But there is definitely a cast of mind we might call mathmind.   As well as the usual suspects, such as Euclid, Newton and Einstein, there are many others with this cast of mind.  For example, Thomas Harriott (c. 1560-1621), inventor of the less-than symbol, and the first person to draw the  moon with a telescope was one.   Newton’s friend, Nicolas Fatio de Duiller (1664-1753), was another.   In the talented 18th-century family of Charles Burney, whose relatives and children included musicians, dancers, artists, and writers (and an admiral), Charles’ grandson, Alexander d’Arblay (1794-1837), the son of Fanny Burney, was 10th wrangler in the Mathematics Tripos at Cambridge in 1818, and played chess to a high standard.  He was friends with Charles Babbage, also a student at Cambridge at the time, and a member of the Analytical Society which Babbage had co-founded; this was an attempt to modernize the teaching of pure mathematics in Britain by importing the rigor and notation of continental analysis, which d’Arblay had already encountered as a school student in France.

And there are people with mathmind right up to the present day.   The Guardian a year ago carried an obituary, written by a family member, of Joan Burchardt, who was described as follows:

My aunt, Joan Burchardt, who has died aged 91, had a full and interesting life as an aircraft engineer, a teacher of physics and maths, an amateur astronomer, goat farmer and volunteer for Oxfam. If you had heard her talking over the gate of her smallholding near Sherborne, Dorset, you might have thought she was a figure from the past. In fact, if she represented anything, it was the modern, independent-minded energy and intelligence of England. In her 80s she mastered the latest computer software coding.”

Since language and text have dominated modern Western culture these last few centuries, our culture’s histories are mostly written in words.   These histories favor the literate, who naturally tend to write about each other.    Clive James’ book of a lifetime’s reading and thinking, Cultural Amnesia (2007), for instance, lists just 1 musician and 1 film-maker in his 126 profiles, and includes not a single mathematician or scientist.     It is testimony to text’s continuing dominance in our culture, despite our society’s deep-seated, long-standing reliance on sophisticated technology and engineering, that we do not celebrate more the matherati.

FOOTNOTE: The image above shows the equivalence classes of directed homotopy (or, dihomotopy) paths in 2-dimensional spaces with two holes (shown as the black and white boxes). The two diagrams model situations where there are two alternative courses of action (eg, two possible directions) represented respectively by the horizontal and vertical axes.  The paths on each diagram correspond to different choices of interleaving of these two types of actions.  The word directed is used because actions happen in sequence, represented by movement from the lower left of each diagram to the upper right.  The word homotopy refers to paths which can be smoothly deformed into one another without crossing one of the holes.  The upper diagram shows there are just two classes of dihomotopically-equivalent paths from lower-left to upper-right, while the lower diagram (where the holes are positioned differently) has three such dihomotopic equivalence classes.  Of course, depending on the precise definitions of action combinations, the upper diagram may in fact reveal four equivalence classes, if paths that first skirt above the black hole and then beneath the white one (or vice versa) are permitted.  Applications of these ideas occur in concurrency theory in computer science and in theoretical physics.

Technorati Tags: Thomas Harriott, Nicolas Fatio de Duiller, Charles Burney, Alexander d’Arblay, Fanny Burney, Charles Babbage, matherati

A drawing of a tree shows, not a tree, but a tree-being-looked-at.  . . .  Within the instant of the sight of a tree is established a life-experience.” (page 71)

Berger asserts that we do not draw the objects our eyes seem to look at.  Rather, we draw some representation, processed through our mind and through our drawing arm and hand, of that which our minds have seen.  And that which our mind has seen is itself a representation (created by mental processing that includes processing by our visual processing apparatus) of what our eyes have seen.    Neurologist Oliver  Sacks, writing about a blind man who had his sight restored and was unable to understand what he saw, has written movingly about the sophisticated visual processing involved in even the simplest acts of seeing, which most of us learn as children (Sacks 1993).

So a drawing of a tree is certainly not itself a tree, and not even a direct, two-dimensional representation of a tree, but a two-dimensional hand-processed manifestation of a visually-processed mental manifestation of a tree.   Indeed, perhaps not even always this, as Marion Milner has reminded us:    A drawing of a tree is in fact a two-dimensional representation of the process of manifesting through hand-drawing a mental representation of a tree.  Is it any wonder, then, that painted trees may look as distinctive and awe-inspiring as those of Caspar David Friedrich (shown above) or Katie Allen?

