Archive for the 'Matherati' Category Page 3 of 5

A salute to Charles Hamblin

This short biography of Australian philosopher and computer scientist Charles L. Hamblin was initially commissioned by the Australian Computer Museum Society.

Charles Leonard Hamblin (1922-1985) was an Australian philosopher and one of Australia’s first computer scientists. His main early contributions to computing, which date from the mid 1950s, were the development and application of reverse polish notation and the zero-address store. He was also the developer of one of the first computer languages, GEORGE. Since his death, his ideas have become influential in the design of computer interaction protocols, and are expected to shape the next generation of e-commerce and machine-communication systems.

Continue reading ‘A salute to Charles Hamblin’

ABS Cadets 1979

Australian Bureau of Statistics graduate cadets, 1979:

  • Phil Aungles
  • Gail Bansemer
  • Penny Barlow
  • Warren Bird
  • Wendy Darr
  • Ken Henry
  • Karen Hyams
  • Debra Keillor
  • Peter McBurney
  • Vivienne Palmer
  • Prue Phillips
  • Suzanne Sheridan
  • Steven Skates
  • John Stroud.

John Bennett RIP

John Bennett AO (1921-2010), first professor of computing in Australia and founder of Sydney University’s Basser Department of Computer Science, died last month.  The SMH obit, from which the lines below are taken, is here.

Emeritus Professor John Bennett AO was an internationally recognised Australian computing pioneer. Known variously as ”the Prof”, ”JMB” or ”Rusty”, he was a man with a voracious appetite for ideas, renowned for his eclectic interests, intellectual generosity, cosmopolitan hospitality and prodigious general knowledge.

As Australia’s first professor of computer science and foundation president of the Australian Computer Society, Bennett was an innovator, educator and mentor but at the end of his life he wished most to be remembered for his contribution to the construction of one of the world’s first computers, the Electronic Delay Storage Automatic Calculator (EDSAC). In 1947 at Cambridge University, as Maurice Wilkes’s first research student, he was responsible for the design, construction and testing of the main control unit and bootstrap facility for EDSAC and carried out the first structural engineering calculations on a computer as part of his PhD. Bennett’s work was critical to the success of EDSAC and was achieved with soldering irons and war-surplus valves in the old Cambridge anatomy dissecting rooms, still reeking of formalin.

. . .

The importance of his work on EDSAC was recognised by many who followed. In Cambridge he also pioneered the use of digital computers for X-ray crystallography in collaboration with John Kendrew (later a Nobel Prize winner), one of many productive collaborations.

He was recruited from Cambridge by Ferranti Manchester in 1950 to work on the Mark 1*. Colleague G. E. ”Tommy” Thomas recalls that when Ferranti’s promise to provide a computer for the 1951 Festival of Britain could not be fulfilled, ”John suggested … a machine to play the game of Nim against all comers … [It] was a great success. The machine was named Nimrod and is the precursor of the vast electronic games industry we know today.”

In 1952, Bennett married Mary Elkington, a London School of Economics and Political Science graduate in economics, who was working in another section at Ferranti. Moving to Ferranti’s London Computer Laboratory in 1953, Bennett worked in a team led by Bill Elliott, alongside Charles Owen, whose plug-in components enabled design of complete computers by non-engineers. Owen went on to design the IBM 360/30. Bennett remembered of the time, ”Whatever we touched was new; it gives you a great lift. We weren’t fully aware of what we were pioneering. We knew we had the best way but we weren’t doing it to convert people – we were doing it because it was a new tool which should get used. We knew we were ploughing new ground.”

Bennett was proud of being Australian and strongly felt the debt he owed for his education. When Harry Messel’s School of Physics group asked him in 1956 to head operations on SILLIAC (the Sydney version of ILLIAC, the University of Illinois Automatic Computer – faster than any machine then commercially available), he declined a more lucrative offer he had accepted from IBM and moved his family to Sydney.

The University of Sydney acknowledged computer science as a discipline by creating a chair for Bennett, the professor of physics (electronic computing) in 1961. Later the title became professor of computer science and head of the Basser department of computer science. Fostering industry relationships and ensuring a flow of graduates was a cornerstone of his tenure.

