Archive for the ‘Art’ Category

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Art as Argument #3: commutative diagrams in Category Theory

Published by peter on 2009-07-09 in Argumentation, Art and Mathematics. 0 Comments

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:
Commutative diagram square
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.

Art as Argument #2

Published by peter on 2009-07-04 in Anthropology, Argumentation, Art and Culture. 0 Comments

Following my earlier post about the possibility of a work of art being an argument, I want to give another example.  This example is also drawn from Australian aboriginal society, and involves a 1997 claim for legal title to land by the Ngurrara people over land in the Great Sandy Desert of Western Australia.  Frustrated by their inability to convince the Native Title Tribunal of their right to the land, the Ngurrara community decided to create a collaborative painting (photographed below) which would demonstrate their traditional rights. The painting was presented, and accepted, as evidence before the Tribunal and is therefore a work of argument, as well as a work of art.
Ngurrara II Canvas 1997
(The Ngurrara Canvas. Painted by Ngurrara artists and claimants, coordinated by Mangkaja Arts Resource Agency, May 1997. 10 metres x 8 metres. Photo: Mangkaja Arts Resource Agency.)
The case is mentioned in a 2003 New Yorker magazine article about aboriginal art by Geraldine Brooks, who says:

“In 1992, the Australian government first recognized the right of Aborigines to claim legal ownership of their ancestral lands – provided they could show evidence of having an enduring connection with them. Before proceeding to court, Aboriginal groups had to make their case before a Native Title Tribunal. Frustrated by their inability to articulate their arguments in courtroom English, the people of Fitzroy Crossing decided to paint their “evidence”. They would set down, on canvas, a document that would show how each person related to a particular area of the Great Sandy Desert – and to the long stories that had been passed down for generations.
“Ngurrara I”, the first attempt, was a canvas that measured sixteen feet by twenty-six feet and was worked on by nineteen artists. It was completed in 1996. But Skipper and Chuguna [two of the artists involved], in particular, didn’t feel that it properly reflected all the important places and stories, so more than forty additional artists were invited to produce a more definitive version. In 1997, “Ngurrara II”, which was twenty-six feet by thirty-two feet, was rolled out before a plenary session of the Native Title Tribunal. It was, one tribunal member said, the most eloquent and overwhelming evidence that had ever been produced there. The Aborigines could proceed to court.” (page 65).

 
References:
Geraldine Brooks [2003]: “The Painted Desert“, New Yorker, 28 July 2003, pp. 60-67.
Australian National Native Title Tribunal [2002]: Native Title Determination Summary – Marty and Ngurrara.  27 September 2002.  Background press release here.
Also, here is a transcript of a radio story (broadcast 1997-07-15) on Australian ABC radio about the submission of the painting as evidence to the Native Title Tribunal.
More on different forms of geographic knowledge here.

Art as argument

Published by peter on 2009-07-01 in Anthropology, Argumentation, Art and Culture. 0 Comments

Can a work of visual art be an argument?  I believe the answer to this question is yes.  In this and in some future posts, I will give examples, drawn from Australian Aboriginal art and from pure mathematics.
In August 1963, the Yolgnu people of Yirrkala  (eastern Arnhem Land in Australia’s Northern Territory) petitioned the Australian Government for legal rights to traditional land.  The petition was in the form of two painted bark panels.  The argument for land rights was made in three ways — in English text, in Gumatj text, and in the surrounding art, which depicted the traditional relations between the Yolgnu people and their land.   It is important to note that the visual images are not mere decoration of the text, but a presentation of the same argument in a different language, a visual language.
Yirrkala Bark Petitions 1963-1
Yirrkala Bark Petition 1963-2
 
 
 
 
 
 
The artwork of the Yolngu Bark Petition (copied above) is a form of argument, for a claim asserting traditional rights to particular land.  The reason that the artwork is an argument derives from the general nature of traditional Australian Aboriginal art, which presents a diagrammatic or iconic description of a particular geographic region, identifying the landscape features of that region (eg, rivers, hills, etc) along with the dreamtime entities (animals, trees, spirits) who are believed to have created the region and may still inhabit it. (The “dreamtime” is the period of the earth’s creation.)    The art derives from stories of creation for the region, which are believed to have been handed down (orally and via artwork) to the current inhabitants from the original dreamtime spirits through all the intermediate generations of inhabitants.
Accordingly, the only people who have the necessary knowledge, and the necessary moral right, to create an artistic depiction of a region are those who have been the recipients of that region’s creation story.   In other words, the fact that the Yolngu people were able to draw this depiction of their region is itself evidence of their long-standing relationship to the specific land in question.   The existence of the art-work depicting the local landscape is thus an argument for their claim to ownership rights to that land.  (Note that the art work’s role as argument arises primarily from the special nature of the claim it supports;  the art is not, and could not easily be, an argument for any other kind of claim.)
In support of this position, I present some quotations, the first several as explanation for people unfamiliar with Australian aboriginal mythology and art.
Judith Ryan (1993, p. 50):

 The term “Dreaming” is difficult for us to comprehend because of its use as noun and adjective in imprecise and ungrammatical ways to refer to the creation period, conception site, totem, Ancestral being, ground of existence, and the notions of supernatural, eternal or uncreated.”

