Thomas Harriott (c. 1560-1621) was an English mathematician, navigator, explorer, linguist, writer, and astronomer. As was the case at that time, he worked in various branches of physics and chemistry, and he was probably the first modern European to learn a native American language. (As far as I have been able to discover, this language was Pamlico (Carolinian Algonquian), a member of the Eastern Algonquian sub-family, now sadly extinct.) He was among those brave sailors and scientists who traversed the Atlantic, in at least one journey in 1585-1586, during the early days of the modern European settlement of North America. Because of his mathematical and navigational skills, he was employed variously by Sir Walter Raleigh and by Henry Percy, 9th Earl of Northumberland, both of whom were rumoured to have interests in the occult and in the hermetic sciences. Harriott was the first person to use a symbol to represent the less-than relationship (“<“), a feat which may seem trivial, until you realize this was not something that Babylonian, Egyptian, Greek, Islamic, Indian, or Chinese mathematicians ever did; none of these cultures were slouches, mathematically.
Yesterday, 26 July 2009, was the 400th anniversary of Harriott’s drawing of the moon using a telescope, the first such drawing known. In doing this, he beat Galileo Galilei by a year. The Observer newspaper yesterday honoured him with a brief editorial.
Interestingly, Harriott was born about the same year as the poet Robert Southwell, although I don’t know if they ever met. Southwell spent most of his teenage years and early adulthood abroad, and upon his return to England was either living in hiding or in prison. So a meeting between the two was probably unlikely. But they would have each known of each other.
Previous posts in this series are here. An index to posts about the Matherati is here.
Monthly Archives: July 2009
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Recursive Capitalism
Never a dull moment under capitalism, where news arrives of Wells Fargo Bank suing itself. (HT: AB)
Poem: Auf Flügeln des Gesanges
Today, an orientalist poem by German romantic, Heinrich Heine (1797-1856), Auf Flügeln des Gesanges, famously set to music by Felix Mendelssohn (published as Opus 34, #3, in 1834-6).
Auf Flügeln des Gesanges
Upon wings of song,
my dearest one, I’ll transport you
to the Ganges plains,
Where I know the most lovely spot.
There is a garden of red blooms,
and in the solemn moonlight,
the lotus flowers await
Their devoted little sister.
The violets giggle and cuddle,
and stare up at the stars above,
Secretly the roses recite
Their fragant fairy tales.
The pious, smart gazelles,
Leap up and listen;
and in the distance whisper
The waves of a holy stream.
There we will lie down,
under the palm-tree,
and drink of love and peace
And dream our sacred dream.
Reference:
Heinrich Heine [1827]: Buch der Lieder: Lyrisches Intermezzo (Translation by SH.)
Mendelssohn’s fascination with Oriental ideas was expressed in an 1840 letter to his brother Paul, urging him to read Friedrich Ruckert’s book of sufist and hindu translations, Erlaubiches and Beschauliches aus dem Morgenlaude (Establishments and Contemplations from the Orient, 1836-1838), which provided Mendelssohn with “delight beyond measure”. He was also a close friend of the first Professor of Oriental Literature at the University of London (the institution later called University College, London), Friedrich August Rosen (1805 – 1837). More on Mendelssohn’s orientalism here.
Previous poetry posts can be found here.
On knowing
I have long thought the many of the members of the cult of militant anti-religionists — people like Richard Dawkins and Christopher Hitchens — have been assailing a straw-man. Their target is religious belief of a particularly narrow, fundamentalist kind, and as Terry Eagleton among others have noted, this target is a gross caricature of most of the people who practice or believe religious ideas. The main argument of the anti-God cult is usually that religious beliefs are held without evidence.
First, as the writer Karen Armstrong discusses today, for most people, religion is about doing, not about knowing. It’s really only philosophers and their street-brawling imitators who obsess over beliefs. Indeed, because doubt and scepticism are integral parts of most of the world’s religions, religious practice may not necessarily start with belief, but in fact end with it: Belief can be what comes after you practice spiritual exercises long enough, not necessarily what causes you to practice them. People do zazen or yoga not because they are already enlightened, but to achieve enlightenment.
