Archive for the 'Matherati' Category Page 4 of 6



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.

References:

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.

References:

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.

APPENDIX

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.”




The cultures of mathematics education

I posted recently about the macho culture of pure mathematics, and the undue focus that school mathematics education has on problem-solving and competitive games.

I have just encountered an undated essay, “The Two Cultures of Mathematics”, by Fields Medallist Timothy Gowers, currently Rouse Ball Professor of Mathematics at Cambridge.    Gowers identifies two broad types of research pure mathematicians:  problem-solvers and theory-builders.  He cites Paul Erdos as an example of the former (as I did in my earlier post), and Michael Atiyah as an example of the latter.   What I find interesting is that Gowers believes the the profession as a whole currently favours theory-builders over problem-solvers.  And domains of mathematics where theory-building is currently more important (such as Geometry and Algebraic Topology) are favoured over domains of mathematics where problem-solving is currently more important (such as Combinatorics and Graph Theory).

I agree with Gowers here, and wonder, then, why the teaching of mathematics at school still predominantly favours problem-solving over theory-building activities, despite a century of Hilbertian and Bourbakian axiomatics.   Is it because problem-solving was the predominant mode of British  mathematics in the 19th century (under the pernicious influence of the Cambridge Mathematics Tripos, which retarded pure mathematics in the Anglophone world for a century) and school educators are slow to catch-on with later trends?  Or, is it because the people designing and implementing school mathematics curricula are people out of sympathy with, and/or not competent at, theory-building?  Certainly, if your over-riding mantra for school education is instrumental relevance than the teaching of abstract mathematical theories may be hard to justify  (as indeed is the teaching of music or art or ancient Greek).   This perhaps explains how I could learn lots of tricks for elementary arithmetic in day-time classes at primary school, but only discover the rigorous beauty of Euclid’s geometry in special after-school lessons from a sympathetic fifth-grade teacher (Frank Torpie).




Vale: Martin Gardner: Defending the honor of the human mind!

The death has just occurred of Martin Gardner (1914-2010), for 25 years (1956-1981) the writer of the superb Mathematical Games column of Scientific American.   I remember eagerly seeking each new copy of SciAm in my local public library to read Gardner’s column each month,  and devouring all of his books that I could find.  His articles interested me despite my general contempt for games and competitions, and for ad hoc approaches to mathematical reasoning.

Scientific American’s tribute page is here, and here is a just-posted transcript of a February 1979 conversation between Gardner and other mathematicians.   This transcript contains a wonderful statement by mathematician Stan Ulam:

In fact, you know, yesterday Ron Graham gave a marvelous, really interesting lecture about some esoteric question; and I was wondering during it, Well, the question sounds very complicated, why devote so much ingenuity? Then I remember what, I think, Fourier or Laplace wrote: That mathematics—one reason for its being—is to defend the honor of the human mind.”

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Vale: Robin Milner

 

The death has just occurred of Robin Milner (1934-2010), one of the founders of theoretical computer science.   Milner was an ACM Turing Award winner and his main contributions were a formal theory of concurrent communicating processes and, more recently, a category-theoretic account of hyperlinks and embeddings, his so-called theory of bigraphs.   As we move into an era where the dominant metaphor for computation is computing-as-interaction, the idea of concurrency has become increasingly important; however, understanding, modeling and managing it have proven to be among the most difficult conceptual problems in modern computer science.  Alan Turing gave the world a simple mathematical model of computation as the sequential writing or erasing of characters on a linear tape under a read/write head, like a single strip of movie film passing back and forth through a projector.  Despite the prevalence of the Internet and of ambient, ever-on, and ubiquitous computing, we still await a similar mathematical model of interaction and interacting processes.  Milner’s work is a major contribution to developing such a model. In his bigraphs model, for example, one graph represents the links between entities while the other represents geographic proximity or organizational hierarchy.

Robin was an incredibly warm, generous and unprepossessing man.   About seven years ago, without knowing him at all, I wrote to him inviting him to give an academic seminar; even though famous and retired, he responded positively, and was soon giving a very entertaining talk on bigraphs (a representation of which is on the blackboard behind him in the photo).  He joined us for drinks in the pub afterwards, buying his round like everyone else, and chatting amicably with all, talking both about the war in Iraq and the problems of mathematical models based on pre-categories.  He always responded immediately to any of my occasional emails subsequently.

The London Times has an obituary here, and the Guardian here (from which the photo is borrowed).

References:

Robin Milner [1989]: Communication and Concurrency. Prentice Hall.

Robin Milner [1999]: Communicating and Mobile Systems: the Pi-Calculus. Cambridge University Press.

Robin Milner [2009]: The Space and Motion of Communicating Agents. Cambridge University Press.

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Macho mathematicians

Pianist and writer Susan Tomes has just published a new book, Out of Silence, which the Guardian has excerpted here.  This story drew my attention:

Afterwards, my husband and I reminisced about our attempts to learn tennis when we were young. I told him that my sisters and I used to go down to the public tennis courts in Portobello. We had probably never seen a professional tennis match; we just knew that tennis was about hitting the ball to and fro across the net. We had a few lessons and became quite good at leisurely rallies, hitting the ball back and forth without any attempt at speed. Sometimes we could keep our rallies going for quite a long time, and I found this enjoyable.

