The value of an education

In a letter to Rupert Hart-Davies on 29 November 1956 George Lyttelton included this statement from William Johnson Cory (1823-1892, Master of Eton 1845-1872) on education:

At school you are engaged not so much in acquiring knowledge as making mental efforts under criticism. A certain amount of knowledge, you can indeed with average faculties acquire so as to retain; nor need you regret the hours you spent on much that is forgotten, for the shadow of lost knowledge at least protects you from many illusions.  But you go to a great school not so much for knowledge as for arts and habits; for the habit of attention, for the art of expression, for the art of assuming at a moment’s notice a new intellectual position, for the art of entering quickly into another person’s thoughts, for the habit of submitting to censure and refutation, for the art of indicating assent or dissent in graduated terms, for the habit of regarding minute points of accuracy, for the art of working out what is possible in a given time, for taste, for discrimination, for mental courage, and for mental soberness.”

Rupert Hart-Davis (Editor) [1978-79]: The Lyttelton Hart-Davis Letters:  Correspondence of George Lyttelton and Rupert Hart-Davis, 1955-1962. London: John Murray.

Suddenly, the fog lifts . . .

Andrew Wiles, prover of Wiles’ Theorem (aka Fermat’s Last Theorem), on the doing of mathematics:

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were.

This describes my experience, over shorter time-frames, in studying pure mathematics as an undergraduate, with each new topic covered: epsilon-delta arguments in analysis; point-set topology; axiomatic set theory; functional analysis; measure theory; group theory; algebraic topology; category theory; statistical decision theory; integral geometry; etc.    A very similar process happens in learning a new language, whether a natural (human) language or a programming language.     Likewise, similar words describe the experience of entering a new organization (either as an employee or as a management consultant), and trying to understand how the organization works, who has the real power, what are the social relationships and dynamics within the organization, etc, something I have previously described here.
One encounters a new discipline or social organization, one studies it and thinks about it from as many angles and perspectives as one can, and eventually, if one is clever and diligent, or just lucky, a light goes on and all is illuminated.    Like visiting a new city and learning its layout by walking through it, frequently getting lost and finding one’s way again,  enlightenment requires work.  Over time, one learns not to be afraid in encountering a new subject, but rather to relish the state of inchoateness and confusion in the period between starting and enlightenment.  The pleasure and wonder of the enlightenment is so great, that it all the prior pain is forgotten.
Andrew Wiles [1996],  speaking in Fermat’s Last Theorem, a BBC documentary by S. Singh and John Lynch: Horizon, BBC 1996,  cited in Frans Oort [2011 ]:  Did earlier thoughts inspire Grothendieck? (Hat tip).

Automating prayer

I have recently re-read Michael Frayn’s The Tin Men, a superb satire of AI.  Among the many wonderful passages is this, on the semantic verification problem of agent communications:

“Ah,” said Rowe, “there’s a difference between a man and a machine when it comes to praying.”   “Aye. The machine would do it better. It wouldn’t pray for things it oughtn’t pray for, and its thoughts wouldn’t wander.”
“Y-e-e-s. But the computer saying the words wouldn’t be the same . . .”
“Oh, I don’t know. If the words ‘O Lord, bless the Queen and her Ministers‘ are going to produce any tangible effects on the Government, it can’t matter who or what says them, can it?”
“Y-e-e-s, I see that. But if a man says the words he means them.”
“So does the computer. Or at any rate, it would take a damned complicated computer to say the words without meaning them. I mean, what do we mean by ‘mean’? If we want to know whether a man or a computer means ‘O Lord, bless the Queen and her Ministers,’ we look to see whether it’s grinning insincerely or ironically as it says the words. We try to find out whether it belongs to the Communist Party. We observe whether it simultaneously passes notes about lunch or fornication. If it passes all the tests of this sort, what other tests are there for telling if it means what it says? All the computers in my department, at any rate, would pray with great sincerity and single-mindedness. They’re devout wee things, computers.” (pages 109-110).

Michael Frayn [1995/1965]: The Tin Men (London, UK: Penguin, originally published by William Collins, 1965)

Evolutionary psychology

In his Little Red Book, Mao Tse Tung said:   “Learn to play the piano.” [Fn #1]  However,  I don’t recall ever seeing a single piano in an African village, although I certainly saw (and heard) piano accordians in the villages and along the mountain paths of Lesotho (along with various hand-drums and mamokhorongs).  And settlements larger than traditional villages – Zimbabwe’s Growth Points, for example – sometimes had pianos in their churches or newly-built school halls.  Of course, the earliest of these pianos could only have been made in these last 300 years.

It seems to me that the historical absence of village pianos in Africa causes a problem for evolutionary psychology, since clearly a daily compulsion to play the piano is not something that has a long-standing evolutionary basis – at least, not for those of us descended from the peoples of the African savannah.  So if evo-psych cannot explain this very real human characteristic, what business does it have explaining any other human characteristic?  Why are some attitudes or characteristics to be explained by evolutionary means yet not others?  What distinguishes the one class of characteristics from the other? And what credence can we possibly give to any evolutionary explanation of phenomena which is not, prima facie, explainable in this way?   Surely, this limitation of the scope of evolutionary explanations completely undermines such arguments, since either all higher-level human characteristics have evolutionary explanations or none at all do.