As it happens, we still know very little, scientifically, about the internal mental representations that our minds have of our bodies.  Recent research, by Matthew Longo and Patrick Hazzard, suggests that, on average, our mental representations of our own hands are inaccurate.   It would be interesting to see if the same distortions are true of people whose work or avocation requires them to finely-control their hand movements:  for example, jewellers, string players, pianists, guitarists, surgeons, snooker-players.   Do virtuoso trumpeters, capable of double-, triple- or even quadruple-tonguing, have sophisticated mental representations of their tongues?  Do crippled artists who learn to paint holding a brush with their toes or in their mouth acquire sophisticated and more-accurate mental representations of these organs, too?  I would expect so.

These thoughts come to mind as I try to imitate the sound of a baroque violin bow by holding a modern bow higher  up the bow.   By thus changing the position of my hand, my playing changes dramatically, along with my sense of control or power over the bow, as well as the sounds it produces.

References:

John Berger [2005]:  Berger on Drawing.  Edited by Jim Savage.  Aghabullogue, Co. Cork, Eire:  Occasional Press.  Second Edition, 2007.

Matthew Longo and Patrick Haggard [2010]: An implicit body representation underlying human position sense. Proceedings of the National Academy of Sciences, USA, 107: 11727-11732.  Available here.

Marion Milner (Joanna Field) [1950]: On Not Being Able to Paint. London, UK:  William Heinemann.  Second edition, 1957.

Oliver Sacks [1993]:  To see and not see.  The New Yorker, 10 May 1993.

Technorati Tags: John Berger, Marion Milner, Marion Milner, Oliver Sacks

• “For the artist drawing is discovery.  And that is not just a slick phrase, it is quite literally true.  It is the actual act of drawing that forces the artist to look at the object in front of him, to dissect it in his mind’s eye and put it together again; or, if he is drawing from memory, that forces him to dredge his own mind, to discover the content of his own store of past observations.” (page 3)
• “It is a platitude in the teaching of drawing that the heart of the matter lies in the specific process of looking.  A line, an area of tone, is not really important because it records what you have seen, but because of what it will lead you on to see.  Following up its logic in order to check its accuracy, you find confirmation or denial in the object itself or in your memory of it.  Each confirmation or denial brings you closer to the object, until finally you are, as it were, inside it:  the contours you have drawn no longer marking the edge of what you have seen, but the edge of what you have become.  Perhaps that sounds needlessly metaphysical.  Another way of putting it would be to say that each mark you make on the paper is a stepping-stone from which you proceed to the next, until you have crossed your subjecct as though it were a river, have put it behind you.” (page 3)
• “A drawing is an autobiographical record of one’s discovery of an event – seen, remembered or imagined.” (page 3)
• “A drawing of a tree shows, not a tree, but a tree-being-looked-at.  . . .  Within the instant of the sight of a tree is established a life-experience.” (page 71)
• “All genuine art approaches something which is eloquent but which we cannot altogether understand.  Eloquent because it touches something fundamental.  How do we know?  We do not know.  We simply recognize.”   (page 80)
• “Art cannot be used to explain the mysterious.  What art does is to make it easier to notice. Art uncovers the mysterious. And when noticed and uncovered, it becomes more mysterious.”  (page 80)
• “The pen with which I’m writing is the one with which I draw.  And there are times, like tonight, when it won’t flow and when it demands a bath or a hand moving differently.  All drawings are a collaboration, like most circus-acts.” (page 110)
• “where are we, during the act of drawing, in spirit?  Where are you at such moments – moments which add up to so many, one might think of them as another life-time?    Each pictorial tradition offers a different answer to this query.  For instance, the European tradition, since the Renaissance, places the model over there, the draughtsman here, and the paper somewhere in between, within arms reach of the draughtsman, who observes the model and notes down what he has observed on the paper in front of him.   The Chinese tradition arranges things differently.  Calligraphy, the trace of things, is behind the model and the draughtsman has to search for it, looking through the model.   On his paper he then repeats the gestures he has seen calligraphically.  For the Paleolithic shaman, drawing inside a cave, it was different again.  The model and the drawing surface were in the same place, calling to the draughtsman to come and meet them, and then trace, with his hand on the rock, their presence.” (page 123)

Reference:

John Berger [2005]:  Berger on Drawing.  Edited by Jim Savage.  Aghabullogue, Co. Cork, Eire:  Occasional Press.  Second Edition, 2007.

Technorati Tags: John Berger

Brian Dillon reviews a British touring exhibition of the art of John Cage, currently at the Baltic Mill Gateshead.

