Symphonic Form

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

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

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

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

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

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

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

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

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

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

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

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


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

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

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

Jane Gregory [2004]:  Subtle signs that divide the public from the privateThe Independent, 2004-05-20.
Hedrick Smith [1975]:  The Russians.  Crown.  pp. 211-213.


Andropov Reoport to the Central Committee of the CPSU on the protests in Red Square. (20 September 1968)
In characterizing the political views of the participants of the group, in particular DELONE, our source notes that the latter, “calling himself a bitter opponent of Soviet authority, fiercely detests communists, the communist ideology, and is entirely in agreement with the views of Djilas. In analyzing the activities . . . of the group, he (DELONE) explained that they do not have a definite program or charter, as in a formally organized political opposition, but they are all of the common opinion that our society is not developing normally, that it lacks freedom of speech and press, that a harsh censorship is operating, that it is impossible to express one’s opinions and thoughts, that democratic liberties are repressed. The activity of this group and its propaganda have developed mainly within a circle of writers, poets, but it is also enveloping a broad circle of people working in the sphere of mathematics and physics. They have conducted agitation among many scholars with the objective of inducing them to sign letters, protests, and declarations that have been compiled by the more active participants in this kind of activity, Petr IAKIR and Pavel LITVINOV. These people are the core around which the above group has been formed . . .. IAKIR and LITVINOV were the most active agents in the so-called “samizdat.”
This same source, in noting the condition of the arrested DELONE in this group, declared: “DELONE . . . has access to a circle of prominent scientists, academicians, who regarded him as one of their own, and in that way he served . . . to link the group with the scientific community, having influence on the latter and conducting active propaganda among them. Among his acquaintances he named academician Sakharov, who was initially cautious and distrustful of the activities of IAKIR, LITVINOV, and their group; he wavered in his position and judgments, but gradually, under the influence of DELONE’s explanations, he began to sign various documents of the group. . . ; [he also named] LEONTOVICH, whose views coincide with those of the group. In DELONE’s words, many of the educated community share their views, but are cautious, fearful of losing their jobs and being expelled from the party.” . . . [more details on DELONE]

Agents’ reports indicate that the participants of the group, LITVINOV, DREMLIUGA, AND DELONE, have not been engaged in useful labor for an extended period, and have used the means of the so-called “private fund,” which their group created from the contributions of individual representatives in the creative intelligentsia and scientists.
The prisoner DELONE told our source: “We are assisted by monetary funds from the intelligentsia, highly paid academicians, writers, who share the views of the Iakir-Litvinov group . . . [Sic] We have the right to demand money, [because] we are the functionaries, while they share our views, [but] fear for their skins, so let them support us with money.”

Maps and territories and knowledge

Seymour Papert, one of the pioneers of Artificial Intelligence, once wrote (1988, p. 3), “Artificial Intelligence should become the methodology for thinking about ways of knowing.”   I would add “and ways of acting”.
Some time back, I wrote about the painting of spirit-dreamtime maps by Australian aboriginal communities as proof of their relationship to specific places:  Only people with traditional rights to the specific place would have the necessary dreamtime knowledge needed to make the painting, an argument whose compelling force has been recognized by Australian courts.  These paintings are a form of map, showing (some of) the spirit relationships of the specific place.  The argument they make is a very interesting one, along the lines of:

What I am saying is true, by virtue of the mere fact that I am saying it, since only someone having the truth would be able to make such an utterance (ie, the painting).

