Archive for the ‘Creativity’ Category

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Mathematical thinking and software

Published by peter on 2013-01-22 in Creativity, Hands, Human intelligence and Mathematics. 0 Comments

Further to my post citing Keith Devlin on the difficulties of doing mathematics online, I have heard from one prominent mathematician that he does all his mathematics now using LaTeX, not using paper or whiteboard, and thus disagrees with Devlin’s (and my) views.   Thinking about why this may be, and about my own experiences using LaTeX, it occurred to me that one’s experiences with thinking-support software, such as word-processing packages such as MS-WORD or  mark-up programming languages such as LaTeX, will very much depend on the TYPE of thinking one is doing.
If one is thinking with words and text, or text-like symbols such as algebra, the right-handed folk among us are likely to be using the left hemispheres of our brains.  If one is thinking in diagrams, as in geometry or graph theory or much of engineering including computing, the right-handed among us are more likely to be using the right hemispheres of our brains.  Yet MS-WORD and LaTeX are entirely text-based, and their use requires the heavy involvement of our left hemispheres (for the northpaws among us).  One doesn’t draw an arrow in LaTeX, for example, but instead types a command such as \rightarrow or \uparrow.   If one is already using one’s left hemisphere to do the mathematical thinking, as most algebraists would be, then the cognitive load in using the software will be a lot less then if one is using one’s right hemisphere for the mathematical thinking.  Activities which require both hemispheres are typically very challenging to most of us, since co-ordination between the two hemispheres adds further cognitive overhead.
I find LaTeX immeasurably better than any other word-processor for writing text:  it and I work at the same speed (which is not true of MS-WORD for me, for example), and I am able to do my verbal thinking in it.  In this case, writing is a form of thinking, not merely the subsequent expression of thoughts I’ve already had.     However, I cannot do my mathematical or formal thinking in LaTeX, and the software is at best a tool for subsequent expression of thoughts already done elsewhere – mentally, on paper, or on a whiteboard.    My formal thinking is usually about structure and relationship, and not as often algebraic symbol manipulation.
Bill Thurston, the geometer I recently quoted, said:

I was interested in geometric areas of mathematics, where it is often pretty hard to have a document that reflects well the way people actually think.  In more algebraic or symbolic fields, this is not necessarily so, and I have the impression that in some areas documents are much closer to carrying the life of the field.”  [Thurston 1994, p. 169]

It is interesting that many non-mathematical writers also do their thinking about structure not in the document itself or as they write, but outside it and beforehand, and often using tools such as post-it notes on boards; see the recent  article by John McPhee in The New Yorker for examples from his long writing life.
References:
John McPhee [2013]: Structure:  Beyond the picnic-table crisis.  The New Yorker, 14 January 2013, pages 46-55.
William F. Thurston [1994]:  On proof and progress in mathematics.  American Mathematical Society, 30 (2):  161-177.

Mathematical hands

Published by peter on 2013-01-16 in Computer technology, Creativity, Design, Hands, Human intelligence and Mathematics. 0 Comments

With MOOCs fast becoming teaching trend-du-jour in western universities, it is easy to imagine that all disciplines and all ways of thinking are equally amenable to information technology.   This is simply not true, and mathematical thinking  in particular requires hand-written drawing and symbolic manipulation.   Nobody ever acquired skill in a mathematical discipline without doing exercises and problems him or herself, writing on paper or a board with his or her own hands.   The physical manipulation by the hand holding the pen or pencil is necessary to gain facility in the mental manipulation of the mathematical concepts and their relationships.
Keith Devlin recounts his recent experience teaching a MOOC course on mathematics, and the deleterious use by students of the word-processing package latex for doing assignments:

We have, it seems, become so accustomed to working on a keyboard, and generating nicely laid out pages, we are rapidly losing, if indeed we have not already lost, the habit—and love—of scribbling with paper and pencil. Our presentation technologies encourage form over substance. But if (free-form) scribbling goes away, then I think mathematics goes with it. You simply cannot do original mathematics at a keyboard. The cognitive load is too great.

Why is this?  A key reason is that current mathematics-producing software is clunky, cumbersome, finicky, and not WYSIWYG (What You See Is What You Get).   The most widely used such software is Latex (and its relatives), which is a mark-up and command language; when compiled, these commands generate mathematical symbols.   Using Latex does not involve direct manipulation of the symbols, but only their indirect manipulation.   One has first to imagine (or indeed, draw by hand!) the desired symbols or mathematical notation for which one then creates using the appropriate generative Latex commands.   Only when these commands are compiled can the user see the effects they intended to produce.   Facility with pen-and-paper, by contrast, enables direct manipulation of symbols, with (eventually), the pen-in-hand being experienced as an extension of the user’s physical body and mind, and not as something other.   Expert musicians, archers, surgeons, jewellers, and craftsmen often have the same experience with their particular instruments, feeling them to be extensions of their own body and not external tools.
Experienced writers too can feel this way about their use of a keyboard, but language processing software is generally WYSIWYG (or close enough not to matter).  Mathematics-making software  is a long way from allowing the user to feel that they are directly manipulating the symbols in their head, as a pen-in-hand mathematician feels.  Without direct manipulation, hand and mind are not doing the same thing at the same time, and thus – a fortiori – keyboard-in-hand is certainly not simultaneously manipulating concept-in-mind, and nor is keyboard-in-hand simultaneously expressing or evoking concept-in-mind.
I am sure that a major source of the problem here is that too many people – and especially most of the chattering classes – mistakenly believe the only form of thinking is verbal manipulation.  Even worse, some philosophers believe that one can only think by means of words.     Related posts on drawing-as-a-form-of-thinking here, and on music-as-a-form-of-thinking here.
[HT:  Normblog]

Imaginary beliefs

Published by peter on 2012-08-21 in Creativity, Forecasting, Mathematics, Planning, Religion and Uncertainty. 0 Comments

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.

Australian improv comedy pre-history

Published by peter on 2011-05-28 in Creativity, History and Matherati. 0 Comments

My father saw a young Melbourne comedian named Barry Humphries try out an act as an ordinary Moonee Ponds housewife in a Review at the Phillip Street Theatre in Sydney in about 1955.   He and 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 blonde hair.

Drawing as thinking, part 2

Published by peter on 2011-01-10 in Art, Creativity, Hands and Human intelligence. 0 Comments

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.

The Matherati

Published by peter on 2010-08-22 in Computer Science, Creativity, History, Human intelligence, Logic, Mathematics, Matherati and Obituaries. 0 Comments

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 writer 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.
On this page you will find an index to Vukutu posts about 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.

Hand-mind-eye co-ordination

Published by peter on 2010-08-18 in Art, Creativity, Hands, Human intelligence and Music. Closed

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 skills involved in even the simplest acts of seeing, skills which most of us learn as young 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.

Berger on drawing

Published by peter on 2010-07-28 in Art, Creativity, Design, Hands and Human intelligence. 0 Comments

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.