The Pedagogical and Epistemological Uses of Analogy in Poetry and Mathematics

oConsciousness, Literature and the Arts

Archive

Volume 2 Number 1, April 2001

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The Pedagogical and Epistemological Uses of Analogy in Poetry and Mathematics

ABSTRACT:

“A common function of poets and mathematicians is to jar us into place, to refocus our vision so as to order the chaos around us and to connect what is familiar with what is beyond” (Priestley 17).

With these words, W. M. Priestly brings together several of the assumptions upon which we have predicated our collaborative work: he asserts quite simply that there is a “common function” of poets and mathematicians; he describes that function as having a pedagogical purpose (“to connect what is familiar with what is beyond”) as well as an epistemological purpose (“to refocus our vision so as to order the chaos around us”); and in order to describe a relationship that may not seem immediately clear or obvious, he employs an analogy (the common function works like a device or an eye).

In our mathematics and poetry classrooms, we often use analogy to teach new material by building upon established concepts. Analogy also allows the student or professor, like the poet or mathematician, to extend the boundaries of what is known and to find new forms of expression. From problem solving, to exploring new realms, to imagining a “virtual classroom,” analogy has a vast range of pedagogical and epistemological uses in higher education. Although the study of analogy has been extended dramatically through research in cognitive science, psychology, linguistics, and artificial intelligence, for the purposes of this paper, we will examine the role played by analogy in our respective disciplines of mathematics and literature.

I. Using Analogy to Solve Problems and Build on Existing Knowledge

Our collaboration began fifteen years ago when we designed and team-taught a course in interdisciplinary problem solving at Rochester Institute of Technology, a highly technical comprehensive university in Rochester, New York. In this course, we emphasized the cross-disciplinary application of “tools,” such as analogy, using a variety of assignments in logic, mathematics, and literature. It was our belief then, and still is today, that students are more likely to succeed in college when they can see relationships between different courses and disciplines and when they can apply what they know to new situations. (For example, students who understand that adding simple fractions in grade school arithmetic requires finding a common denominator will better understand how to add more abstract algebraic fractions in an analogous manner.)

Hofstadter reinforces this idea and emphasizes the connection between analogy-making and high-level perception:

“Analogical thought is dependent on high-level perception in a very direct way. When people make analogies, they are perceiving some aspects of the structures of two situations - the essences of those situations, in some sense - as identical. These structures, of course, are a product of the process of high-level perception” (Hofstadter 179).

Indeed, even standardized college entrance exams emphasize analogy, and intelligence is often measured by testing analogical reasoning skills.

“Tasks involving recognition of structural similarity between objects, an important component of analogical reasoning, figure heavily in tests of human intelligence; an early study of such tests [. . .] goes so far as to state that ‘intellectual ability can be defined as the ability to reason by analogy from awareness of relations between experienced characters.’ “(Russell 1).

Furthermore, using analogy to make connections between the content of different academic disciplines, in this case poetry and mathematics, may also have pedagogical value when students are asked to approach new “problems” in one discipline using knowledge or strategies from the other discipline. For example, one instructor “encourages his students to use their intuitive knowledge of geometry to help them to understand a difficult poem” (Rishel qtd. in Growney “Isolated or Integrated” 65), while another “encourage[s] students to write a poem after the students have seen hyperreal numbers, reacting as a Pythagorean might to the discovery of incommensurable magnitudes. . .” (Haack 2)

II. Using Analogy to Explore the Unknown, Express Abstract Concepts, and Construct New Knowledge

Analogy has another, different role for college students when it is used to demonstrate and foster inventive, open-ended exploration and expression. The history of science is filled with examples of discovery and invention marked by brilliant analogical leaps, including Archimedes’ “Eureka,” Euler’s finding the sum of an infinite series, and Newton’s discovery of gravitational force. And indeed, mathematics itself has been described as “analogical.”

“One can see at once the abstract, gestalt-like and analogical character of mathematics in the way, for example, that one set of equations (Laplace’s) serves as analogue for the phenomena of change in such diverse fields as gravitation, electrostatics, electricity, elasticity, and the permanent flow of liquids” (Leatherdale 28).

Poetry, too, relies in a fundamental way on the creative use of analogy where it serves as the basis for much figurative language. The poet Mary Oliver, when discussing a passage from Milton’s Paradise Lost, describes the power of figurative language:

“What becomes apparent in this passage is that the force of figurative language can be associative and cumulative, as well as astonishing with each individual figure-that we carry an intended swelling residue with us down the page” (69).

Oliver continues,

“It is possible of course to write a poem without figurative language. And yet...What is poetry but, through whatever particular instance seems believably to be occurring, a meditation upon something more general and more profound?” (70).