Another example of this type of argument is given by Rory Stewart, in his account of his walk across Afghanistan.   Stewart does not carry a paper map of the country he is walking through, lest he be thought a foreign spy (p. 211).   Instead, he learns and memorizes a list of the villages and their headmen, in the order he plans to walk through them.  Like the aboriginal dreamtime paintings, mere knowledge of this list provides proof of his right to be in the area.  Like the paintings, the list is a type of map of the territory, a different way of knowing.  And also like the paintings, possession of this knowledge leads others, when they learn of the possession, to act differently towards the possessor.  Here’s Stewart on his map (p. 213):

It was less accurate the further you were from the speaker’s home . . .  But I was able to add details from villages along the way, till I could chant the stages from memory.
Day one:  Commandant Maududi in Badgah.  Day two:  Abdul Rauf Ghafuri in Daulatyar.  Day three:  Bushire Khan in Sang-izard.  Day four:  Mir Ali Hussein Beg of Katlish.  Day five: Haji Nasir-i-Yazdani Beg of Qala-eNau.  Day six:  Seyyed Kerbalahi of Siar Chisme . . .
I recited and followed this song-of-the-places-in-between as a map.  I chanted it even after I had left the villages, using the list as credentials.  Almost everyone recognized the names, even from a hundred kilometres away.  Being able to chant it made me half belong:  it reassured hosts who were not sure whether to take me in and it suggested to anyone who thought of attacking me that I was linked to powerful names. (page 213)

Because AI is (or should be) about ways of knowing and doing in the world, it therefore has close links to the social sciences, particularly anthropology, and to the humanities.
Seymour Papert [1988]: One AI or Many? Daedalus, 117 (1) (Winter 1988):  1-14.
Rory Stewart [2004]: The Places in Between. London, UK:  Picador, pp. 211-214.

Mathematics and proof

One of the great myths of mathematicians is that mathematical knowledge, once proven, is solid, and not subject to later contestation.   Thus, Oxford mathematician Marcus du Sautoy, writing in the New Scientist (2006-08-26), says:

Proof is supposed to be what sets mathematics apart from the other sciences. Traditionally, the subject has not been an evolutionary one in which the fittest theory survives. New insights don’t suddenly overturn the theorems of the previous generation. The subject is like a huge pyramid, with each generation building on the secure foundations of the past. The nature of proof means that mathematicians, to use Newton’s words, really do stand on the shoulders of giants.
In the past, those shoulders have been extremely steady. After all, in no other science are the discoveries of the Ancient Greeks still as valid today as they were at the time. Euclid’s 2300-year-old proof that there are infinitely many primes is perhaps the first great example of a watertight proof.

The reason for this widespread view is that mathematics uses deduction to reach its conclusions.  At least, that is true of pure mathematics, or was so until computers began to be used in proofs (a topic which du Sautoy discusses in that article).  But all deduction does is to show that, given some assumptions and given some rules of inference, a certain conclusion follows from those assumptions by applying those rules of inference.  If either the assumptions are false or the rules of inference not acceptable, then the stated conclusions will not, in fact, follow.
Du Sautoy is quite wrong to claim that new insights do not overturn the theorems of the previous generation.  The history of pure mathematics is replete with examples where proven conclusions were later revealed to depend on assumptions not made explicit, or on assumptions previously thought to be obvious but which were later shown to be false, or on rules of inference later considered invalid.   For over a century, mathematicians thought that everywhere-continuous functions were also everywhere-differentiable, until shown a counter-example.  For a similar period, they thought that the convergent limit of an infinite sequence of continuous functions was itself also continuous, until shown a counter-example.  They thought that there could not exist a one-to-one and onto mapping between the real unit interval and the real unit square, until shown such a mapping (a so-called space-filling curve).  In fact, there are infinitely-many such mappings; indeed, an uncountable infinity of them.  In all these case, “proofs” of the erroneous conclusions existed, which is why the earlier mathematicians believed those conclusions.  The proofs were later shown to be flawed, because they depended on (usually-implicit) assumptions which were false.   For the differential calculus, the fixing effort was begun by Cauchy and Weierstrauss, using epsilon-delta arguments which were more rigorous than the proofs of the earlier generation of analysts.
Not only does Du Sautoy have his history wrong, but there is shurely shome mishtake in his mentioning Euclid here.  The 19th century was consumed by a controversy over the truth-status of Euclidean geometry, and the discovery of apparently-logical alternatives to it.   As clever a man as the logician and philosopher Gottlob Frege (an intellectual hero of Wittgenstein) could not get his head around the idea that these different versions of geometry could all simultaneously be true.   Yet that is the conclusion mathematicians came to: that, depending on the assumptions you made about the surface on which you doing geometry, there were in fact valid alternatives to the discoveries of the Greeks:  draw your triangles on the surface of a sphere, instead of on a flat plane, for example, and you could readily draw triangles whose three angles did not sum to 180 degrees.  You choose your assumptions, you gets your geometry!  This is not a secure pyramid of knowledge, but many pyramids, post-modernist style.
And in the first part of the 20th century, pure mathematics was consumed with a bitter argument over whether a particular rule of inference – reductio ad absurdem (RAA), or reasoning from an assumption thought to be false – was valid in deductive proofs of the existence of mathematical objects.   The dissidents created their own school of pure mathematics, constructivism, which is still being studied.  Indeed, it turns out that a closely-related logic, intuitionistic logic, appears naturally elsewhere in mathematics (as part of the internal structure of a topos). Once again, you choose your rules of inference, you gets your mathematical theorems.
There is no single, massive pyramid of knowledge here, as du Sautoy claims, but lots of smaller pyramids.  Every so often, a great mathematician is able to devise a new conceptual framework which allows some or all of these baby pyramids to appear to be part of some larger pyramid, as Pieri and Hilbert did with geometry in the 1890s, or as Lawvere and others did with category theory as a foundation for mathematics in the 1960s.   But, based on past experience, new baby pyramids will continue to be created by mathematicians arguing about the assumptions or rules of inference used in earlier proofs.    To consider this process of contestation, splitting, and attempted re-unification to be somehow different to what happens in other domains of human knowledge may be comforting to mathematicians, but is myth nonetheless.

