On the categories email list on 5 March 2006, Ronald Brown quoted the following paragraph on mathematical speculation from a 14 June 1983 letter he had received from Alexander Grothendieck:
Archive for the 'Creativity' Category
An excerpt from a 1959 Australian Broadcasting Commission TV programme on the Beats, featuring interviews with Sydney University students, Clive James and Robert Hughes (pictured, image from ABC).
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 : The Works of Sir Thomas Browne (Editor: G. Keynes), Volume 2. London.
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- )
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 : Structure: Beyond the picnic-table crisis. The New Yorker, 14 January 2103, pages 46-55.
William F. Thurston : On proof and progress in mathematics. American Mathematical Society, 30 (2): 161-177.
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.
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.