My father saw Barry Humphries try out an act as an ordinary Moonee Ponds housewife in a Philip Street Theatre Review in Sydney in about 1955. I saw undergraduate mathematician Adam Spencer winning theatre sports improv contests at The Harold Park Hotel in about 1988. As well as being so witty that I would remember his name all this time, he also still had a full head of hair.
Archive for the 'Creativity' Category
I have posted recently on drawing, particularly on drawing as a form of thinking (here, here and here). I have now just read Patricia Cain’s superb new book on this topic, Drawing: The Enactive Evolution of the Practitioner. The author is an artist, and the book is based on her PhD thesis. She set out to understand the thinking processes used by two drawing artists, by copying their drawings. The result is a fascinating and deeply intelligent reflection on the nature of the cognitive processes (aka thinking) that take place when drawing. By copying the drawings of others, and particularly by copying their precise methods and movements, Dr Cain re-enacted their thinking. It is not for nothing that drawing has long been taught by having students copy the works of their teachers and masters – or that jazz musicians transcribe others’ solos, and students of musical composition re-figure the fugues of Bach. This is also why pure mathematicians work through famous or interesting proofs for theorems they know to be true, and why trainee software engineers reproduce the working code of others: re-enactment by the copier results in replication of the thinking of the original enactor.
In a previous post I remarked that 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: 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.
After reading Cain’s book, I realize that one could represent the process of representational drawing as a sequence of transformations, from real object, through to output image (“the drawing”), as follows (click on the image to enlarge it):
It is important to realize that the entities represented by the six boxes here are of different types. Entity #1 is some object or scene in the real physical world, and entity #6 is a drawing in the real physical world. Entities #2 and #3 are mental representations (or models) of things in the real physical world, internal to the mind of the artist. Both these are abstractions; for example, the visual model of the artist of the object may emphasize some aspects and not others, and the intended drawing may do the same. The artist may see the colours of the object, but draw only in black and white, for instance.
Entity #4 is a program, a collection of representations of atomic hand movements, which movements undertaken correctly and in the intended order, are expected to yield entity #6, the resulting drawing. Entity #4 is called a plan in Artificial Intelligence, a major part of which is concerned with the automated generation and execution of such programs. Entity #5 is a label given to the process of actually executing the plan of #4, in other words, doing the drawing.
Of course, this model is itself a simplified idealization of the transformations involved. Drawing is almost never a linear process, and the partially-realized drawings in #6 serve as continuing feedback to the artist to modify each of the other components, from #2 onwards.
References:
Patricia Cain [2010]: Drawing: The Enactive Evolution of the Practitioner. Bristol, UK: Intellect.
Howard Gardner’s theory of multiple intelligences includes an intelligence he called Logical-Mathematical Intelligence, the ability to reason about numbers, shapes and structure, to think logically and abstractly. In truth, there are several different capabilities in this broad category of intelligence – being good at pure mathematics does not necessarily make you good at abstraction, and vice versa, and so the set of great mathematicians and the set of great computer programmers, for example, are not identical.
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.
Last month, I posted some statements by John Berger on drawing. Some of these statements are profound:
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.
Related posts here, here and here.
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.
Following Bridget Riley on drawing-as-thinking, I have been reading Jim Savage’s fascinating collection of writings by John Berger on the topic of drawing. Although Berger does not say so, he is talking primarily about representational drawing – the drawing of things in the world (whether seen or remembered) or things in some imagined world – not abstract drawing. Some excerpts:
- “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 subject 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.
I have written more on the relationships between hand and mind and eye and object here.
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.
Over at Normblog, Norm is thinking about anxiety dreams, and seeks to answer the question: Who is the author of these dreams of ours? Some think it seems not to be us, since the events in the dream come as a surprise to us and trouble us. He concludes that it is the dreamer who is the author. If we think of dreams as being like films that we view in our sleep, then I assume Norm means that the author is the film-director, or perhaps the projectionist.
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.
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
Barry Conyngham
Palle Dahlstedt
David Fanshawe (1942-2010, African Sanctus, Liverpool)
Rolf Hind
Robin Holloway
Keith Humble (1927-1995)
Gerard McBurney
Olivier Messiaen (1908-1992)
Stephen Montague (Piano Concerto)
Nico Muhly (premiere of Electric Violin Concerto, performed by Thomas Gould, and premiere of opera, Two Boys, at ENO)
Loretta Notareschi
Jim Penberthy (1917-1999)
Behzad Ranjbaran (premiere of Violin Concerto, performed by Joshua Bell)
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.
Pianist and writer Susan Tomes has just published a new book, Out of Silence, which the Guardian has excerpted here. This story drew my attention:
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.
I am a great fan of the writing of Richard Brautigan, so I was delighted once to encounter a short reminiscence of Brautigan by that Zelig of the Beats, Pierre Delattre, in his fascinating memoir, Episodes (page 54):
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.