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How Poetry and Mathematics are Intertwined in the Use of Ambiguity

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Ambiguous Forms: Poetry and Mathematics

Because of the diversity of forms in which they manifest themselves, there is no commonly accepted definition of mathematics or poetry. Thus, the only reasonable definitions of poetry and mathematics are ambiguous, and are unrestricted in the ideas they can express and how these ideas are interpreted. Lynn Arthur Steen offers one possible ambiguous definition of mathematics: “the science of significant form”(Stewart 23). In a similar vein, we could say that poetry is “the language of significant form”, where “significant” and “form” are undefined in both definitions. By analyzing the ways in which the “significant form” is able to introduce ideas, mathematics can be shown to model and provide insight into the language of poetry. Although mathematics and poetry appear superficially dissimilar in form, they are linked in their use of ambiguity to connect ideas.

William Empson conducted a rigorous study of the use of ambiguity in poetry in Seven Types of Ambiguity, where he defines ambiguity to be “any verbal nuance, however slight, which gives room for alternate reactions to the same piece of language”. This definition, though, is not broad enough to accommodate a multidisciplinary study of ambiguity. For this purpose, consider the definition proposed by William Byers in his study How Mathematicians Think, where he states: “Ambiguity involves a single situation or idea that is perceived in two self-consistent but mutually incompatible frames of reference” (28). This will be the definition considered throughout this paper.

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In mathematics, a binary relation on a set S is defined as a set of ordered pairs in S. In particular, an equivalence relation is a special type of relation that is reflexive, symmetric, and transitive. For example, to be related by blood is an equivalence relation: one must be related to oneself (symmetry); if person A is related to person B, person B must be related to person A (reflexivity); If person A is related to person B, who is in turn related to person C, person A must be related to person C (transitivity). When one uses a metaphor, he is defining an equivalence relation on two distinct subjects. Yet, one does not generally explain what the relation is, but instead allows the reader to judge what the relation is by context. If one were to write “mathematics is poetry”, one would look for qualities of mathematics that are often associated with poetry, such as elegance and density. The implied meaning of this statement seems to differ from the statement “poetry is mathematics”, which would suggest the poetry possesses qualities like rationality and motivation despite the fact the statements are semantically equivalent. One can see that the degree of elegance, brevity, rationality, and motivation all define equivalence relations. In fact, it is due to the existence of multiple equivalences that the meanings of the two statements are ambiguous. Empson declares this use of “is” to be the first type of ambiguity: “One thing is said to be like another, and they have several different properties of virtue in which they are alike”(Empson 2). Likewise, it is often useful in mathematics to study the general properties of equivalence relations. One often writes a~b to state that a and b are equivalent, where the specific relation need not be given. The absence of specification allows one to consider a general equivalence relation on any set. The ambiguity of what equivalence is can add depth to poetic passages by making the reader consider a multiplicity of relations between ideas rather than observing a single relation. This is embodied by Nicanor Parra’s text in “Watch out for the Gospel of the Times” in his collection Antipoems: How to Look Better & Feel Great:

2+2 doesn’t make 4: once it made 4 buttoday nothing is known in this regard (12-14)

Because of the political commentary that immediately precedes this passage, one is led to believe that this is a comparison of the Pinochet regime to the oppressive government of George Orwell’s 1984. The poem’s title suggests, however, that Parra feels alienated by changes in ideas at the foundation of his knowledge. One more important detail is found in the original Spanish version, which differs slightly but in an important way:

2+2 no son 4: fueron 4 hoy no se sabe nada al respect Parra’s use of the word “ser” instead of “hacer” implies a metaphoric relationship between 2+2 and 4. In fact, the symbol “=” can symbolize many different equivalence relations in mathematics, and hence always possesses a metaphoric quality. Hence, we can interpret “2+2 no son 4” as “there is some way in which 2+2 and 4 are different”. Maybe Parra is lamenting the division of Latin Americans due to political unrest—two and two are refusing to come together—the whole is not the sum of its parts. Parra could be expressing nostalgia for when the world seemed simple and straightforward, or he could be encouraging the reader to take a broader view of the world and its complexities. As a mathematician, Parra knows that the expression 2+2=4 is not at all a trivial result—in fact, it takes 25, 933 steps to prove from applying logic to the base axioms (Metamath).