Bennett was determined that Australia should be part of the world computing scene and devoted much time and effort to international professional organisations. This was sometimes a trial for his staff. Arthur Sale recalls, ”I quickly learnt that John going away was the precursor to him returning with a big new idea. After a period when we could catch up with our individual work, John would tell us about the new thing that we just had to work on. Once it was the ARPAnet [Advanced Research Projects Agency Network] and nothing would suffice until we started to try to communicate with the Aloha satellite over Hawaii that had run out of gas to establish a link to Los Angeles and ARPAnet and lo and behold, the internet had come to Australia in the 1970s.”

In 1983, Bennett was appointed as an officer of the Order of Australia. After his retirement in 1986, he remained active, attending PhD seminars and lectures to ”stay up to date and offer a little advice” while continuing to earn recognition for his contributions to computer science for more than half a century.

Hardy on the Tripos

Lest anyone think I’m uniquely deranged for my criticisms of the  Cambridge University Mathematics Tripos examination, particularly during the 18th- and 19th-centuries, here is GH Hardy – perhaps Britain’s greatest 20th century pure mathematician – speaking in his Presidential Address to the Mathematics Association in 1926:

My own contribution to the discussion consisted merely in an expression of my feeling that the best thing that could happen to English mathematics, and to Cambridge mathematics in particular, would be that the Mathematical Tripos should be abolished. I stated this on the spur of the moment, but it is my considered opinion, and I propose to defend it at length to-day. And I am particularly anxious that you should understand quite clearly that I mean exactly what I say; that by “abolished” I mean “abolished”, and not “reformed”; that if I were prepared to co-operate, as in fact I have co-operated in the past, in “reforming” the Tripos, it would be because I could see no chance of any more revolutionary change; and that my “reforms” would be directed deliberately towards destroying the traditions of the examination and so preparing the way for its extinction.” [p. 134]

. . .

“I suppose that it would be generally agreed that Cambridge mathematics, during the last hundred years, has been dominated by the Mathematical Tripos in a way in which no first-rate subject in any other first-rate university [page-break] has ever been dominated by an examination. It would be easy for me, were the fact disputed, to justify my assertion by a detailed account of the history of the Tripos, but this is unnecessary, since you can find an excellent account, written by a man who was very much more in sympathy with the Tripos than I am, in Mr. Rouse Ball’s History of Mathematics in Cambridge. I must, however, call your attention to certain rather melancholy reflections which the history of Cambridge mathematics suggests. You will understand that when I speak of mathematics I mean primarily pure mathematics, not that I think that anything which I say about pure mathematics is not to a great extent true of applied mathematics also, but merely because I do not want to criticise where my competence as a critic is doubtful.

Mathematics at Cambridge challenges criticism by the highest standards. England is a first-rate country, and there is no particular reason for supposing that the English have less natural talent for mathematics than any other race; and if there is any first-rate mathematics in England, it is in Cambridge that it may be expected to be found. We are therefore entitled to judge Cambridge mathematics by the standards that would be appropriate in Paris or Gottingen or Berlin. If we apply these standards, what are the results?  I will state them, not perhaps exactly as they would have occurred to me spontaneously-though the verdict is one which, in its essentials, I find myself unable to dispute-but as they were stated to me by an outspoken foreign friend.

In the first place, about Newton there is no question; it is granted that he stands with Archimedes or with Gauss. Since Newton, England has produced no mathematician of the very highest rank. There have been English mathematicians, for example Cayley, who stood well in the front rank of the mathematicians of their time, but their number has been quite extraordinarily small; where France or Germany produces twenty or thirty, England produces two or three. There has been no country, of first-rate status and high intellectual tradition, whose standard has been so low; and no first-rate subject, except music, in which England has occupied so consistently humiliating a position. And what have been the peculiar characteristics of such English mathematics as there has been? Occasional flashes of insight, isolated achievements sufficient to show that the ability is really there, but, for the most part, amateurism, ignorance, incompetence, and triviality. It is indeed a rather cruel judgment, but it is one which any competent critic, surveying the evidence dispassionately, will find it uncommonly difficult to dispute.

I hope that you will understand that I do not necessarily endorse my friend’s judgment in every particular. He was a mathematician whose competence nobody could question, and whom nobody could accuse of any prejudice against England, Englishmen, or English mathematicians; but he was also, of course, a man developing a thesis, and he may have exaggerated a little in the enthusiasm of the moment or from curiosity to see how I should reply.  Let us assume that it is an exaggerated judgment, or one rhetorically expressed.  It is, at any rate, not a ridiculous judgment, and it is serious enough that such a condemnation, from any competent critic, should not be ridiculous. It is inevitable that we should ask whether, if such a judgment can really embody any sort of approximation to the truth, some share of the responsibility must not be laid on the Mathematical Tripos and the grip which it has admittedly exerted on English mathematics.