Second, the issue of evidence is problematic in these diatribes against religion. It is simply not the case that there is no evidence for religious or spiritual ideas, or that such ideas are only supported by the irrational or the feeble-minded. Most people who proclaim any adherence to religious or spiritual ideas will assert they have evidence for a realm beyond or outside the material world. This evidence is usually of the form of direct personal contact with a spirit world or with spiritual entities, as for example, in the experience of Janet Soskice or the physicist Oliver Lodge. Anyone who has spent any extended period in Africa or in East Asia will know people — sober, rational, and intelligent — who have had, and continue to have, what they experience as direct contact and interaction with spiritual entities.
Of course, such direct, personal evidence is usually not replicable at will, nor observable to others. That makes it invalid as the basis of science, which is a shared undertaking, but does not make it invalid as evidence for personal beliefs or actions. Knowledge of the existence of things unseen can be obtained by merely being in the presence of such entities, as the Sufi philosopher and founder of Illuminationism, Shahab al-Din Suhrawardi (1155-1191) argued in the 12th century. Knowledge-from-being-in-the-presence-of is a valid form of knowing, just as knowledge-from-tasting is. Our subjective personal tastes in food and drink, say, or our subjective experience of being in love, are also not observable to others, but that does not invalidate them as evidence for our beliefs or as a rational basis for our actions. When I say I prefer coffee to tea, this is an inference based (usually) on my personal, subjective reactions to the tastes of the two different liquids. Only I know whether this inference is based on true reactions or not; if I am a sufficiently-clever actor, no one will ever be able to conclude anything about my reactions to the respective tastes other than what I claim.
It may be that experiences understood subjectively as contact with spiritual entities can be replicated in the laboratory by stimulating particular parts of the brain, as recent experiments appear to show. But it does not follow from such research that all religious experiences are due to similar mental stimulation, just as using implanted electrodes to create the subjective experience of the taste of coffee would not thus imply the non-existence of coffee.
In closing then, I wonder which is more rational: to commit to certain religious beliefs (or undertake a spiritual practice) based on one’s personal subjective experiences with the divine OR to devote one’s career to studying mathematical models of additional space-time dimensions, dimensions for which there is as yet no evidence whatsoever, not even any subjective personal experience? If Dawkings and Hitchens were really worried about irrational beliefs, they should be attacking the practitioners of String Theory and M-Theory.
References:
Mehdi Amin Razavi [1996]: Suhrawardi and the School of Illumination. London, UK: Routledge.
POSTSCRIPT (2017-06-04): In a New Yorker profile of business author Clayton Christensen, he is quoted regarding his daily reading of The Book of Mormon:
One evening in October, 1975, as I sat in the chair and opened the book following my prayer, I felt a marvelous spirit come into the room and envelop my body. I had never before felt such an intense feeling of peace and love. I started to cry, and did not want to stop. I knew then, from a source of understanding more powerful than anything I had ever felt in my life, that the book I was holding in my hands was true.” (Page 90)
Larissa MacFarquhar [2012]: When Giants Fail. The New Yorker. 14 May 2012, pp.84-95.
Belligerent musical ignorance
Via Tom Service, I learn of a new blog seeking to define classical music in such a way as to exclude anything the writers do not themselves like.
I wonder, first, what is the point. Why can’t people be happy with their own preferences, their own choices, and leave other people to be happy also with their respective preferences and choices? What deep sense of anxiety or profound inferiority leads people so often to try to force others to make the same aesthetic choices as themselves, or, if unable to force that, to disparage the choices of others? There has to be something profoundly wrong with a person’s aesthetic philosophy or with their psyche if they undertake rule-mongering in order to defend their own preferences.
Second, seeking to include only the music they like and to exclude the remainder, the writers of this new blog present an axiomatization for what they refer to as “Art Music”. They use the term Art Music, but I think Autistic Music would be a better fit. Putting aside the cultural assumptions inherent in undertaking axiomatizations (something for another post), let’s examine their proposed axioms (numbered for ease of reference). I first list the axioms and then I interpolate my responses.
