Archive for the 'Creativity' Category

London life



All Africa within us

There is all Africa and her prodigies in us; we are that bold and adventurous piece of Nature which he that studies wisely learns in a compendium what others labour at in a divided piece and endless volume  . . . There is no man alone, because every man is a microcosm and carries the whole world about him.”

Thomas Browne [1928]: The Works of Sir Thomas Browne (Editor: G. Keynes), Volume 2. London.

Broers Cafe, Utrecht


Artists concat

Here is a listing of visual artists whose work I like.    Minimalists and geometric abstractionists are over-represented, relative to their population in the world.    In due course, I will add posts about each of them.

  • Carel Fabritius (1622-1654)
  • Tao Chi (1641-1720)
  • Jin Nong (1687-c.1763)
  • Richard Wilson (1714-1782)
  • Thomas Jones (1742-1803)
  • Katsushika Hokusai (1760–1849)
  • Caspar David Friedrich (1774–1840)
  • John Sell Cotman (1782-1842)
  • Utagawa Hiroshige (1797–1858)
  • Thomas Cole (1801-1848)
  • Richard Parkes Bonington (1802–1828)
  • Thomas Chambers (1808-1869)
  • Thomas Moran (1837-1926)
  • Robert Delaunay (1885-1941)
  • Sophie Taeuber-Arp (1889-1943)
  • Ludwig Hirschfeld-Mack (1893-1965)
  • László Moholy-Nagy (1895-1946)
  • Wilhelmina Barns-Graham (1912–2004)
  • Agnes Martin (1912-2004)
  • Jackson Pollock (1912–1956)
  • Gunther Gerzso (1915-2000)
  • Michael Kidner (1917-2009)
  • Guanzhong Wu (1919–2010)
  • Carlos Cruz-Diez (1923- )
  • Fred Williams (1927-1982)
  • Donald Judd (1928-1994)
  • Sol LeWitt (1928-2007)
  • Bridget Riley (1931- )
  • Norval Morrisseau (1932–2007)
  • Dan Flavin (1933-1996)
  • Jean-Pierre Bertrand (1937- )
  • Hélio Oiticica (1937–1980)
  • Prince of Wales Midpul (c.1937-2002)
  • Peter Struycken (1939- )
  • Alighiero e Boetti (1940-1994)
  • Cildo Meireles (1948- )
  • Jeremy Annear (1949- )
  • Louise van Terheijden (1954- )
  • Doreen Reid Nakamarra (1955-2009)
  • Peter Doig (1959- )
  • Katie Allen
  • Els van ‘t Klooster (1985- )

Mathematical thinking and software

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 occured 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.  Acivities 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.


John McPhee [2013]: Structure:  Beyond the picnic-table crisisThe New Yorker, 14 January 2103, pages 46-55.

William F. Thurston [1994]:  On proof and progress in mathematicsAmerican Mathematical Society, 30 (2):  161-177.

Mathematical hands

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]

Piano on the Bay

Another colourful outdoor piano, this one by Yunior Marino, spotted at 595 Bay Street at Dundas Street, Toronto.   This afternoon some great stride piano was being played on it by an elderly gentleman.


Piano in the park

The City of London Festival’s piano-planting project in the Victoria Embankment Gardens, June 2012:

Australian improv comedy pre-history

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.

Drawing as thinking, part 2

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.


Patricia Cain [2010]:  Drawing: The Enactive Evolution of the Practitioner. Bristol, UK: Intellect.