In an elementary algebraic equation (e.g. x – 2 = 4), one is free to interpret x as a placeholder for any number belonging to a certain set (often the real numbers), or as the particular whose substitution for x makes the expression into an identity. It turns out that ambiguities of this form are common in language, where x is a subject instead of a number. Parra makes this relationship explicit in his poem “Advocate for His Own Cause”, where an unnamed man “arrives at a grave marked ‘x’”. The lack of specification of “x” makes the reader wonder whether the author intends the statement to be symbolic of any of a set of people, or a single person the reader is intended to discover through the variable’s relations to other elements of the text. Byers is able to explain the usefulness of this form: “A variable is specific and general at the same time […] By not resolving the ambiguity until the end of the piece of mathematics one is able to use that ambiguity constructively”(43). In algebraic equations, this amounts to solving for x. In Parra’s passage, one is led to think “x” is Parra’s mother because of the unnamed man’s possession of red carnations and the passage in “Something Like That” that states that “A DECREPIT old man/ throws red carnations/ at his beloved mother’s coffin”(66-69). Of course, the solution to algebraic equations does not have to be unique. For example, the equation x2 -1 = 0 has the solutions x = -1 and x = 1. Such structures often arise in poetry, when subjects are unspecified—hinted at, but left to interpretation. One manifestation of this structure is the problem of pronoun interpretation, seen in Parra’s “Something Like That”: they pass the years pass the years at least they seem to be passing hypothesis non fingo everything goes on as if they were passing (4-8)

The pronoun “they” in this passage may be interpreted as referring to “the years” or to some unmentioned other subjects. If one interprets “they” as the years, one is led to the impression that the narrative voice feels unable to hold onto the time—that the years continually slip through his fingers. The other interpretation leads the reader to believe that the unidentified subjects feel unable to enjoy life, and are passive in the proceeding of time. Moreover, the word “passing” can be understood as the passing of time or people. This ambiguity itself represents the idea expressing the apathy of time in the face of our desires. The algebraic equation is replaced with relations on the subject and the solution is an interpretation of the identity of the subject belonging to a certain set of syntactically viable subjects. One could therefore argue that, like many polynomial equations, this sentence has multiple “solutions”. This seems to be what Parra had in mind given his reference to Isaac Newton’s famous phrase “hypothesis non fingo”, used by Newton in “General Scholium”, an essay he attached to his Principia Mathematica. In his essay, Newton uses this phrase to express his refusal to postulate how gravity is able to act on objects. Parra’s use of the phrase suggests he does not want to tell us how he would interpret his own passages. In fact, the title of the poem is perhaps intentionally ambiguous to account for the discontinuous and largely unrelated narratives.

The mathematical idea corresponding to choice of language in poetry is the choice of symbolic representation. In mathematics, the symbols in an expression may be interpreted as different objects with different relations without altering the meaning of the overall expression. For example, for a function f defined on some subset of the real numbers, the indefinite integral is generally written in the notation ∫ f(x)dx .

Many people familiar with calculus would be surprised to learn that this combination of symbols is intrinsically ambiguous. In elementary calculus, one interprets the expression so that “∫” and “dx” are purely formal symbols acting mapping the function f to another function, called its antiderivative. Yet, mathematicians developed a rigorous justification for viewing “dx” as a mathematical object in its own right, called a differential form (Samelson 522-524). This interpretation suggests the generalization of the integral by combining differential forms, which Élie Cartan accomplished in the early 20th century, centuries after the original formulation of the integral. The linguistic counterpart of this ambiguity is morphological ambiguity. This type of ambiguity is largely unused in traditional poetry, but can be found in Cecilia Vicuña’s “el poema cognado/the poem”: respond e sibila ¿cuál es nuestra ver dad? ¿por qué est amos a quí? (39-47)