I am anxious not to fall into exaggeration in my turn and use extravagant language about the damage which the Tripos may have done, and it would no doubt be an extravagance to suggest that the most ruthless of examinations could destroy a whole side of the intellectual life of a nation. On the [page-break] other hand it is really rather difficult to exaggerate the hold which the Tripos has exercised on Cambridge mathematical life, and the most cursory survey of the history of Cambridge mathematics makes one thing quite clear; the reputation of the Tripos, and the reputation of Cambridge mathematics stand in correlation with one another, and the correlation is large and negative.  As one has developed, so has the other declined. As, through the early and middle nineteenth century, the traditions of the Tripos strengthened, and its importance in the eyes of the public grew greater and greater, so did the external reputation of Cambridge as a centre of mathematical learning steadily decay. When, in the years perhaps between 1880 and 1890, the Tripos stood, in difficulty, complexity, and notoriety, at the zenith of its reputation, English mathematics was somewhere near its lowest ebb. If, during the last forty years, there has been an obvious revival, the fortunes of the Tripos have experienced an equally obvious decline.” [pp. 135-137]

. . .

“It has often been said that Tripos mathematics was a collection of elaborate futilities, and the accusation is broadly true. My own opinion is that this is the inevitable result, in a mathematical examination, of high standards and traditions. The examiner is not allowed to content himself with testing the competence and the knowledge of the candidates; his instructions are to provide a test of more than that, of initiative, imagination, and even of some sort of originality. And as there is only one test of originality in mathematics, namely the accomplishment of original work, and as it is useless to ask a youth of twenty-two to perform original research under examination conditions, the examination necessarily degenerates into a kind of game, and instruction for it into initiation into a series of stunts and tricks. It was in any case certainly true, at the time of which I am speaking, that an undergraduate might study mathematics diligently throughout the whole of his career, and attain the very highest honours in the examination, without having acquired, and indeed without having encountered, any knowledge at all of any of the ideas which dominate modern mathematical thought. His ignorance of analysis would have been practically complete. About geometry I speak with less confidence, but I am sure that such knowledge as he possessed would have been exceedingly one-sided, and that there would have been whole fields of geometrical knowledge, and those perhaps the most fruitful and fascinating of all, of which he would have known absolutely nothing. A mathematical physicist, I may be told, would on the contrary have received an appropriate and an excellent education. It is possible; it would no doubt be very impertinent for me to deny it. Yet I do remember Mr. Bertrand Russell telling me that he studied electricity at Trinity for three years, and that at the end of them he had never heard of Maxwell’s equations; and I have also been told by friends whom I believe to be competent that Maxwell’s equations are really rather important in physics. And when I think of this I begin to wonder whether the teaching of applied mathematics was really quite so perfect as I have sometimes been led to suppose.” [p. 138]

. . .

“I shall judge the Tripos by its real or apparent influence on English mathematics. I have already told you that in my judgment this influence has in the past been bad, that the Tripos has done negligible good and by no means negligible harm, and that, so far from being the great glory of Cambridge mathematics, it has gone a very long way towards strangling its development.”  [p. 141]


G. H. Hardy [1926/1948]: Presidential Address: The Case against the Mathematical Tripos. The Mathematical Gazette, 32 (300): 134-145 (July 1948).

Impure mathematics at Cambridge

I have remarked before that the Mathematics Tripos at Cambridge, with its impure emphasis on the calculations needed for mathematical physics to the great detriment of pure mathematical thinking, understanding and rigor, had deleterious consequences across the globe more than a century later.  Even as late as the 1980s, there were few Australian university mathematics degree programs that did not require students to waste at least one year on the prehensile, brain-dead calculations needed for what is wrongly called Applied Mathematics.    I am still angered by this waste of effort.    Marx called traditions nothing more than the collected errors of past generations, and never was this statement more true.  What pure mathematician or statistician or computer scientist with integrity could stomach such nonsense?