Two quibbles:  First, someone who compare’s Cage’s 4′ 33” to a blank gallery wall hasn’t actually listened to the piece.  If Dillon had compared it to a glass window in the gallery wall allowing a view of the outside of the gallery, then he would have made some sense.  But Cage’s composition is not about silence, or even pure sound, for either of which a blank gallery wall might be an appropriate visual representation.  The composition is about ambient sound, and about what sounds count as music in our culture.

Second, Dillon rightly mentions that the procedures used by Cage for musical composition from 1950 onwards (and later for poetry and visual art) were based on the Taoist I Ching.  But he wrongly describes these procedures as being based on “the philosophy of chance.”     Although widespread, this view is nonsense, accurate neither as to what Cage was doing, nor even as to what he may have thought he was doing.   Anyone subscribing to the Taoist philosophy underlying them understands the I Ching procedures as examplifying and manifesting hidden causal mechanisms, not chance.   The point of the underlying philosophy is that the random-looking events that result from the procedures express something unique, time-dependent, and personal to the specific person invoking the I Ching at the particular time they invoke it. So, to a Taoist, the resulting music or art is not “chance” or “random” or “aleatoric” at all, but profoundly deterministic, being the necessary consequential expression of deep, synchronistic, spiritual forces. I don’t know if Cage was himself a Taoist (I’m not sure that anyone does), but to an adherent of Taoist philosophy Cage’s own beliefs or attitudes are irrelevant to the workings of these forces.  I sense that Cage had sufficient understanding of Taoist and Zen ideas (Zen being the Japanese version of Taoism) to recognize this particular feature:  that to an adherent of the philosophy the beliefs of the invoker of the procedures are irrelevant.

In my experience, the idea that the I Ching is a deterministic process is a hard one for many modern westerners to understand, let alone to accept, so entrenched is the prevailing western view that the material realm is all there is.  This entrenched view is only historically recent in the west:  Isaac Newton, for example, was a believer in the existence of cosmic spiritual forces, and thought he had found the laws which governed their operation.    Obversely, many easterners in my experience have difficulty with notions of uncertainty and chance; if all events are subject to hidden causal forces, the concepts of randomness and of alternative possible futures make no sense.  My experience here includes making presentations and leading discussions on scenario analyses with senior managers of Asian multinationals.  

We are two birds swimming, each circling the pond, warily, neither understanding the other, neither flying away.

References:

Kyle Gann [2010]: No Such Thing as Silence.  John Cage’s 4′ 33”.  New Haven, CT, USA:  Yale University Press. 

James Pritchett [1993]:  The Music of John Cage.  Cambridge, UK:  Cambridge University Press.

Technorati Tags: John Cage, I Ching, Zen, Taoism

But there is another explanation of  all our dreams, not only those which cause us angst.  That explanation is that our dreams are just random images flashed before us by some mechanical process in our brain.  Here there is no continuous film, no coherent plot, no themes, no actors, no film-director, and the projectionist is outside having a cigarette while images are being loaded automatically by a random reel selector that management installed to save on staff.   We, however, are not outside.  We are sitting down in the front-row of the stalls of the cinema, being the audience for the film. So its no wonder we are surprised by what we see.   We try our best, both then and after waking, to make sense of the images that flash past us, looking for some narrative coherence.  If we have anxieties, this is when they appear, in our attempts at reconstruction of  a plot or a theme or some identifiable characters.   We are indeed the authors of our dreams, but only in the way that texts are written by their readers, and not their writers.

Technorati Tags: dreams

I was thinking recently about concert performances I have attended where the composer was present, or rather, where I knew the composer to be present.  Here is my list, as best I remember it:

Don Banks (1923-1980)
Richard Rodney Bennett
Pierre Boulez
Palle Dahlstedt
Rolf Hind
Robin Holloway
Keith Humble (1927-1995)
Gerard McBurney
Stephen Montague
Nico Muhly
Loretta Notareschi
Jim Penberthy (1917-1999)
Behzad Ranjbaran
Peter Sculthorpe
Toru Takemitsu (1930-1996)
David Urquhart-Jones
James Wishart
Iannis Xenakis (1922-2001)

Of course, my presence at a performance does not constitute an endorsement of the music performed:  some of the music of these composers I like or appreciate very much, and some I think is unpleasant, boring or otherwise of low quality.   Although I generally prefer downtown and minimalist music, the music of the composers listed here also includes neo-romanticism (eg, Holloway, late Sculthorpe), abstract expressionism (eg, Penberthy, early Sculthorpe, Takemitsu), and uptown complexity (eg, Boulez, Muhly, Xenakis).  And, I have not included in this list jazz performers, who almost always play some of their own compositions.