Jean-Hubert Martin (1993, p. 32):

One can more or less imagine what “Dreaming” is:  that link between the individual and his land, between the clan and its territory.   The paintings [of Aboriginal artists] show figured spaces representing spaces both physical and mental, but it is difficult to go much further than that.
Following Aboriginal explanations one can recognise and name the various elements in these paintings.  The thought structure, the references and the signifance of these words and fragments of speech – which reach us distorted by translation – still remain an enigma despite the valiant attempt to explain them in the ensuing texts.  A not inconsiderable difficulty is posed by the mystery surrounding certain rituals and their formal depiction.  And, one has to remember that what we see today of Aboriginal art is only that which we have been allowed to see.”

Ulrich Krempel (1993, p. 38) quotes  C. Anderson/F. Dussart (1988, p. 18), as follows:

When asked about their paintings, [Australian Aboriginal] artists usually respond that the painting “means” or is “my country”, that is, it is a depiction of the painter’s territory.  When queried further about the “Dreaming” story, the artist will often identify the main Ancestor depicted and perhaps the primary site at which the Ancestor undertook the actions portrayed in the painting.   It is possible for an outsider, especially if working in the local language, to gain further insight into the narrative of events described in the painting, but even then access to the different levels of meaning may be restricted.”

Three quotations from Horward Morphy [1991]:

From a Yolngu perspective, paintings are not so much a means of representing the ancestral past as one dimension of the ancestral past . . .” (page 292)
Yolngu art also provides a framework for ordering the relations between people, ancestors, and land.” (page 293)
Paintings [in Yolngu society] gain value and power through their incorporation in such a process [of cultural definition], through being integral to the way a system (of clan-based gerontocracy) is reproduced, and through being part of its ideological support.  Paintings gain power because they are controlled by powerful individuals, because they are used to discriminate between different areas of owned land, because they are used to mark status, to separate the initiated from the uninitiated and men from women.  Their use in sociopolitical contexts creates part of their value. However, their value is also conceptualized in other terms, in terms of their intrinsic properties.” (page 293)

Janien Schwarz (1999, pp. 56-57):

In the first section, I argue that an understanding of the Bark Petitions is inseparable from an understanding of Yolngu relations to land.  In Yolngu culture, the painting of designs is regarded as constructing an interface between the ground, its spiritual essence, and specific groups of people.  The designs on the Petition are inseparable from these associations, in particular from the geographic locations at which they originated during Creation or wangarr.  Putting the Petitions’ clan designs in a Yolngu cultural context reveals their strong artistic and political links to the Yolngu people and their land.  With mounting pressures on land use from outsiders, Yolngu people have disclosed their designs (and inferred connections to land) through the context of art and art exhibitions as a political means of laying claim to their country which is under threat by bauxite mining.  I present the Petitions as part of a larger history of Aboriginal people negotiating for land rights and cultural recognition through the production and presentation of painted barks and other objects of spiritual significance.  . . .  I contend that the painted motifs on the Bark Petitions merit interpretation as land claims and that, by extension, the paintings are a form of petition.”

References:
C. Anderson/F. Dussart: “Dreamings in Acrylic: Western Desert Art”.  Catalog for Exhibition:  Dreamings:  The Art of Aboriginal Australia. P. Sutton, Editor. Ringwood, Melbourne, 1988. p. 118.
Ulrich Krempel [1993]: “How does one read “Different” Pictures?  Our encounter with the aesthetic product of other cultures”, in Luthi and Lee, pp. 37-40.
Bernhard Luthi and Gary Lee (Editors) [1993]:   Aratjara:  Art of the First Australians. Exhibition Catalog. Dusseldorf, Germany: Kunstsammlung Nordrhein-Westfalen.
Jean-Hubert Martin [1993]:  “A Delayed Communication”, in Luthi and Lee, pp. 32-35.
Horward Morphy [1991]:  Ancestral Connections:  Art and an Aboriginal System of Knowledge.  Chicago, IL, USA: University of Chicago Press.
Judith Ryan [1993]:  “Australian Aboriginal Art:  Otherness or Affinity?”, in Luthi and Lee, pp. 49-63.
Janien Schwarz [1999]: Beyond Familiar Territory: Dissertation: Decentering the Centre. An analysis of visual strategies in the art of Robert Smithson, Alfredo Jaar and the Bark Petitions of Yirrkala; and Studio Report: A Sculptural Response to Mapping, Mining, and Consumption.  PhD Thesis, Canberra School of Art, Australian National University, Canberra,  Australia.    Available from here.