I am not alone in my views. One of the earliest people who opposed Cambridge’s focus on impure, bottom-up, unprincipled mathematics – those three adjectives are each precisely judged – was Charles Babbage, later a computer pioneer and industrial organizer.  I mentioned his Analytical Society here, created while he was still an undergraduate.     Now, I have just seen an article by Harvey Becher [1995] which places Babbage’s campaign for Cambridge University to teach modern pure mathematics within its full radical political and nonconformist religious context.   A couple of nice excerpts from Becher’s article:

As the revolution and then Napoleon swept across Europe, French research mathematicians such as J. L. Lagrange and S. P. Laplace, and French textbook writers such as S. F. Lacroix, made it obvious that British mathematicians who adhered to the geometrically oriented fluxional mathematics and dot notation of Newton had become anachronisms.  The more powerful abstract and generalized analysis developed on the Continent had become the focus of mathematicians and the language of the physical sciences. This mathematical transmutation fused with social revolution.  ‘Lagrange’s treatises on the calculus were written in response to the educational needs of the Revolution’, recounts Ivor Grattan-Guinness, and Lagrange, Laplace and Lacroix were intimately involved with the educational and scientific reorganizations of the earlier revolutionaries and Napoleon.   Thus, French mathematics became associated with revolutionary France.

This confluence of social and mathematical revolution washed into the heart of Cambridge University because the main purpose of the Cambridge mathematics curriculum, as the core of a liberal education, Cambridge’s raison d’etre, was to produce [page-break] educated gentlemen for careers in the Church, the law and academe. With a student clientele such as this, few were disturbed that the Cambridge curriculum stuck to emphasizing Euclidean geometry, geometric optics and Newtonian fluxions, mechanics and astronomy. However, it was not the landed sons (who constituted the largest segment of the undergraduates), but the middle class and professional sons who, though a minority of the student body as a whole, made up the majority of the wranglers.   For them, especially those who might have an interest in mathematics as an end in itself rather than as merely a means to a comfortable career, the currency of the mathematics in the curriculum might be of concern.

Even though a Cambridge liberal education catered to a social/political elite, most nineteenth-century British mathematicians and mathematical physicists graduated from Cambridge University as wranglers. The Cambridge curriculum, therefore, contoured British mathematics, mathematical physics and other scientific fields. Early in the century, the mathematics curriculum underwent an ‘analytical revolution’ aimed at ending the isolation of Cambridge mathematics from continental mathematics by installing continental analytics in place of the traditional curriculum. Although the revolution began before the creation of the undergraduate constituted ‘Analytical Society’ in 1811, and though the revolution continued after the demise of that Society around 1817, the Analytical Society, its leaders – Charles Babbage, John Herschel and George Peacock – and their opponents, set the parameters within which the remodelling of the curriculum would take place.  This essay is an appraisal of their activities within the mathematical/social/political/religious environment of Cambridge.  The purpose is to reveal why the curriculum took the form it did, a form conducive to the education of a liberally educated elite and mathematical physicists, but not necessarily to the education of pure mathematicians.” [pages 405-406]

And later:

As Babbage and Herschel were radicals religiously and socially, they were radicals mathematically. They did not want to reform Cambridge mathematics; rather, they wanted [page-break] to reconstruct it. As young men, they had no interest in mixed mathematics, the focal point of Cambridge mathematics. In mixed mathematics, mathematics was creatively employed to achieve results for isolated, particular, sometimes trivial, physical problems. The mathematics created for a specific problem was intuitively derived from and applied to the problem, and its only mathematical relevance was that the ingenious techniques developed to solve one problem might be applicable to another. The test of mathematical rigour was to check results empirically. Correspondingly, mathematics was taught from ‘the bottom up’ by particular examples of applications.

Babbage’s and Herschel’s concerns lay not in mixed mathematics, but rather, as they put it in the introduction to the Memoirs, ‘exclusively with pure analytics’. In the Memoirs and other of their publications as young men, they devoted themselves to developing mathematics by means of the mechanical manipulation of symbols, a means purely abstract and general with no heuristic intuitive, physical, or geometric content. This Lagrangian formalism was what they conceived mathematics should be, and how it should be taught.  Indeed, they believed that Cambridge mathematicians could not read the more sophisticated French works because they had been taught analysis by means of its applications to the exclusion of general abstract operations. To overcome this, they wanted first to inculcate in the students general operations free of applications to get them to think in the abstract rather than intuitively.  On the theoretical level, they urged that the calculus ought not to be taught from an intuitive limit concept, to wit, as the derivative being generated by the vanishing sides of a triangle defined by two points on a curve approaching indefinitely close to one another; or by instantaneous velocity represented by the limit of time over distance as the quantities of time and distance vanished; or by force defined as the ultimate ratio of velocity to time. Rather, they urged that students start with derived functions of Lagrange, that is, successive coefficients of the expansion of a function in a Taylor Series being defined as the successive derivatives of the function. This was algebra, free of all limiting intuitive or physical encumbrances. It would condition the student to think in the abstract without intuitive crutches. And on the practical level, pure calculus, so defined, should be taught prior to any of its applications. To achieve this would have inverted the traditional Cambridge approach and revolutionized the curriculum, both intellectually and socially, for only a handful of abstract thinkers, pure mathematicians like Babbage and Herschel, could have successfully tackled it.   The established liberal education would have been a thing of the past.” [pages 411-412]

