Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00807 by guest on 29 September 2021 general note Frampton’s Demon A Mathematical Interpretation of Hollis Frampton’s Zorns Lemma
C l i n t E n n s
Hollis Frampton’s much-discussed film Zorns Lemma is a complex T Axiom of Choice and some of its equivalent statements (one and fascinating film that has a labyrinthine structure, alluding to a of which is Zorn’s Lemma), this article provides an interpre- R mathematical reading of the film as a visual metaphor for Max Zorn’s
S tation of Zorns Lemma as a cinematic/poetic application of famous axiom Zorn’s Lemma. In the extensive literature about Zorns ABT AC Lemma, there have been many different interpretations offered; however, this mathematical axiom. In addition, this article explores none of these readings has provided a satisfactory mathematical some of the consequences of such an interpretation. interpretation of the film. After first providing an overview of Zermelo’s In the recent past, Frampton’s films have only been avail- Axiom of Choice and some of its equivalent statements, this article able on 16mm; however, in 2012 the Criterion Collection re- provides a mathematical interpretation of Zorns Lemma that shows leased a DVD boxed set of his work titled A Hollis Frampton the film as a cinematic/poetic demonstration of the Axiom of Choice, Odyssey [4]. This release of his work on DVD and the 2009 a statement that is mathematically equivalent to Zorn’s Lemma. In addition, this paper explores some of the consequences of such release of On the Camera Arts and Consecutive Matters, a col- an interpretation, including one that connects the film to the ideas lection of Frampton’s writings, critical essays, interviews and Frampton was exploring in Magellan. production material edited by Bruce Jenkins, have produced a new generation of film and media scholars interested in discussing, interpreting and analyzing Frampton’s work. For I’m starting to resent Zorns Lemma SLIGHTLY, telling instance, at the 2013 Society of Cinema and Media Studies people that I have in fact made 14 other films etcetera. Conference in Chicago, there were at least nine papers pre- —Hollis Frampton, letter to Sally Dixon, 1970 [1] sented regarding his work. Frampton’s film Zorns Lemma is structured around three Zorns Lemma, a 1970 experimental film by Hollis Framp- sections. The first section consists of a female narrator read- ton (1936–1984), is a complex and fascinating film that has a ing verses from the 18th-century Bay State Primer set to a labyrinthine structure alluding to a mathematical reading of black screen. Each verse focuses on a word beginning with the work as a visual metaphor for mathematician Max Zorn’s a letter from the Roman alphabet—a 24-letter predecessor (1906–1993) famous axiom Zorn’s Lemma [2]. The film, in to the contemporary English alphabet, wherein I/J and U/V addition to Ernie Gehr’s Serene Velocity (1970), Michael are considered equivalent. The second section—the “main Snow’s Wavelength (1967), Paul Sharits’s T,O,U,C,H,I,N,G section” [5] and the portion I focus on here—is set in silence (1968) and Tony Conrad’s The Flicker (1965), belongs to the and consists of “2,700 one-second cuts, one second segments, tradition of American structural film, a “cinema of structure and twenty-four frame segments” [6]. This section begins wherein the shape of the whole film is predetermined and by traversing an iteration of the Roman alphabet, which was simplified, and [wherein] it is that shape that is the primal “typed on tinfoil and photographed in one-to-one close-up” impression of the film” [3]. In the extensive literature writ- [7]. In the following iterations, each letter is sequentially ten about Zorns Lemma, there have been many different replaced by a word that begins with the same letter (for a interpretations offered; however, none of the readings has visual representation of the structure in the second section provided a satisfactory mathematical interpretation of the of Zorns Lemma, see the Article Frontispiece). The word is film. After first providing a brief overview of Ernst Zermelo’s usually embedded in an urban environment and selected using a chance operation [8]. Finally, each letter is gradually Clint Enns (student, artist). Email:
©2016 ISAST doi:10.1162/LEON_a_00807 LEONARDO, Vol. 49, No. 2, pp. 156–160, 2016 157
Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00807 by guest on 29 September 2021 Grosseteste’s On Light, or the Ingression of Forms, an 11th- the right) shoe. In the case of infinitely many socks, however, century text discussing the nature of the universe, at a rate it is not possible to assume that such a choice function ex- of one word per second. ists since there is no intrinsic way of distinguishing between socks. In other words, when the choice is truly arbitrary, one T he Axiom of Choice and Its Equivalents must invoke the Axiom of Choice. Mathematically, Zorn’s Lemma explicitly states: Aah M t ematical Interpretation of Zorns Lemma Let (P, <) be a nonempty, partially ordered set and let every Before providing a mathematical interpretation of Zorns chain in P have an upper bound. Then P has a maximal Lemma, it is worth observing that Frampton did not plan for element [9]. his film to be interpreted this way; however, he did encourage It seems when Frampton discussed Zorn’s Lemma, he was such a reading. Frampton states: actually referring to one of its many equivalent statements. I did not set out to provide a cinematic demonstration of a Frampton explicitly used the following statement: mathematical proposition. On the other, I don’t mind that Every partially ordered set contains a maximal fully or- the work should respire that possibility [18]. dered subset [10]. One of the standard interpretations of this film involves When the word fully is replaced by the correct, mathemati- viewing it in terms of “cuts.” There is evidence indicating that cally loaded term totally, it can be shown that Frampton’s Frampton was interested in exploring this concept in Zorns statement is equivalent to the Hausdorff maximality principle Lemma, as indicated in a 1964 letter to his friend Reno Odlin: [11], a result that was first proved by Felix Hausdorff in 1914 The excernment [sic] of the fully ordered set constitutes a [12]. Furthermore, it can also be shown that the Hausdorff cut. Where there are several possible cuts, the set of all cuts maximality principle is equivalent to Zorn’s Lemma [13]. constitutes the maximal ordered set [19]. Finally, Zermelo’s Axiom of Choice is equivalent to Zorn’s Lemma and is therefore equivalent to Frampton’s version of Melissa Ragona has suggested that the sculptor Carl An- Zorn’s Lemma [14]. In other words, Frampton’s version dre might have first introduced Frampton to thinking about of Zorn’s Lemma is mathematically equivalent to Zermelo’s mathematical “cuts” in a filmic context based on a discussion Axiom of Choice. that the two had in the early 1960s regarding the mathemati- The Axiom of Choice was first formulated by Zermelo in cal concept known as the Dedekind Cut. Ragona goes on to 1904 and formally states the following: suggest that this idea sparked Frampton’s “interest in analyz- ing the ‘cut’ in film as the ‘cut’ in the order of film” [20]. For every family F of non-empty sets, there is a function In my interpretation of Zorns Lemma, I am less concerned f defined on F such that f(S) S for each set S in F [15]. ϵ with the act of cutting and more concerned with the act of In other words, for each set in F there is a function that substitution. In order to interpret Zorns Lemma as a visual chooses an element of that set. For this reason, the function f demonstration of the Axiom of Choice, let us first think of is referred to as a choice function on F. It is worth noting that the letters as enumerating the bins. For the first substitution, invoking the Axiom of Choice was at one time considered that is, the first application of the Axiom of Choice, the fam- controversial among mathematicians, since “the Axiom of ily of sets consists of the set of all the words that begin with Choice can be used to prove theorems which are to a certain A, the set of all the words that begin with B, the set of all the extent ‘unpleasant’, and even theorems which are not exactly words that begin with C, and so on. By applying the Axiom of in line with our ‘common sense’ intuition” [16]. Choice, a word is selected from each of the above finite sets. Here is one way to conceptualize the Axiom of Choice. Through each iteration, different words are selected, dem- Suppose we are given bins each containing at least one object. onstrating the arbitrary nature of the choice. It is important The Axiom of Choice states that it is always possible to select to note that evoking the Axiom of Choice is excessive in this exactly one object from each bin. Of course, only in special case since there are only a finite number of letters and a finite cases do we actually need to evoke such a powerful tool. For number of words to choose from. In fact, this was observed instance, in cases where there are only a finite number of by Frampton: bins, we do not need to use the Axiom of Choice, since there Most words (not all, but most) were from the environment; is an obvious procedure for selecting objects from the bins. they’re store signs and posters and things like that, and one That is, we can enumerate the bins and then select an object finds out very quickly that very many words begin with c from the first bin, the second bin and so on. As there