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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.

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Great mathematical ideas

Normblog has a regular feature, Writer’s Choice, where writers give their opinions of books which have influenced them.   Seeing this led me recently to think of the mathematical ideas which have influenced my own thinking.   In an earlier post, I wrote about the writers whose  books (and teachers whose lectures) directly influenced me.  I left many pure mathematicians and statisticians off that list because most mathematics and statistics I did not receive directly from their books, but indirectly, mediated through the textbooks and lectures of others.  It is time to make amends. 

Here then is a list of mathematical ideas which have had great influence on my thinking, along with their progenitors.  Not all of these ideas have yet proved useful in any practical sense, either to me or to the world – but there is still lots of time.   Some of these theories are very beautiful, and it is their elegance and beauty and profundity to which I respond.  Others are counter-intuitive and thus thought-provoking, and I recall them for this reason.

  • Euclid’s axiomatic treatment of (Euclidean) geometry
  • The various laws of large numbers, first proven by Jacob Bernoulli (which give a rational justification for reasoning from samples to populations)
  • The differential calculus of Isaac Newton and Gottfried Leibniz (the first formal treatment of change)
  • The Identity of Leonhard Euler:  exp ( i * \pi) + 1 = 0, which mysteriously links two transcendental numbers (\pi and e), an imaginary number i (the square root of minus one) with the identity of the addition operation (zero) and the identity of the multiplication operation (1).
  • The epsilon-delta arguments for the calculus of Augustin Louis Cauchy and Karl Weierstrauss
  • The non-Euclidean geometries of Janos Bolyai, Nikolai Lobachevsky and Bernhard Riemann (which showed that 2-dimensional (or plane) geometry would be different if the surface it was done on was curved rather than flat – the arrival of post-modernism in mathematics)
  • The diagonalization proof of Gregor Cantor that the Real numbers are not countable (showing that there is more than one type of infinity) (a proof-method later adopted by Godel, mentioned below)
  • The axioms for the natural numbers of Guiseppe Peano
  • The space-filling curves of Guiseppe Peano and others (mapping the unit interval continuously to the unit square)
  • The axiomatic treatments of geometry of Mario Pieri and David Hilbert (releasing pure mathematics from any necessary connection to the real-world)
  • The algebraic topology of Henri Poincare and many others (associating algebraic structures to topological spaces)
  • The paradox of set theory of Bertrand Russell (asking whether the set of all sets contains itself)
  • The Fixed Point Theorem of Jan Brouwer (which, inter alia, has been used to prove that certain purely-artificial mathematical constructs called economies under some conditions contain equilibria)
  • The theory of measure and integration of Henri Lebesgue
  • The constructivism of Jan Brouwer (which taught us to think differently about mathematical knowledge)
  • The statistical decision theory of Jerzy Neyman and Egon Pearson (which enabled us to bound the potential errors of statistical inference)
  • The axioms for probability theory of Andrey Kolmogorov (which formalized one common method for representing uncertainty)
  • The BHK axioms for intuitionistic logic, associated to the names of Jan Brouwer, Arend Heyting and Andrey Kolmogorov (which enabled the formal treatment of intuitionism)
  • The incompleteness theorems of Kurt Godel (which identified some limits to mathematical knowledge)
  • The theory of categories of Sam Eilenberg and Saunders Mac Lane (using pure mathematics to model what pure mathematicians do, and enabling concise, abstract and elegant presentations of mathematical knowledge)
  • Possible-worlds semantics for modal logics (due to many people, but often named for Saul Kripke)
  • The topos theory of Alexander Grothendieck (generalizing the category of sets)
  • The proof by Paul Cohen of the logical independence of the Axiom of Choice from the Zermelo-Fraenkel axioms of Set Theory (which establishes Choice as one truly weird axiom!)
  • The non-standard analysis of Abraham Robinson and the synthetic geometry of Anders Kock (which formalize infinitesimal arithmetic)
  • The non-probabilistic representations of uncertainty of Arthur Dempster, Glenn Shafer and others (which provide formal representations of uncertainty without the weaknesses of probability theory)
  • The information geometry of Shunichi Amari, Ole Barndorff-Nielsen, Nikolai Chentsov, Bradley Efron, and others (showing that the methods of statistical inference are not just ad hoc procedures)
  • The robust statistical methods of Peter Huber and others 
  • The proof by Andrew Wiles of The Theorem Formerly Known as Fermat’s Last (which proof I don’t yet follow).