POSTSCRIPT (Added 2010-11-03):

I have just seen the short paper by David Forfar [1996], reporting on the subsequent careers of the Cambridge Tripos Wranglers.    The paper has two flaws.  First, he includes in his Tripos alumni Charles Babbage, someone who refused to sit the Tripos, and who actively and bravely campaigned for its reform.  Forfar does, it is true, mention Babbage’s non-sitting, but only a page later after first listing him, and then without reference to his principled opposition.  Second, Forfar presents overwhelming evidence for the failure of British pure mathematics in the 19th- and early 20th-centuries, listing just Cayley, Sylvester, Clifford, Hardy and Littlewood as world-class British pure mathematicians – I would add Babbage, Boole and De Morgan – against 14 world-class German and 17 world-class French mathematicians that he identifies.   But then, despite this pellucid evidence, Forfar can’t bring himself to admit the obvious cause of the phenomenon – the Tripos exam.  He concludes:  “The relative failure of British pure mathematics during this period in comparison with France and Germany remains something of a paradox.” No, Mr Forfar,  there is no paradox here; there is not even any mystery.    (En passant, I can’t imagine any pure mathematician using the word “paradox” in the way Forfar does here.)

Forfar says:  “While accepting these criticisms [of GH Hardy], it seems curious that those who became professional pure mathematicians apparently found difficulty in shaking off the legacy of the Tripos.” The years which Tripos students spent on the exam were those years generally judged most  productive for pure mathematicians – their late teens and early twenties.  To spend those years practising mindless tricks like some performing seal, instead of gaining a deep understanding of analysis or geometry, is why British pure mathematics was in the doldrums during the whole of the Georgian, Victorian and Edwardian eras, the whole of the long nineteenth century, from 1750 to 1914.


Harvey W. Becher [1995]:  Radicals, Whigs and conservatives:  the middle and lower classes in the analytical revolution at Cambridge in the age of aristocracy.   British Journal for the History of Science, 28:  405-426.

David O. Forfar [1996]:  What  became of the Senior Wranglers?  Mathematical Spectrum, 29 (1).

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On birds and frogs

I have posted before about the two cultures of pure mathematicians – the theory-builders and the problem-solvers.  Thanks to string theorist and SF author Hannu Rajaniemi, I have just seen a fascinating paper by Freeman Dyson, which draws a similar distinction – between the birds (who survey the broad landscape, making links between disparate branches of mathematics) and the frogs (who burrow down in the mud, solving particular problems in specific branches of the discipline).   This distinction is analogous to that between a focus on breadth and a focus on depth, respectively, as strategies  in search.   As Dyson says, pure mathematics as a discipline needs both personality-types if it is to make progress.   Yet, a tension often exists between these types:  in my experience, frogs are often disdainful of birds for lacking deep technical expertise.   I have less often encountered disdain from birds, perhaps because that is where my own sympathies are.

A similar tension exists in computing – a subject which needs both deep technical expertise AND a rich awareness of the breadth of applications to which computing may be put.  This need arises because the history of the subject shows an intricate interplay of theory and applications, led almost always by the application.    Turing’s abstract cineprojector model of computing arrived a century after Babbage’s calculating machines, for example, and we’ve had programmable devices since at least Jacquard’s loom in 1804, yet only had a mathematical theory of programming since the 1960s.  In fact, since computer science is almost entirely a theory of human artefacts (apart from that part – still small – which looks at natural computing), it would be strange indeed were the theory to divorce itself from the artefacts which are its scope of study.

A story which examplifies this division in computing is here.


Freeman Dyson [2009]:  Birds and frogs.  Notices of the American Mathematical Society, 56 (2): 212-223, February 2009.   Available here.