Research funding myopia

The British Government, through its higher education funding council, is currently considering the use of socio-economic impact factors when deciding the relative rankings of university departments in terms of their research quality, the Research Assessment Exercise (RAE), held about every five years.   These impact factors are intended to measure the social or economic impact of research activities in the period of the RAE (ie, within 5 years). Since the RAE is used to allocate funds for research infrastructure to British universities these impact factors, if implemented, will thus indirectly decide which research groups and which research will be funded.    Some academic reactions to these proposals are here and here.
From the perspective of the national economy and technological progress, these proposals are extremely misguided, and should be opposed by us all.    They demonstrate a profound ignorance of where important ideas come from, of when and where and how they are applied, and of where they end up.  In particular, they demonstrate great ignorance of the multi-disciplinary nature of most socio-economically-impactful research.
One example will demonstrate this vividly.  As more human activities move online, more tasks can be automated or semi-automated.    To enable this, autonomous computers and other machines need to be able to communicate with one using shared languages and protocols, and thus much research effort in Computer Science and Artificial Intelligence these last three decades has focused on designing languages and protocols for computer-to-computer communications.  These protocols are used in various computer systems already and are likely to be used in future-generation mobile communications and e-commerce systems.
Despite its deep technological nature, research in this area draws fundamentally on past research and ideas from the Humanities, including:

  • Speech Act Theory in the Philosophy of Language (ideas due originally to Adolf Reinach 1913, John Austin 1955, John Searle 1969 and Jurgen Habermas 1981, among others)
  • Formal Logic (George Boole 1854, Clarence Lewis 1910, Ludwig Wittgenstein 1922, Alfred Tarski 1933, Saul Kripke 1959, Jaakko Hintikka 1962, etc), and
  • Argumentation Theory (Aristotle c. 350 BC, Stephen Toulmin 1958, Charles Hamblin 1970, etc).

Assessment of the impacts of research over five years is laughable when Aristotle’s work on rhetoric has taken 2300 years to find technological application.   Even Boole’s algebra took 84 years from its creation to its application in the design of electronic circuits (by Claude Shannon in 1938).  None of the humanities scholars responsible were doing their research to promote technologies for computer interaction or to support e-commerce, and most would not have even understood what these terms mean.  Of the people I have listed, only John Searle (who contributed to the theory of AI), and Charles Hamblin (who created one of the first computer languages, GEORGE, and who made major contributions to the architecture of early computers, including invention of the memory stack), had any direct connection to computing.   Only Hamblin was afforded an obituary by a computer journal (Allen 1985).
None of the applications of these ideas to computer science were predicted, or even predictable.  If we do not fund pure research across all academic disciplines without regard to its potential socio-economic impacts, we risk destroying the very source of the ideas upon which our modern society and our technological progress depend.
M. W. Allen [1985]: “Charles Hamblin (1922-1985)”. The Australian Computer Journal, 17(4): 194-195.