For Peter Sculthorpe, one occasion (of several where we have both been present) was a performance near Patonga, Broken Bay, Sydney, of his profound and achingly-beautiful Sun Music III, in which I had the great good fortune, as second percussionist, to play the guiro (pictured).    One has to wonder how the same person could compose the innovative Sun Music series of the 1960s and also the derivative, warmed-over, late-romantic tosh that Sculthorpe has written in recent years.  Bill Burroughs would have seen it as a clear case of spirit possession.

Technorati Tags: Peter Sculthorpe, guiro, Sun Music

Afterwards, my husband and I reminisced about our attempts to learn tennis when we were young. I told him that my sisters and I used to go down to the public tennis courts in Portobello. We had probably never seen a professional tennis match; we just knew that tennis was about hitting the ball to and fro across the net. We had a few lessons and became quite good at leisurely rallies, hitting the ball back and forth without any attempt at speed. Sometimes we could keep our rallies going for quite a long time, and I found this enjoyable.

Then our tennis teacher explained that we should now learn to play “properly”. It was only then that I realised we were meant to hit the ball in such a way that the other person could not hit it back. This came as an unpleasant surprise. As soon as we started “playing properly”, our points became extremely short. One person served, the other could not hit it back, and that was the end of the point. It seemed to me that there was skill in hitting the ball so that the other person could hit it back. If they could, the ball would flow, one got to move about and there was not much interruption to the rhythm of play. It struck me that hitting the ball deliberately out of the other person’s reach was unsportsmanlike. When I tell my husband all this, he laughs and says: “There speaks a true chamber musician.”

This story resonated strongly with me.  Earlier this year, I had a brief correspondence with mathematician Alexandre Borovik, who has been collecting accounts of childhood experiences of learning mathematics, both from mathematicians and from non-mathematicians.  After seeing a discussion on his blog about the roles of puzzles and games in teaching mathematics to children, I had written to him:

Part of my anger & frustration at school was that so much of this subject that I loved, mathematics, was wasted on what I thought was frivolous or immoral applications:   frivolous because of all those unrealistic puzzles, and immoral because of the emphasis on competition (Olympiads, chess, card games, gambling, etc).   I had (and retain) a profound dislike of competition, and I don’t see why one always had to demonstrate one’s abilities by beating other people, rather than by collaborating with them.  I believed that “playing music together”, rather than “playing sport against one another”, was a better metaphor for what I wanted to do in life, and as a mathematician.

Indeed, the macho competitiveness of much of pure mathematics struck me very strongly when I was an undergraduate student:  I switched then to mathematical statistics because the teachers and students in that discipline were much less competitive towards one another.  For a long time, I thought I was alone in this view, but I have since heard the same story from other people, including some prominent mathematicians.  I know one famous category theorist who switched from analysis as a graduate student because the people there were too competitive, while the category theory people were more co-operative.

Perhaps the emphasis on puzzles & tricks is fine for some mathematicians – eg, Paul Erdos seems to have been motivated by puzzles and eager to solve particular problems.  However, it is not fine for others – Alexander Grothendieck comes to mind as someone interested in abstract frameworks rather than puzzle-solving.  Perhaps the research discipline of pure mathematics needs people of both types.  If so, this is even more reason not to eliminate all the top-down thinkers by teaching only using puzzles at school.”

More on the two cultures of mathematics here.

Technorati Tags: Paul Erdos, Alexander Grothendieck

The last time I saw him [RB], we were walking past the middle room of his house. There was a table in there with a typewriter on it.   ”Quiet,” he whispered, pushing me ahead of him into the kitchen. “My new novel’s in there. I kind of stroll in occasionally, write a quick few paragraphs, and get out before the novel knows what I’m doing.  If novels ever find out you’re writing them, you’re done for.”

Reference:

Pierre Delattre [1993]:  Episodes. St. Paul, MN, USA:  Graywolf Press.

Technorati Tags: Richard Brautigan, Pierre Delattre

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)

References:

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.