are and s, and so forth; very few begin with x or q, or what have only a finite number of bins, this procedure will eventually you. One quickly begins to run out of q’s and x’s and z’s [21]. end. The selection process becomes a bit trickier, however, in the case of an infinite number of bins. Such a scenario was For the second substitution, the family of sets consists of the impetus for a famous aphorism attributed to Bertrand the set of all moving images of pages being turned in a book Russell that states “to select one sock from each of infinitely (replacement for A), the set of all moving images of an egg many pairs of socks requires the Axiom of Choice; but for frying (replacement for B), the set of all moving images of a shoes the Axiom is not needed” [17]. In the case of infinitely red ibis flapping its wings (replacement for C), et cetera [22]. many shoes, you can always select the left (or, equivalently, By applying the Axiom of Choice, a moving image is selected
158 Enns, Frampton’s Demon
Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00807 by guest on 29 September 2021 from each of the above infinite sets. Once again, invoking the tween two chambers, allowing only hot molecules to pass, Axiom of Choice is superfluous since there are only a finite causing one of the chambers to heat up and the other to cool number of substitutions to be made, despite the fact that each down. In essence, the demon acts as a molecular bouncer of the moving image sets is infinite. that allows the Second Law of Thermodynamics to be vio- In theory, we can imagine a case in which the Axiom of lated. Although it has been shown that Maxwell’s Demon Choice is needed. For instance, consider the infinite family of is a physical impossibility, it still provides an interesting sets that consists of all cinematic images. Each of these mov- example of the personification of an abstract concept. In ing image sets would contain infinitely many elements, since fact, the Axiom of Choice has even been referred to as the it is possible to shoot an image from any given angle at any “mathematicians’ Maxwell’s Demon” [25]. In other words, given time. By applying the Axiom of Choice to the family through Zorns Lemma, Frampton is assuming the role of the of sets that consists of all cinematic images, one obtains a mathematical Maxwell’s Demon, since he is deciding on the film. In fact, this is one of the main concepts that Frampton choice function. explored in his unfinished film cycle Magellan, an idea that Zorns Lemma was Frampton’s attempt to bridge the gap be- I explore in the next section. tween mathematics and cinema. Mathematics is often thought of as the underlying structure of the universe, and Frampton’s Consu eq ences of This Interpretation unfinished film cycle Magellan can be seen as an attempt to If we view Zorns Lemma as a demonstration of the Axiom use cinema to create an epistemological model of the universe, of Choice, we find that the artist decides on the choice func- a cinematic Library of Babel, “more poignant” [26] than its tion. In other words, Frampton chose the shot, the content text-based predecessor. Although Frampton realized it was within shots, the sequence in which each shot is shown, the impossible to create an “infinite film,” he believed he could duration of the shot, etc. In fact, in Zorns Lemma, Frampton “generate a grammatically complete synopsis of it” [27]. At even chose “to incorporate deliberately a series of kinds of first glance this may all seem ridiculous; however, physicists errors” [23]. Applying the Axiom of Choice to the family of are using mathematics to develop an epistemological model sets that is all cinematic images provides us with a conceptual of the universe. Zorns Lemma provides a theoretical frame- procedure for eliminating what Frampton refers to as “nom- work for constructing an “infinite film,” namely, by invoking inally subjective, ‘thumbprint’ procedures” [24]—one of the the Axiom of Choice. Although Frampton didn’t believe he goals of Frampton’s most ambitious projects, Magellan—an was creating a cinematic demonstration of a mathematical unfinished 36-hour film cycle. principle, this interpretation demonstrates that many of the This interpretation can also be read as the personification concepts being worked through in Zorns Lemma formed a of an abstract concept, an idea that interested Frampton, conceptual basis for Magellan. The ultimate strength ofZorns as demonstrated by his homage to Scottish physicist James Lemma is that it successfully creates a link between math- Clerk Maxwell in his 1968 film Maxwell’s Demon. Maxwell’s ematical reasoning and filmic reasoning—one of the many Demon is a thought experiment in which a demon sits be- reasons this film remains one of Frampton’s most discussed.