1. In: Chapter 10:  Leadership of Party CommitteesQuotations from Mao Tse-Tung.  Peking, PRC:  Foreign Languages Press, 1966.

Think again, Helen Vendler, think differently!

Helen Vendler wrote a superb and indispensible commentary on Shakespeare’s Sonnets, deconstructing the poems’ complex and subtle verbal gymnastics and providing a guide to the unmatched mental ingenuity Shakespeare manifests.   As her exegesis clearly shows, Vendler, as well as Shakespeare, is a master of verbal intelligence.   However, she seems to believe that the only intelligence that is, is linguistic.  

In a recent article in Harvard Magazine, Vendler presents a case for primary school education to centre around reading and words, with just a nod to mathematics.    It is good that she included mathematics there somewhere, since I presume she would like her electricity network to keep humming with power, her sewers flushed, her phones connected, her air-travel crash-free, her food and drink and flowers freshly delivered, her weather forecasted, her borders defended, and her online transactions safely encrypted.    None of these, in our modern, technologically-centred world, would be sure to happen if our schools produced only literati.  

But 15 periods per day – 1 of mathematics and 14 for reading – and yet no time for children to draw or paint?  They can look at art and discuss it (periods #7 and #10) but not do it!  How revealing is THAT about Ms Vendler’s opinions of the relative importance of words and images!    And no time in those 15 periods for learning or playing music, apart from group singing?  The only singing allowed in her day is the “choral singing of traditional melodic song (folk songs, country songs, rounds)” ?  Why should traditional melodies be so privileged?   That is like saying that children should only read books written before 1900.   Surely, a person so concerned with words and reading would be delighted if children engaged in rap, that most verbal and linguistically-intellectually-challenging of musics?    This list of activities begins to look merely like an anti-contemporary-world tirade of the sort we have seen before.

Not only does her syllabus have an anti-modern bias, but there is also a bias against other forms of human thinking, such as drawing-as-thought, and music-as-thought.   The philosopher Stephen Toulmin noted the pro-text tendency our culture has evidenced these last four centuries.  While this tendency still dominates us all, we are at last seeing the rise of minority tendencies:  an increasing role for film and video and image in our culture generally; the use of GUIs in devices which interact with humans; the use of graphically-oriented software development tools (so that no longer do all programmers have to be left-brained text manipulators); an attention to design in product development;  and the rise – for the first time since Euclid’s geometry – of a western mathematical discipline where reasoning occurs over diagrams.  

We are just at the beginning at understanding, modeling, systematizing, and using visual thinking and reasoning over diagrams, or musical and sonic reasoning.  We’ve hardly started this effort for the other types of human intelligence we know about:  spatial, kinesthetic, interpersonal, intrapersonal, naturalistic, and existential.   And all the non-human forms of intelligence await even recognition and discovery.   What a great shame if all this rich diversity of intelligent modes of thought were to be squeezed out by a narrow school syllabus favouring just one-and-a-bit types of thinking.  


Rebecca Donner [2021]: All the Frequent Troubles of Our Days: The True Story of the Woman at the Heart of the German Resistance to Hitler. Canongate Books.

Helen Vendler [1999]:  The Art of Shakespeare’s Sonnets.  Cambridge, MA: Belknap Press.

Helen Vendler [2011]:  Reading is elemental.  Harvard Magazine, September-October 2011.

Postscript 1 (Added 2021-08-09):
In case of linkrot, here is an excerpt from the Harvard Magazine article by Vendler:

“In a utopian world, I would propose, for the ultimate maintenance of the humanities and all other higher learning, an elementary-school curriculum that would make every ordinary child a proficient reader by the end of the fourth grade—not to pass a test, but rather to ensure progressive expansion of awareness. Other than mathematics, the curriculum of my ideal elementary school would be wholly occupied, all day, every day, with “reading” in its very fullest sense. Let us imagine the day divided into short 20-minute “periods.” Here are 14 daily such periods of “reading,” each divisible into two 10-minute periods, or extended to a half-hour, as seems most practical to teachers in different grades. Many such periods can be spent outside, to break up the tedium of long sitting for young children. The pupils would:

1. engage in choral singing of traditional melodic song (folk songs, country songs, rounds);
2. be read to from poems and stories beyond their own current ability to read;
3. mount short plays—learning roles, rehearsing, and eventually performing;
4. march or dance to counting rhymes, poems, or music, “reading” rhythms and sentences with their bodies;
5. read aloud, chorally, to the teacher;
6. read aloud singly to the teacher, and recite memorized poems either chorally or singly;
7. notice, and describe aloud, the reproduced images of powerful works of art, with the accompanying story told by the teacher (Orpheus, the three kings at Bethlehem, etc.);
8. read silently, and retell in their own words, for discussion, the story they have read;
9. expand their vocabulary to specialized registers through walks where they would learn the names of trees, plants, flowers, and fruits;
10. visit museums of art and natural history to learn to name exotic or extinct things, or visit an orchestra to discover the names and sounds of orchestral instruments;
11. learn conjoined prefixes, suffixes, and roots as they learn new words;
12. tell stories of their own devising;
13. compose words to be sung to tunes they already know; and
14. if they are studying a foreign language, carry out these practices for it as well.