A computer pioneer

I have posted before about how the history of commercial computing is intimately linked with the British tea-shop, via LEO, a successful line of commercial computers developed by the Lyons tea-shop chain.    The first business application run on a Lyons computer was almost 60 years ago, in 1951.   Today’s Grauniad carries an obituary for John Aris (1934-2010), who had worked for LEO on the first stage of an illustrious career in commercial IT.  His career included a period as Chief Systems Engineer with British computer firm ICL (later part of Fujitsu).    Aris’ university education was in Classics, and he provides another example to show that the matherati represent a cast of mind, and not merely a collection of people educated in mathematics.

John’s career in computing began in 1958 when he was recruited to the Leo (Lyons Electronic Office) computer team by J Lyons, then the major food business in the UK, and initiators of the notion that the future of computers lay in their use as a business tool. At the time, the prevailing view was that work with computers required a trained mathematician. The Leo management thought otherwise and recruited using an aptitude test. John, an Oxford classics graduate, passed with flying colours, noting that “the great advantage of studying classics is that it does not fit you for anything specific”. “

Of course, LEO was not the first time that cafes had led to new information industries, as we noted here in a post about the intellectual and commercial consequences of the rise of coffee houses in Europe from the mid-17th century.  The new industries the first time round were newspapers, insurance, and fine art auctions (and through them, painting as a commercial activity aimed at non-aristocrat collectors); the new intellectual discipline was the formal modeling of uncertainty (then aka probability theory).

UPDATE (2012-05-22):  The Telegraph of 2011-11-10 ran an article about the Lyons Tea Shop computer business, here, to celebrate the 60th anniversary of the the LEO (1951-11-17).

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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 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.

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AI’s first millenium: prepare to celebrate

A search algorithm is a computational procedure (an algorithm) for finding a particular object or objects in a larger collection of objects.    Typically, these algorithms search for objects with desired properties whose identities are otherwise not yet known.   Search algorithms (and search generally) has been an integral part of artificial intelligence and computer science this last half-century, since the first working AI program, designed to play checkers, was written in 1951-2 by Christopher Strachey.    At each round, that program evaluated the alternative board positions that resulted from potential next moves, thereby searching for the “best” next move for that round.

The first search algorithm in modern times apparently dates from 1895:  a depth-first search algorithm to solve a maze, due to amateur French mathematician Gaston Tarry (1843-1913).  Now, in a recent paper by logician Wilfrid Hodges, the date for the first search algorithm has been pushed back much further:  to the third decade of the second millenium, the 1020s.  Hodges translates and analyzes a logic text of Persian Islamic philosopher and mathematician, Ibn Sina (aka Avicenna, c. 980 – 1037) on methods for finding a proof of a syllogistic claim when some premises of the syllogism are missing.   Representation of domain knowledge using formal logic and automated reasoning over these logical representations (ie, logic programming) has become a key way in which intelligence is inserted into modern machines;  searching for proofs of claims (“potential theorems”) is how such intelligent machines determine what they know or can deduce.     It is nice to think that automated theorem-proving is almost 990 years old.


B. Jack Copeland [2000]:  What is Artificial Intelligence?

Wilfrid Hodges [2010]: Ibn Sina on analysis: 1. Proof search. or: abstract state machines as a tool for history of logic.  pp. 354-404, in: A. Blass, N. Dershowitz and W. Reisig (Editors):  Fields of Logic and Computation. Lecture Notes in Computer Science, volume 6300.  Berlin, Germany:  Springer.   A version of the paper is available from Hodges’ website, here.

Gaston Tarry [1895]: La problem des labyrinths. Nouvelles Annales de Mathématiques, 14: 187-190.

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Five minutes of freedom

Jane Gregory, speaking in 2004, on the necessary conditions for a public sphere:

To qualify as a public, a group of people needs four characteristics. First, it should be open to all and any: there are no entry qualifications. Secondly, the people must come together freely. But it is not enough to simply hang out – sheep do that. The third characteristic is common action. Sheep sometimes all point in the same direction and eat grass, but they still do not qualify as a public, because they lack the fourth characteristic, which is speech. To qualify as a public, a group must be made up of people who have come together freely, and their common action is determined through speech: that is, through discussion, the group determines a course of action which it then follows. When this happens, it creates a public sphere.