Vicuña takes ambiguity one step further, limiting our ability to distinguish between the beginning and end of each word. Like dividing the integral expression into its constituent symbols, the divisions of each fragment of Vicuña’s work allows the reader to explore each meaning of the resultant addition of fragments. For example, in the first stanza, one can construct “cuál es nuestra verdad”, or “which is our truth”. One can construct “cuáles nuestra verdad”, almost “which of our truths”. One can construct “cuál es nuestra ver dad”, or “which is ours to see dad”. In the second stanza, one can construct “porqué estamos aquí”, or “why are we here”. One can also construct “porqué est Amos aquí”, which is almost “why is Amos here” (referring to the Hebrew prophet Amos), or “why is it necessary for God to send his prophets to us”. This would be consistent with presence of sibila, or oracle, in the first stanza. One can construct “porqué est amos a quí”, almost “why is love to who”, throwing Latin in the mix. This seemingly rhetorical question is central to Vicuña’s message—love should not be selective. The fact that these questions all occur simultaneously in these fragments makes the reader consider their connection: do we share some universal truth, and is this the key to finding our common origin—our dad? Is the reason prophets are sent to us because we do not love each other unconditionally? Are we here to discover the truth we share?

The fact that Vicuña is able to successfully write poetry in such an unconventional way suggests a reexamination of what is and isn’t poetry. Looking at Parra’s work, for example, one can see the disappearance of elements of traditional poetry like regularity of form. This is reflected in the language in Parra’s poetry—he often uses words that reflect emptiness and death, seen in poems like “He’s dead” and “Apropos of Nothing”. Parra makes explicit that he is sending a message with what his poetry lacks in his “Final Poem”, where he writes (and doesn’t):

……………………………………

……………………………………

……………………………………

……………………………

…………………………………………

……………………………

I’m not sure I’m making myself clear:

What I really mean to say is the following: (1-9).

Parra’s empty spaces in both his choice of choice of form and his message blur the line between figure and ground. His use of punctuation frustrates interpretation—the colon in the second to last line could imply that his message is the last line of the poem, or that the elaborations are “nested”. Perhaps Parra is trying to make a statement about how language is constructed through an analogy to set theory. In set theory, the empty set Ø is the set containing no other sets, and the set { Ø } is the set containing the empty set. If one lets the sets be a metaphor for statements, the set { Ø } corresponds to Parra’s passage. This is significant because of its relation to Von-Neumann’s construction of the ordinal numbers, where Ø = 0 and { Ø } = 1. The set { Ø } contains the mathematical representation of nothing, yet it is representative of existence. The construction of the remainder of the natural numbers is an inductive extension of this principle: {{ Ø }, Ø } = 2, {{{ Ø }, Ø }, { Ø }, Ø } = 3, etc. Possibly, Parra is bringing to our attention fact that it is impossible to construct poetry without making a statement with what is missing. Joan Retallack is also aware of the ambiguity of figure and form, and makes use of this idea several times in Procedural Elegies/ Western Civ Cont’d: In part 18 of the collection’s eponymous poem, Retallack leaves a hole in her otherwise homogenous paragraph and inserts the caption “FIG.18” on the bottom line of the hole. One is free, of course, to interpret the caption as referring writing or the gap. Retallack also uses this in her poem AID/I/SAPPERANCE”, an elegy for Stefan Fitterman where she removes the letters “a”, “i”, “d”, and “s” from an original poem, followed by the letters in the alphabet adjacent to these, until each line is blank. The first copy of the poem is primarily Retallack’s original figure, while the final copy is the ground revealed by the disappearance of the letters. But what can one say about the penultimate copy? It is the previous copy without the letters “n” and “x”, but, because it consists of all y’s, it could be interpreted as a symbol for the chromosomes of the victim—the essential information about his genetic identity. The fact that it can be interpreted as figure or ground by itself raises several questions about human identity: When does a victim of a terminal disease cease to be a living human being? The complete disappearance of Stefan’s name before this stage hints that Retallack believes he was finished long before he was dead. Furthermore, the apparent symmetry of the importance of figure and ground highlights the ambiguity of the meaning of an elegy. Is an elegy a passage about someone’s life or their death?