To count as Art Music, a work must meet ALL* the following criteria:
1. It must be written for acoustic instruments and/or unamplified voices (mechanical and electr(on)ic devices may also be employed for textural effect)
2. It must be the original work of a single author (texts notwithstanding)
3. It must be preserved and transmitted as a score, written in orthodox musical notation, alterable only by the composer (unless the composer dies before completion)
4. It must stand on, or peer over, the shoulders of giants, i.e. acknowledge, build on or work from 1000 years of fundamentally accumulative history from the so-called Medieval, Renaissance, Baroque, Classical, Romantic and Modern (see right) eras (or their equivalents in non-Western cultures)
5. It must be conceived for performance according to the instructions and faithful to the intent of the composer (performers always following the score precisely in as much detail as the composer provides; improvisations and ornamentations permitted where the composer allows or expects)
6. It must be musically and intellectually complex, coherent and sophisticated, i.e. display and encode, in various permutations, originality, discursiveness, subtlety, intricacy, symbolism, logic, humour etc through the use (in various combinations) of development-over-time (through-composition), advanced harmony, modulation, variation, variance of musical phrase length, counterpoint, polyphony etc. It will therefore:
6.1 Require a high level of musicianship (concentration, insight, accomplishment) on the part of performers, who must draw on musical education, personal experience and imagination, knowledge of a work’s idiom, and the accumulated body of historical performance practices even for a merely competent performance
6.2 Require relatively high levels of concentration, understanding and competence from listeners for appreciation and (even basic) comprehension
6.3 Be susceptible to detailed theoretical analysis
7. It must aspire to provide the listener with emotional and intellectual enjoyment and satisfaction, by communicating through musical complexity, sophistication and coherence exceptional and/or transcendent reflections on the human condition
In reality this is an anti-modern, anti-jazz, anti-downtown, anti-world music, anti-rock, anti-pop, anti-folk, anti-hip hop manifesto. Most of the music excluded is music by non-white peoples. Perhaps it is just a coincidence that the music which passes these rules is mostly written by dead, white, European males, or perhaps the authors really are the racists that these rules would suggest.
It is hard to know where to start with such an absurd list, so let us proceed in order.
1. It must be written for acoustic instruments and/or unamplified voices (mechanical and electr(on)ic devices may also be employed for textural effect)
First the restriction to written texts (#1, #3) excludes all of improvised music – that’s music by people like Bach, Buxtehude, Mozart, Beethoven, Mendelssohn, etc, not to mention jazz, klezmer, gypsy, indian music, gamelan, etc.
Axiom #1 also requires that mechanical and electronic instruments only be used for textural effects. There goes the organ repertoire! Baroque organs were perhaps the most sophisticated mechanical devices in the pre-modern era, and the large ones required at least two human operators – one person to play the keyboards and one or more to pump the bellows. And every modern trumpet, cornet, French Horn, tenor horn, euphonium and tuba uses a mechanical device called a valve, while Bb/F trombones use a switch to change from tenor to bass. So all the brass repertoire since about 1800 disappears too. And, indeed, keyboard instruments like the harpsichord and the piano use mechanical devices to transfer action executed by the performer to actions executed by the instrument. Even the sound and the means of performance of string instruments have been changed with new technologies, such as new materials for bows and the invention of shoulder-rests. A violin shoulder rest, for instance, means the left-arm of the violinist is no longer required to maintain the violin in position under the player’s chin. That in turn means the performer’s left hand can zip up and down the finger board with far greater rapidity and flexibility. The 19th and 20th century violin repertoire would be mostly unplayable without shoulder rests.
The irony in using a web-page to argue for acoustic instruments seems to have escaped these authors. I honestly don’t understand the mentality of people who favour so-called acoustic instruments. The instrument with the cleanest interface between human action and sound output is undoubtedly the theremin, where the performer touches nothing, and merely (after long practice!) waves his or her hands in the air. Technologically, this instrument is as unsophisticated as stone-age fire in comparison to the sophistication involved in the design, construction and maintenance of a modern piano or, for that matter, a baroque organ. So an intellectually-coherent set of musical axioms could hardly include the piano while excluding the theremin – unless there is something immoral about using electricity to aid sound production.
But in that case (as I have long argued contra to the authentic performance movement), why perform in air-conditioned halls lit by electric light? If you limit yourself to acoustic instruments, then surely intellectual consistency would require performance in halls or rooms without any other modern convenience. The actual sound – as produced by the musician, and as perceived by the listener – will be influenced by the ambient temperatures in the performance venue. If you think this comment is a trivial one, then you have never played a brass instrument in a cold hall or outside on a winter’s day.
