Archive for the ‘Mathematics’ Category

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Precision as the enemy of knowledge

Published by peter on 2010-08-09 in Africa, Computer Science, Economics and Mathematics. 0 Comments

I have posted previously about the different ways in which knowledge may be represented.  A key learning of the discipline of Artificial Intelligence in its short life thus far is that not all representations are equal.    Indeed, more precise representations may provide less information, as in this example from cartography (from a profile of economist Paul Krugman):

Again, as in his [Krugman’s] trade theory, it was not so much his idea [that regional ecomomic specializations were essentially due to historical accidents] that was significant as the translation of the idea into a mathematical language.  “I explained this basic idea” – of economic geography – “to a non-economist friend,” Krugman wrote, “who replied in some dismay, ‘Isn’t that pretty obvious?’  And of course it is.”  Yet, because it had not been well modelled, the idea had been disregarded by economists for years.  Krugman began to realize that in the previous few decades economic knowledge that had not been translated into [tractable analytical mathematical] models had been effectively lost, because economists didn’t know what to do with it.  His friend Craig Murphy, a political scientist at Wellesley, had a collection of antique maps of Africa, and he told Krugman that a similar thing had happened in cartography.  Sixteenth century maps of Africa were misleading in all kinds of ways, but they contained quite a bit of information about the continent’s interior – the River Niger, Timbuktu.  Two centuries later, mapmaking had become more accurate, but the interior of Africa had become a blank.  As standards for what counted as a mappable fact rose, knowledge that didn’t meet those standards – secondhand travellers’ reports, guesses hazarded without compasses or sextants – was discarded and lost.  Eventually, the higher standards paid off – by the nineteenth century the maps were filled in again – but for a while the sharpening of technique caused loss as well as gain. ” (page 45)

Reference:
Larissa MacFarquhar [2010]:  The deflationist:  How Paul Krugman found politics.  The New Yorker, 2010-03-01, pp. 38-49.

Symphonic Form

Published by peter on 2010-08-03 in Argumentation, Mathematics and Music. 0 Comments
Composer and musicologist Kyle Gann has an interesting post citing David Fanning’s quotation of Russian musicologist Mark Aranovsky’s classification of the movements of the typical symphony, a classification which runs as follows:
  • Movement #1:  Homo agens: man acting, or in conflict (Allegro)
  • Movement #2: Homo sapiens: man thinking (Adagio)
  • Movement #3:  Homo ludens: man playing (Scherzo), and
  • Movement #4:  Homo communis: man in the community (Allegro)
This makes immense sense, and provides a neat explanation of the structure of symphonic form.  Many of my long-standing questions are answered with this classification.    Why normally 4 movements?  Why is the first one normally louder and faster and more serious than the next two?  And why does the first movement often seem more like an ending movement than a beginning one?   In other words, why is the climax to the first movement so often more impressive and more compelling than that for the other movements?  Why is there usually a middle movement that is noticeably less serious than the outer movements?  Why is the last movement often in rondo form?  Why do some composers (eg, Mozart, Mendelssohn) include a fugue in their last movements?  Why do some composers include a song to brotherly love  (Beethoven) or a hymn (Mendelssohn)  in their last movements?
Of great relevance here is that the German word for movement (of a musical work) is Satz, meaning “sentence”.  In the German art-musical tradition, a musical work first makes some claim or states some musical position, and then (in Sonata form) argues the case for that claim by exploring the musical consequences of the theme (or themes), or of its  component musical parts, before returning to a re-statement of the initial claim (theme) at the end of the movement.  In this tradition, the theme, being a claim which is developed, does not have to be very interesting or melodious in itself, since its purpose is not to please the ear but to announce a position.   Beethoven, for example, was notorious for not writing good melodies:  his most famous theme, that of the first movement of the 5th Symphony, has just 4 notes, of which 3 are identical and are repeated together.  But he was a superb developer, perhaps one of the best, of themes, even of such apparently insignificant ones as this one.
The distinction between writing good melodies and developing them well strikes me as very similar to that between problem-solving and theory-building mathematicians – both these cases essentially involve a difference between exploring and exploiting.

Five minutes of freedom

Published by peter on 2010-08-01 in Argumentation, Heroes, History, Intelligence and Espionage, Literature, Mathematics, Matherati, Poetry and Politics. 0 Comments

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 private.  The 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.”

Bayesian statistics

Published by peter on 2010-07-08 in AI, Decision theory, Mathematics and Uncertainty. Closed

One of the mysteries to anyone trained in the frequentist hypothesis-testing paradigm of statistics, as I was, and still adhering to it, as I do, is how Bayesian approaches seemed to have taken the academy by storm.   One wonders, first, how a theory based – and based explicitly – on a measure of uncertainty defined in terms of subjective personal beliefs, could be considered even for a moment for an inter-subjective (ie, social) activity such as Science.