The only homework, in addition to mathematics, would be additional reading practices over the weekends (to be checked by a brief Monday discussion by students). If such a curriculum were carried out—with additional classroom support and needed modification for English-language learners or pupils in special education—I believe that by the end of the fourth grade, the majority of the class would enjoy, and do well in, reading. Then, in middle school and high school, armed with the power of easy and pleasurable reading, students could be launched not only into appropriate world literature, but also into reading age-appropriate books of history or geography or civics or science—with much better results than at present. If reading—by extensive exposure and intensive interaction—cannot be made enjoyable and easy, there is no hope for students in their later education.”

Postscript 2 (Added 2021-08-09):
Items #1 and #4 in Vendler’s list reminded me of the German Nazi Ministry of Education’s school manual, Erziehung und Unterricht in der Höheren Schule: Amtliche Ausgabe des Reichs und Preuszischen Ministeriums für Wissenschaft, Erziehung, und Volksbildung (Education and Instruction in the Higher Schools: Official Publication of the Reich and Prussian Ministry of Knowledge, Education, and National Culture) that included requirements that:

Boys of fourteen should study songs of medieval foot soldiers, modern soldier songs, marching songs. . . . Boys of sixteen are to learn military folk songs and an opera by Wagner.

Source: Donner (2021).

Music as thought

I have remarked before that music is a form of thinking.  It is a form of thinking for the composer and may also be for the listener.  If the performers are to transmit its essence effectively and well, it will be a form of thinking for them also. Listening recently to the music of Prokofiev,  I realize I don’t think in the way he does, and so I find his music alien.

But what is the nature of this musical thought?
Continue reading ‘Music as thought’

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.

Distributed cognition

Some excerpts from an ethnographic study of the operations of a Wall Street financial trading firm, bearing on distributed cognition and joint-action planning:

This emphasis on cooperative interaction underscores that the cognitive tasks of the arbitrage trader are not those of some isolated contemplative, pondering mathematical equations and connected only to to a screen-world.  Cognition at International Securities is a distributed cognition.  The formulas of new trading patterns are formulated in association with other traders.  Truly innovative ideas, as one senior trader observed, are slowly developed through successions of discreet one-to-one conversations.
. . .
An idea is given form by trying it out, testing it on others, talking about it with the “math guys,” who, significantly, are not kept apart (as in some other trading rooms),  and discussing its technical intricacies with the programmers (also immediately present).”   (p. 265)
The trading room thus shows a particular instance of Castell’s paradox:  As more information flows through networked connectivity, the more important become the kinds of interactions grounded in a physical locale. New information technologies, Castells (2000) argues, create the possibility for social interaction without physical contiguity.  The downside is that such interactions can become repititive and programmed in advance.  Given this change, Castells argues that as distanced, purposeful, machine-like interactions multiply, the value of less-directd, spontaneous, and unexpected interactions that take place in physical contiguity will become greater (see also Thrift 1994; Brown and Duguid 2000; Grabhar 2002).  Thus, for example, as surgical techniques develop together with telecommunications technology, the surgeons who are intervening remotely on patients in distant locations are disproportionately clustering in two or three neighbourhoods of Manhattan where they can socialize with each other and learn about new techniques, etc.” (p. 266)
“One examplary passage from our field notes finds a senior trader formulating an arbitrageur’s version of Castell’s paradox:
“It’s hard to say what percentage of time people spend on the phone vs. talking to others in the room.   But I can tell you the more electronic the market goes, the more time people spend communicating with others inside the room.”  (p. 267)
Of the four statistical arbitrage robots, a senior trader observed:
“We don’t encourage the four traders in statistical arb to talk to each other.  They sit apart in the room.  The reason is that we have to keep diversity.  We could really hammered if the different robots would have the same P&L [profit and loss] patterns and the same risk profiles.”  (p. 283)

Daniel Beunza and David Stark [2008]:  Tools of the trade:  the socio-technology of arbitrage in a Wall Street trading room.  In:  Trevor Pinch and Richard Swedborg (Editors):  Living in a Material World:  Economic Sociology Meets Science and Technology Studies. Cambridge, MA, USA: MIT Press.  Chapter 8, pp. 253-290.
M. Castells [1996]:  The Information Age:  Economy, Society and Culture. Blackwell, Second Edition.

The Matherati

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

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.
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 seeThe New Yorker, 10 May 1993.