Some of these ideas are among the most sublime and beautiful thoughts of humankind.  Not having an education which has equipped one to appreciate these ideas would be like being tone-deaf.

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Nicolas Fatio de Duillier

Fatio de DuillierNicolas Fatio de Duillier (1664-1753) was a Genevan mathematician and polymath, who for a time in the 1680s and 1690s, was a close friend of Isaac Newton. After coming to London in 1687, he became a Fellow of the Royal Society (on 1688-05-15), as later did his brother Jean-Christophe (on 1706-04-03).  He played a major part in Newton’s feud with Leibniz over who had invented the differential calculus, and was a protagonist all his life for Newton’s thought and ideas.

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Guerrilla logic: a salute to Mervyn Pragnell

When a detailed history of computer science in Britain comes to be written, one name that should not be forgotten is Mervyn O. Pragnell.  As far as I am aware, Mervyn Pragnell never held any academic post and he published no research papers.   However, he introduced several of the key players in British computer science to one another, and as importantly, to the lambda calculus of Alonzo Church (Hodges 2001).  At a time (the 1950s and 1960s) when logic was not held in much favour in either philosophy or pure mathematics, and before it became to be regarded highly in computer science, he studied the discipline not as a salaried academic in a university, but in a private reading-circle of his own creation, almost as a guerrilla activity.

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Alan Turing

Yesterday, I reported on the the restoration of the world’s oldest, still-working modern computer.  Last night, British Prime Minister Gordon Brown apologized for the country’s treatment of Alan Turing, computer pioneer.  In the words of Brown’s statement:

Turing was a quite brilliant mathematician, most famous for his work on breaking the German Enigma codes. It is no exaggeration to say that, without his outstanding contribution, the history of World War Two could well have been very different. He truly was one of those individuals we can point to whose unique contribution helped to turn the tide of war. The debt of gratitude he is owed makes it all the more horrifying, therefore, that he was treated so inhumanely. In 1952, he was convicted of ‘gross indecency’ – in effect, tried for being gay. His sentence – and he was faced with the miserable choice of this or prison – was chemical castration by a series of injections of female hormones. He took his own life just two years later.”

It might be considered that this apology required no courage of Brown.  This is not the case.  Until very recently, and perhaps still today, there were people who disparaged and belittled Turing’s contribution to computer science and computer engineering.  The conventional academic wisdom is that he was only good at the abstract theory and at the formal mathematizing (as in his “schoolboy essay” proposing a test to distinguish human from machine interlocuters), and not good for anything practical.   This belief is false.  As the philosopher and historian  B. Jack Copeland has shown, Turing was actively and intimately involved in the design and construction work (mechanical & electrical) of creating the machines developed at Bletchley Park during WWII, the computing machines which enabled Britain to crack the communications codes used by the Germans.

Turing-2004-Poster

Perhaps, like myself, you imagine this revision to conventional wisdom would be uncontroversial.  Sadly, not.  On 5 June 2004, I attended a symposium in Cottonopolis to commemorate the 50th anniversary of Turing’s death.  At this symposium, Copeland played a recording of an oral-history interview with engineer Tom Kilburn (1921-2001), first head of the first Department of Computer Science in Britain (at the University of Manchester), and also one of the pioneers of modern computing.   Kilburn and Turing had worked together in Manchester after WW II.  The audience heard Kilburn stress to his interviewer that what he learnt from Turing about the design and creation of computers was all high-level (ie, abstract) and not very much, indeed only about 30 minutes worth of conversation.  Copeland then produced evidence (from signing-in books) that Kilburn had attended a restricted, invitation-only, multi-week, full-time course on the design and engineering of computers which Turing had presented at the National Physical Laboratories shortly after the end of WW II, a course organized by the British Ministry of Defence to share some of the learnings of the Bletchley Park people in designing, building and operating computers.   If Turing had so little of practical relevance to contribute to Kilburn’s work, why then, one wonders, would Kilburn have turned up each day to his course.

