year by Sir Robert Tredgold);  Vernon Lewis was married to Ethel Amy Jameson, daughter of Leander Starr Jameson (1853-1917), Prime Minister of the Cape Colony between 1904-1908, who had led a failed attack against the Transvaal in 1895-1896 (later called the Jameson Raid).  Another son of Vernon Lewis, John Vernon Radcliffe Lewis (1917-?), was also a Rhodesian and Zimbabwean judge.

Christopher Lewis’s maternal grandfather Leonard Ray Morgan (1894-1967), also a Rhodes Scholar, was a lifelong friend of Robert Graves whom he met at Oxford, and was Permanent Secretary for Education in the Federation of Rhodesia and Nyasaland; and Christopher’s uncle by marriage was the chiShona linguist, George Fortune (1915-2012).  Ian Hancock’s interesting history of liberal white opposition to the racist policies of the Rhodesia Front is dedicated to the memory of Pat Lewis. Christopher’s sister Annette was married to lawyer Anthony Eastwood (1940-2015), whose first wife Ruth Fischer (later Ruth Fischer-Rice) (b. 1939) was the daughter of Bram Fischer (1908-1975), lead defence counsel at the Rivonia Trial of Nelson Mandela and others. Bram Fischer was the son of a Judge-President of the Orange Free State and grandson of the only Prime Minister of the Orange River Colony, Abraham Fischer (Prime Minister 1907-1910); his wife Molly was a niece of Jan Smuts.

Well before the fall of communism, Anthony Eastwood once told me of visiting the USSR as an honoured guest and asking, as a lawyer, if he could meet a fellow lawyer.  The next day he was ushered into a meeting with the Chief Justice of the Supreme Court of the USSR.

A wonderfully well-written but very sad memoir by Hayden Eastwood, son of Anthony and Annette Eastwood, of his upbringing was published in 2018.


Hayden Eastwood [2018]: Like Sodium in Water: A Memoir of Home and Heartache. Cape Town, RSA: Jonathan Ball.

Ian Hancock [1984]: White Liberals, Moderates and Radicals in Rhodesia 1953-1980. New York, USA: St Martin’s Press.