Vale: Stephen Toulmin

The Anglo-American philosopher, Stephen Toulmin, has just died, aged 87.   One of the areas to which he made major contributions was argumentation, the theory of argument, and his work found and finds application not only in philosophy but in computer science.
For instance, under the direction of John Fox, the Advanced Computation Laboratory at Europe’s largest medical research charity, Cancer Research UK (formerly, the Imperial Cancer Research Fund) applied Toulmin’s model of argument in computer systems they built and deployed in the 1990s to handle conflicting arguments in some domain.  An example was a system for advising medical practitioners with the arguments for and against prescribing a particular drug to a patient with a particular medical history and disease presentation.  One company commercializing these ideas in medicine is Infermed.    Other applications include the automated prediction of chemical properties such as toxicity (see for example, the work of Lhasa Ltd), and dynamic optimization of extraction processes in mining.
S E Toulmin
For me, Toulmin’s most influential work was was his book Cosmopolis, which identified and deconstructed the main biases evident in contemporary western culture since the work of Descartes:

  • A bias for the written over the oral
  • A bias for the universal over the local
  • A bias for the general over the particular
  • A bias for the timeless over the timely.

Formal logic as a theory of human reasoning can be seen as example of these biases at work. In contrast, argumentation theory attempts to reclaim the theory of reasoning from formal logic with an approach able to deal with conflicts and gaps, and with special cases, and less subject to such biases.    Norm’s dispute with Larry Teabag is a recent example of resistance to the puritanical, Descartian desire to impose abstract formalisms onto practical reasoning quite contrary to local and particular sense.
Another instance of Descartian autism is the widespread deletion of economic history from graduate programs in economics and the associated privileging of deductive reasoning in abstract mathematical models over other forms of argument (eg, narrative accounts, laboratory and field experiments, field samples and surveys, computer simulation, etc) in economic theory.  One consequence of this autism is the Great Moral Failure of Macroeconomics in the Great World Recession of 2008-onwards.
S. E. Toulmin [1958]:  The Uses of Argument.  Cambridge, UK: Cambridge University Press.
S. E. Toulmin [1990]: Cosmopolis:  The Hidden Agenda of Modernity.  Chicago, IL, USA: University of Chicago Press.

Social surveys in the developing world

Robert Chambers, sociologist of development, writing about social science surveys in the developing world:

As data collection is completed, processing begins. Coding, punching and some simple programming present formidable problems. Consistency checks are too much to contemplate. Funds begin to run out because the costs of this stage have been underestimated. Reports are due before data are ready. There has been an overkill in data collection; there is enough information for a dozen Ph.D. theses but no one to use it. Much of the material remains unprocessed, or if processed, unanalysed, or if analysed, not written-up, or if written-up, not read, or if read, not remembered, or if remembered, not used or acted upon. Only a minuscule proportion, if any, of the findings affect policy and they are usually a few simple totals. These totals have often been identified early on through physical counting of questionnaires or coding sheets and communicated verbally, independently of the main data processing.”

Robert Chambers [1983]: Rural Development: Putting the Last First. London, UK: Longman. p. 53.

Knowing ways

Further to my post about different ways of knowing and recent posts on religion, is this statement from a story in The Melbourne Age today, about Indigenous Australian footballers:

In her book Yuendumu Everyday, Yashmine Musharbash refers to the Aboriginal notion that knowledge is acquired through ”doing” rather than questions.”

Yasmine Musharbash [2009]: Yuendumu Everyday: Contemporary Life in Remote Aboriginal Australia.  Aboriginal Studies Press.