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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).

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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.

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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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