But Milner’s book is replete with some wonderful insights, many of which express a Zen sensibility.     Herewith a sample:

But if willed effort to create a “good’” picture or a “good’” person only, so far as I could see, led to something which had a counterfeit quality, surely this did not mean that one should never try to learn what a good picture or a good person was like?  It seemed rather than one must do two things.  One must certainly work at hammering out internally one’s ideal, know as far as possible what one wanted or liked. But then one must forget it, plunge into a kind of action in which the acting and the end were not separate”  [p. 92]

Also it now seemed possible to say more about where the learning of the rules did come in, in learning how to paint.  For the beginner, the chief obstacle emphasised is his lack of skill in managing the medium.  Thus he is often expected to spend a long time discovering and learning the laws of optics, finding out how certain arrangements of shapes and colours produce certain regular effects on the eye.   If one is going to be a professional painter probably this is all right.   But for the Sunday-painter I thought it was not at all all right, at least not for me; for one might [page-break] spend a lifetime of Sundays and get very little way, if one did not also how to become more used to taking the plunge, more able to throw the rules to the winds and forget the separateness of oneself and the object.   I thought of children’s drawings in this connection, how often with so little knowledge of proper methods of depicting visual experiences they can yet take the plunge and the results delight us.   Probably they can do this because the plunge itself is less of plunge to them, since they live so much of their  lives, through play, in a state where dream and external reality are fused; it is a familiar element for them, they are like birds and can live both on land and in the sky without complicated machinery to get there.” [pp. 92-93]

Thus the phrase “expression of’” suggested too much that the feeling to be expressed was there beforehand, rather than an experience developing as one made the drawing.” [p. 116]

Now another question had to be answered about the free drawings. For the fact that the ideas in them were obviously in part  determined by the circumstances of a Freudian analysis did not, I thought alter another fact; that was that they embodied a form of  knowing that traditional education of the academic kind largely ignores, and one that I myself was unaware of using - until I  began to study the drawings in detail.  But when I had done this there had been no doubt that many of the drawings did represent thinking of [page-break] some sort, reflections about the human situation, as well as the experiences with a medium.  So, the question arose, why had it not been possible to think out such ideas directly in words?  This raised a more general question of thinking in the private language of one’s own subjective images, as against thinking in the public language of words.  It also brought to the fore the problem of the academic and over-linguistic bias of traditional education.” [pp. 122-123]

Such ideas about what one might be trying to do in one’s painting pointed the way to settling certain very practical doubts I had had about the relation between painting and living.  For years I had had to decide each week-end, should I shut myself away and paint or should I just live?  It was perhaps less of a problem for the professional painter who could live in his spare time. But for the Sunday-painter it brought the need to balance up the various renunciations and gains. I had so often come away from a morning spent painting with a sense of futility, a sense of how much better it would have been to get on with something practical that really needed doing.  And I had often felt, when out painting, both exalted and yet guilty, as if I were evading something that the people round me, all busy with their daily lives, were facing, that their material was real life and mine was dreams.” [p. 135]

“There was another reason why it was now possible to paint. It was because there was one central fact that made it seem worthwhile going on, whatever the objective value of the pictures to other people. It was that I had discovered in painting a bit of experiences that made all other usual occupations unimportant by comparison.  It was the discovery that when painting something from nature there occurred, at least sometimes, a fusion into a never-before-known wholeness; not only were the object and oneself no longer felt to be separate, but neither were thought and sensation and feeling and action.  All one’s visual perceptions of colour, shape, texture, weight, as well as thought and memory, ideas about the object and action towards it, the movement of one’s hand together with the feeling of delight in the “thusness” of the thing, all seemed to fuse into a wholeness of being which was different from anything else that ever happened to me.  It was different because thought was not drowned in feeling, they were somehow all there together.  Moreover, when this state of concentration was really achieved one was no longer aware of oneself doing it, one no longer acted from a centre to an object as remote; in fact, something quite special happened to one’s sense of self.  And when the bit of painting was finished there was before one’s eyes a permanent record of the experience, giving a constant sense of immense surprise at how it had ever happened; it did not seem something that oneself had done at all, certainly not the ordinary everyday self and way of being.” [p. 142]

“The central certainty that this process [the investigation and writing of this book] does not work from purpose to deed, in the way that expedient activities do, is easy to put into words now, at the end, but was not there with effective conviction from the beginning . . . . There had been nothing in the beginning but vague uneasy feelings and an urge to follow certain trickles of curiosity wherever they might lead. All the same, I have left the introduction as it was originally written, partly because books  need introductions, partly because the fact it had seemed, retrospectively, that that was what I had set out to do from the beginning, was in itself an illustration of the later discovered truth that activity creates purpose.” [p. 145]

References:

Alfred Gell [1998]: Art and Agency:  An Anthropological Theory.  Oxford, UK: Clarendon Press.

Marion Milner (Joanna Field) [1950]: On Not Being Able to Paint. London, UK:  William Heinemann.  Second edition, 1957.

Alan Rusbridger [2002]: On not being able to play the piano. The Guardian, 2002-01-05.

Technorati Tags: Alan Rusbridger, Marion Milner