Manhattanhenge

Published by peter on 2009-06-04 in Art. 0 Comments

manhattanhenge-20090531
Sunset over Manhattan on 31 May 2009, Manhattenhenge, when the sun aligns with 42nd street.  The photo was taken at 42nd and 3rd Avenue, by jonbell has no h, and posted here.

Vale George Brecht

Published by peter on 2008-12-16 in Art, Culture, Music and Obituaries. 0 Comments Tags: , .

Fluxus artist George Brecht (born George MacDiarmid in 1926) has just died, aged 82. He was a student in the Experimental Composition class which John Cage gave at the New School in New York in the late 1950s, Regretting that I was born too late to join this class*, I took the next best step, which was to track down a copy of Brecht’s notebooks in order to pore over his lecture notes taken in this class.  His most recent exhibition was at MACBA, the Museum of Contemporary Art in Barcelona, in 2006.

The photo that was here showed Brecht performing “Drip Music” (1959): “For single or multiple performance. A source of dripping water and an empty vessel are arranged so that the water falls into the vessel.”

* I did once take a composition class with Gentleman Jim Penberthy (1917-1999), which makes me a grand-pupil of Nadia Boulanger. That class focused mainly on Penberthy’s compositional method of expressionist serialism.

Complexity of communications

Published by peter on 2008-08-17 in Advertising, Anthropology, Art, Mathematics and Telecommunications. 0 Comments

Recently, I posted about probability theory, and mentioned its modern founder, Andrei Kolmogorov.  In addition to formalizing probability theory,  Kolmogorov also defined an influential approach to assessing the complexity of something.
He reasoned that a more complex object should be harder to create or to re-create than a simpler object, and so you could “measure” the degree of complexity of an object by looking at the simplest computer program needed to generate it.  Thus, in the most famous example used by complexity scientists, the 1915 painting called “Black Square” of Kazimir Malevich, is allegedly very simple, since we could recreate it with a very simply computer program:
Paint the colour black on every pixel until the surface is covered, say.

But Kolmogorov’s approach ignores entirely the context of the actions needed to create the object.   Just because an action is simple or easily described, does not make it easy to do, or even easy to decide to do.   Art objects, like most human artefacts, are created with deliberate intent by specific creators, as anthropologist Alfred Gell argued in his theory of art.  To understand a work of art (or indeed any human artefact) we need to assess its effects on the audience in the light of its creator’s intended effects, which means we need to consider the intentions, explicit or implicit, of its creators.  To understand these intentions in turn requires us to consider the context of its creation, what a philosopher of language might call its felicity conditions.
Malevich’s Black Sqare can’t be understood, in any sense, without understanding why no artist before him created such a painting.  There is no physical or technical reason that Rembrandt, say, or Turner, could not have painted a canvas consisting only of one colour, black.  But they did not, and could not have, and could not even have imagined doing so. (Perhaps only the 18th-century Welsh painter Thomas Jones could have imagined doing so, with his subtle paintings of near-monochrome Neapolitan walls.)  It is not a coincidence that Malevich’s painting appeared in the historical moment when it did, and not anytime before nor anyplace else.   For instance, Malevich worked at a time when educated people were fascinated with notions of a fourth or even further dimensions, and Malevich himself actively tried to represent these other dimensions in his art.  To imagine that such a painting could be adequately described without reference to any art-historical background, or socio-political context, or the history of ideas is to confuse the syntax of the painting with its semantics and pragmatics.  We understand nothing about the painting if all we understand is that every pixel is colored black.
We have been here before.  The mathematical theory of communications of Claude Shannon and Warren Weaver has been very influential in the design of the physical layers of telecommunications and computer communications networks.   But this theory explicitly ignores the semantics – the meanings – of messages. (To be fair to Shannon and Weaver they do tell us explicitly early on that they will be ignoring the semantics of messages.)    Their theory is therefore of no use to anyone interested in communications at layers above the physical transmission of signals, that is, anyone interested in understanding or using communication to communicate with other people or machines.
References:
M. Dabrowski [1992]: “Malevich and Mondrian:  nonobjective form as the expression of the “absolute”. ” pp. 145-168, in: G. H. Roman and V. H. Marquardt (Editors): The Avant-Garde Frontier:  Russia Meets the West, 1910-1930. Gainesville, FL, USA: University Press of Florida.
Alfred Gell [1998]: Art and Agency:  An Anthropological Theory.  Oxford, UK: Clarendon Press.
L. D. Henderson [1983]: The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton, NJ, USA: Princeton University Press.
Claude E. Shannon and Warren Weaver [1963]: The Mathematical Theory of Communication. Chicago, IL, USA:  University of Illinois Press.