It is clear that the poetry of Parra, Retallack, and Vicuña are in some ways very different from traditional poetry. In this context, traditional poetry is meant to mean poetry prominently featuring regular form and rhythm, eloquent language, or appeal to aesthetics. The alternative poetry of these Parra and others focuses more on philosophical ideas, and advancement in language and form. The appearance of prominent alternative poetry in the mainstream will be shown to be in strong correspondence with the appearance of alternative geometries in mathematics and the subsequent expansion of the study of abstract space.

Most of the theorems taught in high school geometry were developed by an ancient Greek mathematician named Euclid using a set of five axioms. Yet, in the 19th century, mathematicians began to discover new types of geometry that are self-consistent but distinct from Euclidean geometry (Greenberg 177-189). All of these geometries agree with Euclid’s first four axioms, but differ in the fifth axiom, often called the parallel postulate, which can be stated as follows:

For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.

One often uses the terms “point”, “line”, “lie on”, “between”, “congruent”, and “plane” when describing geometric objects. Most people think they know what these building blocks of geometry mean despite the fact that the terms must be undefined for a consistent geometric system. They would most likely find the points and lines of spherical geometry to be incompatible with their definitions. In spherical geometry, a “point” is a pair of diametrically opposed points on a sphere, and a “line” is a circle that lies on the intersection of the sphere with a plane that passes through the sphere’s center. Theorems generally accepted as true in Euclidean geometry are often false in spherical geometry, like the Pythagorean Theorem. Yet, the results of non-Euclidean geometries are just as valid as those of traditional geometry. It is the ambiguity of the interpretations of terms like “line” and “point” that allows for this expansion to take place.

Where non-Euclidean geometries lack the parallel postulate, Parra’s poetry lacks some aspects of conventional poetry. He frequently uses colloquial language, unconventional forms, and arguably subversive themes. Can the poetry of Nicanor Parra be called non-Euclidean? Parra’s works in some ways possess the necessary qualities of poetry, in the same way non-Euclidean geometries possess the necessary features of geometry. What qualifies this association of seemingly distinct forms? In mathematics, sets (including geometric objects) are often classified according to topological equivalence, an equivalence relation that looks at the fundamental structure of each set. For example, even though each geometry differs in its specific properties, it can always be described using coordinates—ordered n-tuplets of numbers that describe objects in any geometrical space. This ability to interpret objects in a distinct but entirely different way is the ambiguity that connects the geometries. What kind of structure can we find that is found in both traditional and alternative poetry? The examples investigated thus far, have featured semantic structures which allow for ambiguity, as well as the intentionality of creating these structures to ask the reader questions and search for ideas. This structure is certainly prevalent in traditional poetry, as Empson explores. Thus the “topological isomorphism” that is the transition into alternative poetry preserves this structure. This is visible even in the some of Parra’s least traditional poems, like “Mission Accomplished”:

literary gems 1

fathers of the church 1

hot air balloons 17 .

sum total 149

tears 0

drops of blood 0 .

sum total 0

Once again, Parra uses the disappearance of expected elements to thwart attempts to unambiguously interpret his writing. The most conspicuous of these is the plus sign: the left column reads sum total, yet there is no summation sign. It must be noted that the Spanish version reads “total” instead of “sum total”— the translator’s choice to translate to “sum total” suggests that she was trying to preserve this ambiguity. Is Parra arguing against the calculation of life events using simple summation? Also, Parra never specifies what the numbers mean—he implies that they are his experiences or possessions at times, but not always. Through this ambiguity, Parra could be raising the question of what it means to possess an idea or experience. Maybe an idea cannot be quantified. An experience cannot be measured. Is this a poetic statement? A poetic statement must be ambiguously defined, or else we would be able to unambiguously define poetry as “literature featuring the use of poetic statements”. The appearance of these statements in the analysis of Parra’s writing suggest they are correspondent to “point”, “line”, and the other undefined objects of geometry. Like these objects, poetic statements build the meaning of a poem, and may appear even in very distinct forms, such as that found in “Mission Accomplished”.

