Archive for the 'Creativity' Category

Rigour and speculation in pure math

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:

Your idea of writing a “frantically speculative” article on groupoids seems to me a very good one.  It is the kind of thing which has traditionally been lacking in mathematics since the very beginnings, I feel, which is one big drawback in comparison to all other sciences, as far as I know. Of course, no creative mathematician can afford not to ‘speculate’, namely to do more or less daring guesswork as an indispensable source of inspiration. The trouble is that, in obedience to a stern tradition, almost nothing of this appears in writing, and preciously little even in oral communication. The point is that the disrepute of  ‘speculation’ or ‘dream’ is such, that even as a strictly private (not to say secret!) activity, it has a tendency to vegetate – much like the desire and drive of love and sex, in too repressive an environment.  Despite the ‘repression’, in the one or two years before I unexpectedly was led to withdraw from the mathematical milieu and to stop publishing, it was more or less clear to me that, besides going on pushing ahead with foundational work in SGA and EGA, I was going to write a wholly science-fiction kind [of] book on ‘motives’, which was then the most fascinating and mysterious mathematical being I had come to meet so far. As my interests and my emphasis have somewhat shifted since, I doubt I am ever going to write this book – still less anyone else is going to, presumably. But whatever I am going to write in mathematics, I believe a major part of it will be ‘speculation’ or ‘fiction’, going hand in hand with painstaking, down-to-earth work to get hold of the right kind of notions and structures, to work out comprehensive pictures of still misty landscapes.  The notes I am writing up lately are in this spirit, but in this case the landscape isn’t so remote really, and the feeling is rather that, as for the specific program I have been out for is concerned, getting everything straight and clear shouldn’t mean more than a few years work at most for someone who really feels like doing it, maybe less. But of course surprises are bound to turn up on one’s way, and while starting with a few threads in hand, after a while they may have multiplied and become such a bunch that you cannot possibly grasp them all, let alone follow.”

In a subsequent posting (2006-03-14), Brown wrote this about rigour in category theory:

The situation is more complicated in that what could be classed as speculation may get published as theorem and proof.  For example, in algebraic topology, sometimes proofs of continuity are omitted as if this was an exercise for the reader, yet the formulation of why the maps are continuous (if they are necessarily so) may contain a key aspect of what should be a complete proof. This difficulty was pointed out to me years ago by Eldon Dyer in relation to results on local fibration implies global fibration (for paracompact spaces) where he and Eilenberg felt Dold’s paper on this contained the first complete proof. I have been unable to complete the proof in Spanier’s book, even the second edition. (I sent a correction to Spanier as the key function in the first edition was not well defined, after Spanier had replied `Isn’t it continuous?’) Eldon speculated (!) that perhaps 50% of published algebraic topology was seriously wrong!

van Kampen’s original 1935 ‘proof’ of what is called his theorem is incomprehensible today, and maybe was then also.

Efforts to give full details of a major result, i.e. to give a proof, are sometimes derided.  Of course credit should be given to the originator of the major steps towards a proof.

Grothendieck’s efforts to develop structures and language which would reduce proofs to a sequence of tautologies are notable here. Colin McLarty’s excellent article on “The rising sea: Grothendieck on simplicity and generality” is relevant.

Some scientists snear at the mathematical notion of rigour and of proof. On the other hand many are attracted to math because it can give explanations of why something is true. But ‘explanations’ need a higher level of structural language than for what might be called proofs.

I can’t resist mentioning that one student questionaire on my first year analysis wrote “Professor Brown puts in too many proofs.”   So I determined to rectify the situation, and next year there were no theorems, and no proofs.  However there were lots of statements labelled ‘FACT” followed by several paragraphs labelled ‘EXPLANATION’. This did modify the course because something labelled ‘explanation’ ought really to explain something! I leave you all to puzzle this out!

In homotopy theory, many matters, such as the homotopy addition lemma, had clear proofs only years after they were well used.

Surely much early algebraic topology is speculative, in that the language has not yet been developed to express concepts with rigour so that a clear proof can be written down.  It would be a curious ahistorical assumption that there is not at this date another future level of concepts which require a similar speculative approach to reach towards them.”

Is it because I trained as a pure mathematician that I find this lack of rigor and completeness in what should be caregorial fundamentals immensely disturbing?

The Beats: Australian responses

Hughes Robert 1959

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).

London life


Scooter Caffe, Lower Marsh Street, Waterloo, London SE1 7AE.

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]

Imaginary beliefs

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.