There is no public sphere in a totalitarian regime – for there, there is insufficient freedom of action; and difference is not tolerated. So there are strong links between the idea of a public sphere and democracy.”

I would add that most totalitarian states often force their citizens to participate in public events, thus violating two basic human rights:  the right not to associate and the right not to listen.

I am reminded of a moment of courage on 25 August 1968, when seven Soviet citizens, shestidesiatniki (people of the 60s), staged a brave public protest at Lobnoye Mesto in Red Square, Moscow, at the military invasion of Czechoslovakia by forces of the Warsaw Pact.   The seven (and one baby) were:  Konstantin Babitsky (mathematician and linguist), Larisa Bogoraz (linguist, then married to Yuli Daniel), Vadim Delone (also written “Delaunay”, language student and poet), Vladimir Dremlyuga (construction worker), Victor Fainberg (mathematician), Natalia Gorbanevskaya (poet, with baby), and Pavel Litvinov (mathematics teacher, and grandson of Stalin’s foreign minister, Maxim Litvinov).  The protest lasted only long enough for the 7 adults to unwrap banners and to surprise onlookers.  The protesters were soon set-upon and beaten by “bystanders” – plain clothes police, male and female – who  then bundled them into vehicles of the state security organs.  Ms Gorbanevskaya and baby were later released, and Fainberg declared insane and sent to an asylum.

The other five faced trial later in 1968, and were each found guilty.   They were sent either to internal exile or to prison (Delone and Dremlyuga) for 1-3 years; Dremlyuga was given additional time while in prison, and ended up serving 6 years.  At his trial, Delone said that the prison sentence of almost three years was worth the “five minutes of freedom” he had experienced during the protest.  

Delone (born 1947) was a member of a prominent intellectual family, great-great-great-grandson of a French doctor, Pierre Delaunay, who had resettled in Russia after Napoleon’s defeat.   Delone was the great-grandson of a professor of physics, Nikolai Borisovich Delone (grandson of Pierre Delaunay), and grandson of a more prominent mathematician, Boris Nikolaevich Delaunay (1890-1980), and son of physicist Nikolai Delone (1926-2008).  In 1907, at the age of 17, Boris N. Delaunay organized the first gliding circle in Kiev, with his friend Igor Sikorski, who was later famous for his helicopters.   B. N. Delaunay was also a composer and artist as a young man, of sufficient talent that he could easily have pursued these careers.   In addition, he was one of the outstanding mountaineers of the USSR, and a mountain and other features near Mount Belukha in the Altai range are named for him.

Boris N. Delaunay was primarily a geometer – although he also contributed to number theory and to algebra – and invented Delaunay triangulation.  He was a co-organizer of the first Soviet Mathematics Olympiad, a mathematics competition for high-school students, in 1934.   One of his students was Aleksandr D. Alexandrov (1912-1999), founder of the Leningrad School of Geometry (which studies the differential geometry of curvature in manifolds, and the geometry of space-time).   Vadim Delone also showed mathematical promise and was selected to attend Moskovskaya Srednyaya Fiz Mat Shkola #2, Moscow Central Special High School No. 2 for Physics and Mathematics (now the Lyceum “Second School”). This school, established in 1958 for mathematically-gifted teenagers, was famously liberal and tolerant of dissent. (Indeed, so much so that in 1971-72, well after Delone had left, the school was purged by the CPSU.  See Hedrick Smith’s 1975 account here. Other special schools in Moscow focused on mathematics are #57 and #179. In London, in 2014, King’s College London established a free school, King’s Maths School, modelled on FizMatShkola #2.) Vadim Delone lived with Alexandrov when, serving out a one-year suspended sentence which required him to leave Moscow, he studied at university in Novosibirsk, Siberia.   At some risk to his own academic career, Alexandrov twice bravely visited Vadim Delone while he was in prison.

