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A Mathematical Interpretation of Hollis Frampton's Zorns Lemma 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: <[email protected]>. replaced by a moving, wordless image that runs for one See <www.mitpressjournals.org/toc/leon/49/2> for supplemental files associated with this issue. second per iteration until the letters are all replaced, con- cluding the section. In the final section, a man, a woman Article Frontispiece. Image montage from Hollis Frampton’s and a dog cross a snow-covered field from foreground to Zorns Lemma (Hollis Frampton, 1970). (© Clint Enns) background while six women read sections from Robert ©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: Ah 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.
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