2. It must be the original work of a single author (texts notwithstanding)
Axiom #2 requires that the work be single-authored. What of Bach’s reworking then of Vivaldi’s music? What of Gounod’s “Ave Maria”, a melody famously set to a prelude by Bach? I rather like that setting, as indeed I expect the authors of these axioms would. Axiom #2 also excludes most of jazz, world music, rock, etc.
3. It must be preserved and transmitted as a score, written in orthodox musical notation, alterable only by the composer (unless the composer dies before completion)
The restriction to orthodox notation (#3) excludes some of the greatest music of the last 50 years, which is perhaps the authors’ intention. But what of figured bass notation? Is this traditional? It was once, but has not been so much used these last 150 years. Since its use implies an improvisational stance to music, perhaps its loss is also
intentional (as per #1).
But anyway, what is so special about orthodox notation? Elsewhere on the site, the authors say they aim “to repudiate cultural relativism in music”. But what is more culturally-relative than musical notation? The standard notation we use in the west today is culturally and historically-specific. It is by no means the only notation. It is not even necessarily the best notation – it fails, for example, to adequately represent divisions of the octave into other than 12 pitch-classes; it does not deal well with unequal temperament or with dynamic pitches or with polyrhythms or allow precise gradations of dynamics; it ignores timbre; it mostly overlooks sound production (ask Morty Feldman about that!) and it is harder to learn than some other notations (eg, popular guitar chords symbols), etc. Like any system of representation of human knowledge it has strengths and it has weaknesses. But these authors proclaim “Art Music is in many ways objectively superior to Pop ‘Music’ “ and yet insist on using a culturally-specific notation with known weaknesses.
4. It must stand on, or peer over, the shoulders of giants, i.e. acknowledge, build on or work from 1000 years of fundamentally accumulative history from the so-called Medieval, Renaissance, Baroque, Classical, Romantic and Modern (see right) eras (or their equivalents in non-Western cultures)
Well all music (and indeed all art) does this, even when it ignores the giants. This axiom reveals the ignorance of the authors, since even their hated pop musicians “build on or work from” the music of their predecessors. And here suddenly here we have an allowance for non-Western music. Most of these musics were excluded by Axioms 1 (which requires music to be “written”) and 2, so including them here would seem to be just some weak attempt to prove the authors are not racists after all. Nothing in the other axioms would lead one to think that the authors really like or understand, for example, Javanese gamelan or Shona mbira music, or perhaps even know what they are.
5. It must be conceived for performance according to the instructions and faithful to the intent of the composer (performers always following the score precisely in as much detail as the composer provides; improvisations and ornamentations permitted where the composer allows or expects).
Oh dear. Here and in Axiom #3 we have the romantic fallacy that performing musicians are mere slaves to the will of the god-composer. Have none of the authors ever listened to Chopin’s music? Almost every performer of Chopin’s solo piano music — INCLUDING CHOPIN HIMSELF – plays with rubato, an elongation and compression of time, like a natural breathing, rather than a rigid adherence to a beat. None of this breathing is marked on the score, but is always and everywhere decided by the performer, as if on-the-fly.
But the performer is only part of the story. A musical work also requires an audience. It is the complete trio – composer, performers, audience – who interpret a piece of work, not any one of the three. Go read the books of Mark Evan Bonds to see how crucial the audience is for understanding the meaning of a musical work, and understanding how it should be read and performed. The ignorance the authors reveal here of western music history – ie, the history of the very music the authors claim to be promoting – is simply stunning.
6. It must be musically and intellectually complex, coherent and sophisticated, i.e. display and encode, in various permutations, originality, discursiveness, subtlety, intricacy, symbolism, logic, humour etc through the use (in various combinations) of development-over-time (through-composition), advanced harmony, modulation, variation, variance of musical phrase length, counterpoint, polyphony etc. It will therefore:
Well, all music is “musically and intellectually complex, coherent and sophisticated”. Because the authors first require “advanced harmony”, I suspect the intent here is to exclude minimalist, downtown and rock music. If the authors think that any of these musics is not complex and sophisticated, they are simply not listening. (It is something truly strange to ponder why so many trained uptown musicians can hear downtown or pop or non-western music without actually listening to it; I guess the answer is in their training.)