Although mathematics does not offer much insight into the macroscopic deviations from traditional form that is often characteristic of alternative poetry (Vicuña’s drawings, for example), the field of Computational Linguistics offers a glimpse of how structural ambiguity can arise when one attempts to parse poetry in a systematic way by using formal languages to analyze natural languages (Eijck and Unger 180-182). A formal language is a set of symbols, called atomic formulas, and a set of operations on these symbols that create new formulas, also called well-formed strings. These symbols have no specified meanings of their own, but one may use formal languages to model the construction of grammatically correct sentences using recursive transformation rules. Similarly, one may use these rules to deconstruct a sentence in a natural language into relations between the atomic formulas. Yet, even in this strictly mathematical framework, ambiguities frequently arise. The most interesting case from the viewpoint of Computational Semantics is that of the structural ambiguity, which occurs when the deconstruction of a sentence results in two or more syntactic structures. This type of ambiguity has been featured before in the first passage from Parra. However, whereas most poets use this ambiguity to link and create ideas, language poets like Joan Retallack often use this ambiguity to force the reader to create his own meaning. Empson classifies this as the sixth type of ambiguity: “When a statement says nothing […] so that the reader is forced to invent statements of his own”. These passages often incorporate parataxis, as in Joan Retallack’s “Not a Cage” from Procedural Elegies/Western Civ Cont’d: “around them in us we were very they”(11). It is clear that a direct analysis of this passage will not give any well-formed strings. The reader can insert punctuation and connectives or edit the morphology of words so that such analysis is possible, but in such extreme cases this requires major changes. One could interpret this as “when we were around them, we felt alienated from ourselves”, or “we felt aligned with them when we were near them”. Yet, there are no other indications in the passage supporting either claim—in fact, the pronouns “we” and “they” do not appear anywhere else in the stanza. The sole method of arriving at any kind of semantic meaning in much of Retallack’s text is through operating on the text.

Yet, it is unlikely that Retallack intended for the entirety of her work to lack a message. On what level, then should we look for meaning? This problem is closely intertwined with the central problems of Topology, the most recently developed of the major branches of mathematics. Topology is a way of preserving the structure and essential characteristics of a set without paying attention to some of the details. A topological space is a set A together with a set containing subsets of A, called a topology, that defines a type of structure on A (Adams and Franzosa 42). The ambiguity arises in that the sets in any topology over A are called “open sets”. Because of the way topologies are defined, all open sets of a set have similar qualities, so one may prove theorems about open sets of a space in general. This idea can be used to analyze Retallack’s poem “Not a Cage”, a work constructed from the beginnings and endings of books. Because of the heterogeneous and restrictive origin of the individual lines, it would be of limited use to look at the poem line by line. Yet, its structure and origin show the importance of experimentalist philosophy to Retallack. Also, the fact that Retallack created a poem with unconventional form in the process of discarding antiquated texts could demonstrate the importance she places on advancing poetry through new structural elements. After this large-scale understanding has been developed, the “topology” of the poem may be refined. The poem begins with the phrase “scientific inquiry, seen in a very broad perspective”, probably referring to Retallack’s beliefs about John Cage’s or her own work. Retallack’s choice to combine the sentences “Not a building, this earth, not a cage, The artist” likely reflect her ideas about the responsibility of the artist. This demonstration of the poem’s multiple layers highlights the ambiguity of “on what level the text is meant to be understood”, or “what are the open sets of the poem”.

In what respect, then, are mathematics and poetry the science and language of significant form? One may interpret the significance of a form to be the quality that allows for multiple interpretations, and thus the creation of ideas through reconciliation of these ambiguities. The “form” would correspond to the meaning behind a symbolic expression in mathematics, or the semantics of a poetic composition. Then significant forms are the birthplace of ideas, and that math and poetry are the languages in which we explore these ideas.

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