That these issues were still fresh in the minds of some people was shown by the Q&A session at the end of Copeland’s presentation.  Several elderly members of the audience, clearly supporters of Kilburn, took strident and emotive issue with Copeland’s argument, with one of them even claiming that Turing had contributed nothing to the development of computing.   I repeat: this took place in Manchester 50 years after Turing’s death!    Clearly there were people who did not like Turing, or in some way had been offended by him, and who were still extremely upset about it half a century later.  They were still trying to belittle his contribution and his practical skills, despite the factual evidence to the contrary.

I applaud Gordon Brown’s courage in officially apologizing to Alan Turing, an apology which at least ensures the historical record is set straight for what our modern society owes this man.

POSTSCRIPT #1 (2009-10-01): The year 2012 will be a centenary year of celebration of Alan Turing.

POSTSCRIPT #2 (2011-11-18):  It should also be noted, concerning Mr Brown’s statement, that Turing died from eating an apple laced with cyanide.  He was apparently in the habit of eating an apple each day.   These two facts are not, by themselves, sufficient evidence to support a claim that he took his own life.

POSTSCRIPT #3 (2013-02-15):  I am not the only person to have questioned the coroner’s official verdict that Turing committed suicide.    The BBC reports that Jack Copeland notes that the police never actually tested the apple found beside Turing’s body for traces of cyanide, so it is quite possible it had no traces.     The possibility remains that he died from an accidental inhalation of cyanide or that he was deliberately poisoned.   Given the evidence, the only rational verdict is an open one.

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Social forecasting: Doppio Software

Five years ago, back in the antediluvian era of Web 2.0 (the web as enabler and facilitator of social networks), we had the idea of  social-network forecasting.  We developed a product to enable a group of people to share and aggregate their forecasts of something, via the web.  Because reducing greenhouse gases were also becoming flavour-du-jour, we applied these ideas to social forecasts of the price for the European Union’s carbon emission permits, in a nifty product we called Prophets-360.  Sadly, due mainly to poor regulatory design of the European carbon emission market, supply greatly outstripped demand for emissions permits, and the price of permits fell quickly and has mostly stayed fallen.  A flat curve is not difficult to predict, and certainly there was little value in comparing one person’s forecast with that of another.  Our venture was also felled.

But now the second generation of social networking forecasting tools has arrived.  I see that a French start-up, Doppio Software, has recently launched publicly.   They appear to have a product which has several advantages over ours:

  • Doppio Software is focused on forecasting demand along a supply chain.  This means the forecasting objective is very tactical, not the long-term strategic forecasting that CO2 emission permit prices became.   In the present economic climate, short-term tactical success is certainly more compelling to business customers than even looking five years hence.
  • The relevant social network for a supply chain is a much stronger community of interest than the amorphous groups we had in mind for Prophets-360.  Firstly, this community already exists (for each chain), and does not need to be created.  Secondly, the members of the community by definition have differential access to information, on the basis of their different positions up and down the chain.  Thirdly, although the interests of the partners in a supply chain are not identical, these interests are mutually-reinforcing:  everyone in the chain benefits if the chain itself is more successful at forecasting throughput.
  • In addition, Team Doppio (the Doppiogangers?) appear to have included a very compelling value-add:  their own automated modeling of causal relationships between the target demand variables of each client and general macro-economic variables, using  semantic-web data and qualitative modeling technologies from AI.  Only the largest manufacturing companies can afford their own econometricians, and such people will normally only be able to hand-craft models for the most important variables.  There are few companies IMO who would not benefit from Doppio’s offer here.

Of course, I’ve not seen the Doppio interface and a lot will hinge on its ease-of-use (as with all software aimed at business users).  But this offer appears to be very sophisticated, well-crafted and compelling, combining social network forecasting, intelligent causal modeling and semantic web technologies.

Well done, Team Doppio!  I wish you every success with this product!

PS:  I have just learnt that “doppio” means “double”, which makes it a very apposite name for this application – forecasts considered by many people, across their human network.  Neat!  (2009-09-16)

Article in The Observer (UK) about Doppio 2009-09-06 here. And here is an AFP TV news story (2009-09-15) about Doppio co-founder, Edouard d’Archimbaud.  Another co-founder is Benjamin Haycraft.

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