The complexities in these musics often lie in places elsewhere than in music in the main thread of western classical music — for example, in the interplay of multiple, intersecting rhythms rather than in harmonies. But complexities there certainly are. If you limit yourself to music which is only harmonically complex, for example, you’d also have to forget the pre- and early-Baroque, along with 20th century composers like Shostakovich, Orff, Satie, or the Ravel of Bolero. Of course, you’d get all the Wagner you could possibly want, although that trade would not satisfy me at all.
6.1 Require a high level of musicianship (concentration, insight, accomplishment) on the part of performers, who must draw on musical education, personal experience and imagination, knowledge of a work’s idiom, and the accumulated body of historical performance practices even for a merely competent performance
See comment to #6.
6.2 Require relatively high levels of concentration, understanding and competence from listeners for appreciation and (even basic) comprehension
See comment to #6.
6.3 Be susceptible to detailed theoretical analysis
See comment to #6. Anyone who thinks that popular music, for example, is not susceptible to detailed theoretical analysis, is simply ignorant.
7. It must aspire to provide the listener with emotional and intellectual enjoyment and satisfaction, by communicating through musical complexity, sophistication and coherence exceptional and/or transcendent reflections on the human condition
See comment to #6.
The contemptible views expressed on the site are very similar to those I’ve heard expressed before by uptown composers such as Harrison Birtwistle. Is this website the uptown response to downtown and popular music? Shoot-out at autistic musical gulch, perhaps? It is hard to imagine that people with such views still exist, let alone that they have heard of the web. But that is enough dragon-slaying for now. I will sure have to more to say in a future post.
Update (2018-03-25): The site was musoc.org, but it now seems to have disappeared.
References:
Mark Evan Bonds [2006]: Music as Thought: Listening to the Symphony in the Age of Beethoven. Princeton, NJ, USA: Princeton University Press.
Mrs Palin
Peggy Noonan was a speech-writer for Ronald Reagan, and she can still throw a serrated-edged dagger at high-speed with a precision of millimetres. Here she is (in her column of 2009-07-10) on one Mrs Palin:
In television interviews she was out of her depth in a shallow pool. She was limited in her ability to explain and defend her positions, and sometimes in knowing them. She couldn’t say what she read because she didn’t read anything. She was utterly unconcerned by all this and seemed in fact rather proud of it: It was evidence of her authenticity. She experienced criticism as both partisan and cruel because she could see no truth in any of it. She wasn’t thoughtful enough to know she wasn’t thoughtful enough. Her presentation up to the end has been scattered, illogical, manipulative and self-referential to the point of self-reverence. “I’m not wired that way,” “I’m not a quitter,” “I’m standing up for our values.” I’m, I’m, I’m.”
I doubt I could ever support a party of irrational economic policies, incompetent war-mongering, and class resentments such as the GOP. But if I were inclined, then I’d agree also with these words:
Here’s why all this matters. The world is a dangerous place. It has never been more so, or more complicated, more straining of the reasoning powers of those with actual genius and true judgment. This is a time for conservative leaders who know how to think.
Here are a few examples of what we may face in the next 10 years: a profound and prolonged American crash, with the admission of bankruptcy and the spread of deep social unrest; one or more American cities getting hit with weapons of mass destruction from an unknown source; faint glimmers of actual secessionist movements as Americans for various reasons and in various areas decide the burdens and assumptions of the federal government are no longer attractive or legitimate.
The era we face, that is soon upon us, will require a great deal from our leaders. They had better be sturdy. They will have to be gifted. There will be many who cannot, and should not, make the cut. Now is the time to look for those who can. And so the Republican Party should get serious, as serious as the age, because that is what a grown-up, responsible party—a party that deserves to lead—would do.
It’s not a time to be frivolous, or to feel the temptation of resentment, or the temptation of thinking next year will be more or less like last year, and the assumptions of our childhoods will more or less reign in our future. It won’t be that way.
We are going to need the best.”
Art as Argument #3: commutative diagrams in Category Theory
Following these posts on whether art could be understood as arguments, I turn attention to diagrams in pure mathematics. I know of only two areas of mathematics where diagrams are used frequently as arguments in proofs, rather than simply as illustrations of arguments or proofs expressed in algebraic symbols. One area is Euclidean Geometry, which most of us learn in school. The other is category theory (CT). It is interesting that one of the oldest and one of the youngest branches of pure mathematics should be the only ones using diagrams in this way. Perhaps the rise of CT is another signal of the decline of the three-centuries-long dominance of the written word over western culture.