Then our tennis teacher explained that we should now learn to play “properly”. It was only then that I realised we were meant to hit the ball in such a way that the other person could not hit it back. This came as an unpleasant surprise. As soon as we started “playing properly”, our points became extremely short. One person served, the other could not hit it back, and that was the end of the point. It seemed to me that there was skill in hitting the ball so that the other person could hit it back. If they could, the ball would flow, one got to move about and there was not much interruption to the rhythm of play. It struck me that hitting the ball deliberately out of the other person’s reach was unsportsmanlike. When I tell my husband all this, he laughs and says: “There speaks a true chamber musician.”

This story resonated strongly with me.  Earlier this year, I had a brief correspondence with mathematician Alexandre Borovik, who has been collecting accounts of childhood experiences of learning mathematics, both from mathematicians and from non-mathematicians.  After seeing a discussion on his blog about the roles of puzzles and games in teaching mathematics to children, I had written to him:

Part of my anger & frustration at school was that so much of this subject that I loved, mathematics, was wasted on what I thought was frivolous or immoral applications:   frivolous because of all those unrealistic puzzles, and immoral because of the emphasis on competition (Olympiads, chess, card games, gambling, etc).   I had (and retain) a profound dislike of competition, and I don’t see why one always had to demonstrate one’s abilities by beating other people, rather than by collaborating with them.  I believed that “playing music together”, rather than “playing sport against one another”, was a better metaphor for what I wanted to do in life, and as a mathematician.

Indeed, the macho competitiveness of much of pure mathematics struck me very strongly when I was an undergraduate student:  I switched then to mathematical statistics because the teachers and students in that discipline were much less competitive towards one another.  For a long time, I thought I was alone in this view, but I have since heard the same story from other people, including some prominent mathematicians.  I know one famous category theorist who switched from analysis as a graduate student because the people there were too competitive, while the category theory people were more co-operative.

Perhaps the emphasis on puzzles & tricks is fine for some mathematicians – eg, Paul Erdos seems to have been motivated by puzzles and eager to solve particular problems.  However, it is not fine for others – Alexander Grothendieck comes to mind as someone interested in abstract frameworks rather than puzzle-solving.  Perhaps the research discipline of pure mathematics needs people of both types.  If so, this is even more reason not to eliminate all the top-down thinkers by teaching only using puzzles at school.”

More on the two cultures of mathematics here.

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Theatre Lakatos

Last night, I caught a new Australian play derived from the life of logician Kurt Godel, called Incompleteness.  The play is by playwright Steven Schiller and actor Steven Phillips, and was peformed at Melbourne’s famous experimental theatrespace, La Mama, in Carlton. Both script and performance were superb:  Congratulations to both playwright and actor, and to all involved in the production.

Godel was famous for having kept every piece of paper he’d ever encountered, and the set design (pictured here) included many file storage boxes.  Some of these were arranged in a checkerboard pattern on the floor, with gaps between them.  As the Godel character (Phillips) tried to prove something, he took successive steps along diagonal and zigzag paths through this pattern, sometimes retracing his steps when potential chains of reasoning did not succeed.   This was the best artistic representation I have seen of the process of attempting to do mathematical proof:  Imre Lakatos’ philosophy of mathematics made theatrical flesh.

The photograph of the La Mama billboard is from Paola’s site.

Incompleteness- lamama 2009

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The Mathematical Tripos at Cambridge

From the 18th century until 1909, students at Cambridge University took a compulsory series of examinations, called the Mathematical Tripos, named after the three-legged stool that candidates originally sat on.  Until the mid-18th century, these examinations were conducted orally, and only became written examinations over faculty protests.   Apparently, not everyone believed that written examinations were the best or fairest way to test mathematical abilities, a view which would amaze many contemporary people  – although oral examinations in mathematics are still commonly used in some countries with very strong mathematical traditions, such as Russia and the other states of the former USSR.

The Tripos became a notable annual public event in the 19th century, with The Times newspaper publishing articles and biographies before each examination on the leading candidates, and then, after each examination, the results.   There was considerable public interest in the event each year, not just in Cambridge or among mathematicians, and widespread betting on the outcomes.

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Australian logic: a salute to Malcolm Rennie

Recently, I posted a salute to Mervyn Pragnell, a logician who was present in the early days of computer science.  I was reminded of the late Malcolm Rennie, the person who introduced me to formal logic, and whom I acknowledged here.   Rennie was the most enthusiastic and inspiring lecturer I ever had, despite using no multi-media wizardry, usually not even an overhead projector.  Indeed, he mostly just sat and spoke, moving his body as little as possible and writing only sparingly on the blackboard, because he was in constant pain from chronic arthritis.   He was responsible for part of an Introduction to Formal Logic course I took in my first year (the other part was taken by Paul Thom, for whom I wrote an essay on the notion of entailment in a system of Peter Geach).   The students in this course were a mix of first-year honours pure mathematicians and later-year philosophers (the vast majority), and most of the philosophers struggled with non-linguistic representations (ie, mathematical symbols).  Despite the diversity, Rennie managed to teach to all of us, providing challenging questions and discussions with and for both groups.   He was also a regular entrant in the competitions which used to run in the weekly Nation Review (and a fellow-admirer of the My Sunday cartoons of Victoria Roberts), and I recall one occasion when a student mentioned seeing his name as a competition winner, and the class was then diverted into an enjoyable discussion of tactics for these competitions.

Continue reading ‘Australian logic: a salute to Malcolm Rennie’