One wonders, second, how a theory justified by appeals to such socially-constructed, culturally-specific, and readily-contestable activities as gambling (ie, so-called Dutch-book arguments) could be taken seriously as the basis for an activity (Science) aiming for, and claiming to achieve, universal validity.   One wonders, third, how the fact that such justifications, even if gambling presents no moral, philosophical or other qualms,  require infinite sequences of gambles is not a little troubling for all of us living in this finite world.  (You tell me you are certain to beat me if we play an infinite sequence of gambles? Then, let me tell you, that I have a religion promising eternal life that may interest you in turn.)

One wonders, fourthly, where are recorded all the prior distributions of beliefs which this theory requires investigators to articulate before doing research.  Surely someone must be writing them down, so that we consumers of science can know that our researchers are honest, and hold them to potential account.   That there is such a disconnect between what Bayesian theorists say researchers do and what those researchers demonstrably do should trouble anyone contemplating a choice of statistical paradigms, surely. Finally, one wonders how a theory that requires non-zero probabilities be allocated to models of which the investigators have not yet heard or even which no one has yet articulated, for those models to be tested, passes muster at the statistical methodology corral.

To my mind, Bayesianism is a theory from some other world – infinite gambles, imagined prior distributions, models that disregard time or requirements for constructability,  unrealistic abstractions from actual scientific practice – not from our own.

So, how could the Bayesians make as much headway as they have these last six decades? Perhaps it is due to an inherent pragmatism of statisticians – using whatever techniques work, without much regard as to their underlying philosophy or incoherence therein.  Or perhaps the battle between the two schools of thought has simply been asymmetric:  the Bayesians being more determined to prevail (in my personal experience, to the point of cultism and personal vitriol) than the adherents of frequentism.  Greg Wilson’s 2001 PhD thesis explored this question, although without finding definitive answers.

Now, Andrew Gelman and the indefatigable Cosma Shalizi have written a superb paper, entitled “Philosophy and the practice of Bayesian statistics”.  Their paper presents another possible reason for the rise of Bayesian methods:  that Bayesianism, when used in actual practice, is most often a form of hypothesis-testing, and thus not as untethered to reality as the pure theory would suggest.  Their abstract:

A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico-deductivism.  We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science.

Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.

References:
Andrew Gelman and Cosma Rohilla Shalizi [2010]:  Philosophy and the practice of Bayesian statistics.  Available from Arxiv.  Blog post here.

Gregory D. Wilson [2001]:   Articulation Theory and Disciplinary Change:  Unpacking the Bayesian-Frequentist Paradigm Conflict in Statistical Science.  PhD Thesis,  Rhetoric and Professional Communication Programme, New Mexico State University.  Las Cruces, NM, USA.  July 2001.

The cultures of mathematics education

Published by peter on 2010-07-05 in Culture, Mathematics and Matherati. Closed

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 that 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).

The glass bead game of mathematical economics

Published by peter on 2010-07-01 in Decision theory, Economics and Mathematics. Closed