References and Notes 11 The Hausdorff maximality principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered 1 Melissa Ragona, “Hidden Noise: Strategies of Sound Montage in the subset. Films of Hollis Frampton,” October, No. 109 (Summer 2004): 104. 12 Gregory Moore, Zermelo’s Axiom of Choice: Its Origins, Development 2 Note the spelling difference between Zorn’s axiom and the title of and Influence (New York: Springer-Verlag, 1982), 140. For an explicit Frampton’s film. proof that Frampton’s statement is equivalent to the Hausdorff max- imality principle see: John L. Kelley, General Topology (New York: 3 P. Adams Sitney, “Structural Film,” Film Culture Reader, ed. P. Adams Springer-Verlag, 1955), 33. Sitney (New York: Praeger, 1970), 327. (Originally published in Film 13 For an explicit proof that the Hausdorff maximality principle is Culture 47, Summer 1969.) equivalent to Zorn’s Lemma see Kelley [12] 33. 4 A 16mm print of Zorns Lemma is still available for rent from the 14 For an explicit proof that Zorn’s Lemma is equivalent to the Axiom Film-Makers’ Cooperative in New York. of Choice see Jech [9] 10. 5 Peter Gidal, “Interview with Hollis Frampton,” October, No. 32 15 Jech [9] 1. For further historical remarks see Jech [9] 8. (Spring 1985): 94. 16 Jech [9] 2. 6 Gidal [5]. 17 Eric Schechter, Handbook of Analysis and Its Foundations (New York: 7 Gidal [5]. Academic Press, 1997), 140. 8 Gidal [5] 96. 18 Scott MacDonald, A Critical Cinema: Interviews with Independent Filmmakers (Berkeley: University of California Press, 1988), 52. 9 Thomas J. Jech, The Axiom of Choice (Amsterdam: North-Holland Publishing Company, 1973), 9. 19 Hollis Frampton, “Letters from Framp,” October, No. 32 (Spring 1985): 47. 10 Hollis Frampton, “Zorns Lemma: Scripts and Notations,” On the 20 Ragona [1] 101. Camera Arts and Consecutive Matters: The Writings of Hollis Framp- ton, ed. Bruce Jenkins (Cambridge: MIT Press, 2009), 195. 21 Gidal [5] 96.
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00807 by guest on 29 September 2021 22 Frampton [10] 193. Clint Enns is a video artist and filmmaker from Toronto, 23 Gidal [5] 97. Ontario, whose work primarily deals with moving images cre- 24 Frampton, “Statement of Plans for Magellan,” in Frampton [10] 226. ated with broken and/or outdated technologies. His work has been shown both nationally and internationally at festivals, al- 25 Schechter [17] 140. ternative spaces and microcinemas. He has an M.Sc. in mathe- 26 Gidal [5] 98. matics from the University of Manitoba and an M.A. in cinema 27 Frampton with Bill Simon, “Talking about Magellan: An Interview,” and media from York University, where he is currently pursu- in Frampton [10] 241. ing a Ph.D. in cinema and media studies. His writings and in- terviews have appeared in Millennium Film Journal, INCITE Manuscript received 23 May 2013. Journal of Experimental Media and Spectacular Optical.
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T he Role of Artists and Scientists in Times of War
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We are living in an age of conflict, tension, wars and terrorism. Every day we see the slaughter of innocents, kidnapping and murder; we hear of conflicts whose causes have never been resolved and which continue to generate more conflicts and misery.
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