Delone’s wife, Irina Belgorodkaya, was also active in dissident circles, being arrested both in 1969 and again in 1973, and was sentenced to prison terms each time.  She was the daughter of a senior KGB official.  After his release in 1971 and hers in 1975, Delone and his wife emigrated to France in 1975, and he continued to write poetry.   In 1983, at the age of just 35, he died of cardiac arrest.   Given his youth, and the long lives of his father and grandfather, one has to wonder if this event was the dark work of an organ of Soviet state security.  According to then-KGB Chairman Yuri Andropov’s report to the Central Committee of the CPSU on the Moscow Seven’s protest in September 1968, Delone was the key link between the community of dissident poets and writers on the one hand, and that of mathematicians and physicists on the other.    Andropov even alleges that physicist Andrei Sakharov’s support for dissident activities was due to Delone’s personal persuasion, and that Delone lived from a so-called private fund, money from voluntary tithes paid by writers and scientists to support dissidents.   (Sharing of incomes in this way sounds suspiciously like socialism, which the state in the USSR always determined to maintain a monopoly of.)  That Andropov reported on this protest to the Central Committee, and less than a month after the event, indicates the seriousness with which this particular group of dissidents was viewed by the authorities.  That the childen of the nomenklatura, the intelligentsia, and even the KGB should be involved in these activities no doubt added to the concern.  If the KGB actually believed the statements Andropov made about Delone to the Central Committee, they would certainly have strong motivation to arrange his early death.

Several of the Moscow Seven were honoured in August 2008 by the Government of the Czech Republic, but as far as I am aware, no honour or recognition has yet been given them by the Soviet or Russian Governments.   Although my gesture will likely have little impact on the world, I salute their courage here.

I have translated a poem of Delone’s here.   An index to posts on The Matherati is here.


M. V. Ammosov [2009]:  Nikolai Borisovich Delone in my Life.  Laser Physics, 19 (8): 1488-1490.

Yuri Andropov [1968]: The Demonstration in Red Square Against the Warsaw Pact Invasion of Czechoslovakia. Report to the Central Committee of the CPSU, 1968-09-20. See below.

N. P. Dolbilin [2011]: Boris Nikolaevich Delone (Delaunay): Life and Work. Proceedings of the Steklov Institute of Mathematics, 275: 1-14.  Published in Russian in Trudy Matematicheskogo Instituta imeni V. A. Steklov, 2011, 275:  7-21.  Pre-print here.

Jane Gregory [2004]:  Subtle signs that divide the public from the privateThe Independent, 2004-05-20.

Hedrick Smith [1975]:  The Russians.  Crown.  pp. 211-213.


Andropov Reoport to the Central Committee of the CPSU on the protests in Red Square. (20 September 1968)

In characterizing the political views of the participants of the group, in particular DELONE, our source notes that the latter, “calling himself a bitter opponent of Soviet authority, fiercely detests communists, the communist ideology, and is entirely in agreement with the views of Djilas. In analyzing the activities . . . of the group, he (DELONE) explained that they do not have a definite program or charter, as in a formally organized political opposition, but they are all of the common opinion that our society is not developing normally, that it lacks freedom of speech and press, that a harsh censorship is operating, that it is impossible to express one’s opinions and thoughts, that democratic liberties are repressed. The activity of this group and its propaganda have developed mainly within a circle of writers, poets, but it is also enveloping a broad circle of people working in the sphere of mathematics and physics. They have conducted agitation among many scholars with the objective of inducing them to sign letters, protests, and declarations that have been compiled by the more active participants in this kind of activity, Petr IAKIR and Pavel LITVINOV. These people are the core around which the above group has been formed . . .. IAKIR and LITVINOV were the most active agents in the so-called “samizdat.”

This same source, in noting the condition of the arrested DELONE in this group, declared: “DELONE . . . has access to a circle of prominent scientists, academicians, who regarded him as one of their own, and in that way he served . . . to link the group with the scientific community, having influence on the latter and conducting active propaganda among them. Among his acquaintances he named academician Sakharov, who was initially cautious and distrustful of the activities of IAKIR, LITVINOV, and their group; he wavered in his position and judgments, but gradually, under the influence of DELONE’s explanations, he began to sign various documents of the group. . . ; [he also named] LEONTOVICH, whose views coincide with those of the group. In DELONE’s words, many of the educated community share their views, but are cautious, fearful of losing their jobs and being expelled from the party.” . . . [more details on DELONE]

Agents’ reports indicate that the participants of the group, LITVINOV, DREMLIUGA, AND DELONE, have not been engaged in useful labor for an extended period, and have used the means of the so-called “private fund,” which their group created from the contributions of individual representatives in the creative intelligentsia and scientists.

The prisoner DELONE told our source: “We are assisted by monetary funds from the intelligentsia, highly paid academicians, writers, who share the views of the Iakir-Litvinov group . . . [Sic] We have the right to demand money, [because] we are the functionaries, while they share our views, [but] fear for their skins, so let them support us
with money.”