First, here is an example of a typical commutative diagram from category theory:
This particular diagram expresses an equivalence: that in traveling from P to S, it does not matter whether we travel via Q or we travel via R, the end-result will be the same. (Category theory makes this notion of “same-ness” or equivalence quite precise; indeed, in some sense CT is a formal theory about different notions of equivalence and their relationship to one another.) Thus, the diagram is making a claim about the (mathematical) world, a claim which may include its own proof: that executing function (or action) w followed by function y is the same as executing function x followed by function z.
Let us see what CT textbooks say to justify the subject’s use of diagrams. The standard reference on CT for mathematicians is the book by Saunders Mac Lane. An easier introduction is the book by Lawvere and Schanuel. Both of these simply start using diagrams as proofs without any justification for the practice, although they both formally define the diagrams concerned. In the book by Barr & Wells, we find:
When the target graph of a diagram is the underlying graph of a category some new possibilities arise, in particular the concept of commutative diagram, which is the categorist’s way of expressing equations.” (page 93)
Later in the same chapter they say:
This point of view provides a pictorial proof that the composite of two graph homomorphisms is a graph homomorphism. . . . . . The verification process just described is called “chasing the diagram”. Of course, one can verify the required fact by writing the equations (4.14) and (4.15) down, but these equations hide the source and target information given in Diagram (4.13) and thus provide a possibility of writing an impossible composite down. For many people, Diagram (4.13) is much easier to remember than equations (4.14) and (4.15). However, diagrams are more than informal aids; they are formally-defined mathematical objects just like automata and categories.“ (page 96)
Mac Lane says (p. 29) that the use of arrows as a graphical representation of functions was introduced by Hurewicz in about 1940, and that he also probably first used commutative diagrams. Like many practices in mathematics, one learns about the use of diagrams as proofs in CT in the classroom. Despite the textual (ie, non-diagrammatic) nature of most pure mathematical writing, parts of applied mathematics and theoretical physics (e.g. Feynman diagrams) use diagrams although pure mathematicians may question whether these disciplines are actually doing “proving”.
References:
Michael Barr and Charkes Wells [1999]: Category Theory for Computing Science. Montreal: Les Publications CRM, 3rd edition.
W. Hurewicz [1941]: On duality theorems. Bulletin of the American Mathematics Society, 47: 562-563.
F. W. Lawvere and S. H. Schanuel [1997]: Conceptual Mathematics: A First Introduction to Categories. Cambridge: CUP.
Saunders Mac Lane [1998]: Categories for the Working Mathematician. Berlin: Springer, 2nd edition.
Heroes: The People of Iran
The Guardian has created an online memorial to all those murdered or detained in Iran.
The French Member of the English Language Committee of the IMO
The International Mathematics Olympiad is a famous international competition for mathematics students. This is an excerpt from the diary of the Chairman of the English Language Committee (ELC) of the 2005 IMO , Geoff Smith:
In the evening I prepare for the English language committee which I will chair next day. This means I slope off to my room early and try to cast the questions in perfect English myself, in order to have something to start with. The committee meets first thing in the morning. These days everyone is welcome in the ELC, including its most important member, the leader of France. We like to have simple sentences in IMO questions; ones which ideally can be translated almost word for word into as many languages as possible. French is rather special, and does not allow the rather free word order and grammatical latitude of English. Therefore the English language version has to be designed so that it can be easily translated into French. As each English sentence is suggested, we turn to FRA7, Claude Deschamps, to receive either a blessing (a shrug which indicates that all is well) or a sad shaking of the head which indicates that a particular piece of Anglo-Saxon thuggery simply cannot be expressed in French.”
HR runs amuck in NSW
The Sydney Morning Herald tomorrow reports that staff at the New South Wales Law Reform Commission are being invited to apply for their own positions. Apparently, the current staff there are staying in their posts too long, thereby reducing staff turnover, with the serious consequence that:
Zero turnover means that no opportunities arise to attract, develop or retain highly skilled employees.”
It’s always the damn employees who cause trouble for the proper functioning of the Human Resources Department. How can HR possibly execute policies to retain valuable staff if nobody ever threatens to leave!?