Over at the economics blog, A Fine Theorem, there is a post about economic modelling.
My first comment is that the poster misunderstands the axiomatic method in pure mathematics.  It is not the case that “axioms are by assumption true”.  Truth is a bivariant relationship between some language or symbolic expression and the world.  Pure mathematicians using axiomatic methods make no assumptions about the relationship between their symbolic expressions of interest and the world.   Rather they deduce consequences from the axioms, as if those axioms were true, but without assuming that they are.    How do I know they do not assume their axioms to be true?  Because mathematicians often work with competing, mutually-inconsistent, sets of axioms, for example when they consider both Euclidean and non-Euclidean geometries, or when looking at systems which assume the Axiom of Choice and systems which do not.   Indeed, one could view parts of the meta-mathematical theory called Model Theory as being the formal and deductive exploration of multiple, competing sets of axioms.
On the question of economic modeling, the blogger presents the views of Gerard Debreu on why the abstract mathematicization of economics is something to be desired.   One should also point out the very great dangers of this research program, some of which we are suffering now.  The first is that people — both academic researchers and others — can become so intoxicated with the pleasures of mathematical modeling that they mistake the axioms and the models for reality itself.  Arguably the widespread adoption of financial models assuming independent and normally-distributed errors was the main cause of the Global Financial Crisis of 2008, where the errors of complex derivative trades (such as credit default swaps) were neither independent nor as thin-tailed as Normal distributions are.  The GFC led, inexorably, to the Great Recession we are all in now.
Secondly, considered only as a research program, this approach has serious flaws.  If you were planning to construct a realistic model of human economic behaviour in all its diversity and splendour, it would be very odd to start by modeling only that one very particular, and indeed pathological, type of behaviour examplified by homo economicus, so-called rational economic man.   Acting with with infinite mental processing resources and time, with perfect knowledge of the external world, with perfect knowledge of his own capabilities, his own goals, own preferences, and indeed own internal knowledge, with perfect foresight or, if not, then with perfect knowledge of a measure of uncertainty overlaid on a pre-specified sigma-algebra of events, and completely unencumbered with any concern for others, with any knowledge of history, or with any emotions, homo economicus is nowhere to be found on any omnibus to Clapham.  Starting economic theory with such a creature of fiction would be like building a general theory of human personality from a study only of convicted serial killers awaiting execution, or like articulating a general theory of evolution using only a hand-book of British birds.   Homo economicus is not where any reasonable researcher interested in modeling the real world would start from in creating a theory of economic man.
And, even if this starting point were not on its very face ridiculous, the fact that economic systems are complex adaptive systems should give economists great pause.   Such systems are, typically, not continuously dependent on their initial conditions, meaning that a small change in input parameters can result in a large change in output values.   In other words, you could have a model of economic man which was arbitrarily close to, but not identical with, homo economicus, and yet see wildly different behaviours between the two.  Simply removing the assumption of infinite mental processing resources creates a very different economic actor from the assumed one, and consequently very different properties at the level of economic systems.  Faced with such overwhelming non-continuity (and non-linearity), a naive person might expect economists to be humble about making predictions or giving advice to anyone living outside their models.   Instead, we get an entire profession labeling those human behaviours which their models cannot explain as “irrational”.
My anger at The Great Wen of mathematical economics arises because of the immorality this discipline evinces:   such significant and rare mathematical skills deployed, not to help alleviate suffering or to make the world a better place (as those outside Economics might expect the discipline to aspire to), but to explore the deductive consequences of abstract formal systems, systems neither descriptive of any reality, nor even always implementable in a virtual world.

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

Published by peter on 2010-05-25 in Mathematics and Matherati. Closed

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.”

Combinatorics of some musical objects

Published by peter on 2010-05-05 in Mathematics and Music. Closed

Excerpts from Appendix C (page 164) from Keith [1991].  All results assume a 12-tone equal-tempered scale.

Number of diatonic scale classes: 3
Number of note names (A-G); number of notes in a common scale; number of white keys per octave on a piano:  7
Number of scales one note different from the Major scale: 9
Number of notes in the most common equal-tempered scale:  12
Number of common musical keys (C + 1-6 flats/sharps):  13
Number of 7-note diatonic scales (=7 * 3):  21
Number of elementary 2-fold polychords: 23
A k-fold polychord is an n-note chord sub-divided into k non-empty subchords, for k=1,  . . ., n.  For example, the 6-note chord <C, D, E, F#, G, A> can be subdivided into the 3-note 2-fold polychords, <C, E, G> and <D, F#, A>.
Number of 7-note chords:  66
Number of distinct interval sets (partitions of 12):  77
Number of 7-note triatonic scales (=7*35):  245
Number of notationally-distinct diatonic scales (=13 *21):  273
Number of distinct chord-types (= N(12) – 1):  351
Number of 7-note musical scales (=7*66):  462
Number of scales (=Number of n-note scales, summed over all n)  (=2^(12-1) = 2^11):   2048
Number of chords without rotational isomorphism (= 2^12 – 1):  4095
Number of notationally-distinct scales (=13 * 462):  6006
Number of non-syncopated 8-bar 1/4-note rhythmic patterns:  458,330
Number of non-syncopated 8-bar 1/8-note rhythmic patterns:  210,066,388,901

Reference:
Michael Keith [1991]:  From Polychords to Polya:  Adventures in Musical Combinatorics.  (Princeton, NJ:  Vinculum Press.)

This Much I Know (about CS and AI)

Published by peter on 2010-05-05 in Computer Science, Computing-as-interaction, Economics and Mathematics. 0 Comments

Inspired by The Guardian column of the same name, I decided to list here my key learnings of the last several years regarding Computer Science and Artificial Intelligence (AI). Few of these are my own insights, and I welcome comments and responses. From arguments I have had, I know that some of these statements are controversial; this fact surprises me, since most of them seem obvious to me. Statements are listed, approximately, from the more general to the more specific.

Macho mathematicians

Published by peter on 2010-03-23 in Creativity, Human intelligence, Mathematics, Matherati, Music and Team working. 0 Comments

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.