Whether we are referring to the “gestalt-like” qualities of mathematics or the “associative force” of figurative language in poetry, we can encourage students to look beyond the present and the literal by helping them see the profound role analogy plays in sparking the creative leap to new knowledge, new uses of language.

What occurs in this process of creativity? Contemporary research on analogy has attracted scholars from across the disciplines, and many have addressed the process involved in the creative use of analogy. Some scholars have described the conditions in which analogy is employed:

“. . . [T]he analogical act in so far as it is an illuminating perception of resemblance is momentary and instantaneous. However, considered in all its fullness, it should be connected with a whole complex of antecedent behavior and knowledge and a similar complex of consequent implication and association” (Leatherdale 16).

The powerful nature of cognitive blending has also been well documented: “Human beings routinely - and with the support of cultures and arts - put together things that do not belong together. This creates not confusion, but deep insight” (Turner, UB lecture). However, this “putting together” is not haphazard.

“Although making an analogy requires a leap, that leap need not be blind or random. In all its manifestations, analogy is guided by the basic constraints of similarity, structure, and purpose. . .” (Holyoak and Thagard 7).

Indeed, psychologists and cognitive scientists have explored the extent to which human cognition itself is defined by this ability to put things together and “fundamentally shaped by various poetic or figurative processes” (Gibbs 1). One thing is certain, when creative, curious, intuitive human beings grapple with new ideas, they often turn to analogy.

And yet there are differences between the use of analogy when proving a mathematical theorem and when expressing an abstract idea in poetry. Not surprisingly, students may be much more at ease in analyzing analogy in one discipline than in the other. Thagard and Holyoak have made the following distinction:

“Analogy may be the cognitive basis of metaphor, but, [. . . ] metaphor is often extended by an associative aura created by metonymy and other figurative devices. Hence the meaning of a poetic metaphor is not fully captured by the kind of analysis that would suffice for a more rigorous scientific analogy” (223).

Students should be encouraged to engage in both kinds of analysis and may be surprised to see how, sometimes, the two disciplines seem to change roles. Scott Buchanan has masterfully described the ways in which poetry and mathematics “exchange disguises.”

“Very simply, poetry and mathematics are two very successful attempts to deal with ideas. In this respect they are genuine. Both employ sets of symbols and systems of notation. In this respect they have very interesting and illuminating comparisons and contrasts[. . . ]. They exchange disguises so that mathematics, commonly accepted for its hard-headedness, rigor and accuracy is often poetry creating a realm of fancy; and poetry, commonly loved for its playful spontaneity and utter ineffectualness, becomes the mathematical demiurge joining words and images into a world of hard persuasive fact” (19-20).

An excellent example of mathematics in its disguise as “poetry creating a realm of fancy” is the concept of infinity. As early as the sixth century BCE, the ancient Greeks acknowledged infinity as a central issue in mathematics, but they did not have a system of notation for dealing with such a novel idea (Maor 3). Prior to this time mathematics dealt with more practical issues of weighing, measuring, and time. According to Eli Maor, “Infinity had to wait until mathematics would make the transition from a strictly practical discipline to an intellectual one, where knowledge for its own sake became the main goal” (3).

Infinity is not a fixed number, but rather, infinity refers to the concept of numbers growing larger and larger without end. Infinity is often linked to the divine or to a Supreme Being. Infinity has puzzled, confounded, and tantalized mathematicians, by allowing for this more “poetic” or fanciful way of thinking. Unlike the solid, indisputable axioms of arithmetic, infinity has inspired terror as well as wonder. Mathematicians like Leonhard Euler and Georg Cantor stretched mathematical thought about infinity. While they certainly used known results from finite mathematics to think about the infinite, it was their unexpected twists and creative thinking that opened up the possibility for new ideas about infinity.

Euler was the first to find the answer to the infinite sum shown below:

The surprise in Euler’s discovery is the unexpected appearance of the number in the limit of a series involving only the natural numbers. To this day, Euler’s series is regarded as one of the most beautiful results in mathematical analysis. Using similar methods, Euler was able to find the sum of many other infinite series involving the natural numbers” (Maor 35).

Euler’s work, which expanded thinking in finite sums to infinite ones, occurred in the eighteenth century, but it wasn’t until the nineteenth century that Georg Cantor showed that there are different classes of infinity, or an actual hierarchy where some infinities are greater than others. Cantor set up a correspondence between the set of even integers and the set of all integers, showing that both of these sets had the same number of elements, despite the apparent contradiction that the even integers are a subset of all integers. These two sets are analogous, exactly the same in size, but different in their respective elements. Further, he showed that there are infinite sets, like the real numbers, which are so dense they cannot be counted. Such sets are also infinite in size, but much larger than the infinite set of countable integers. “Cantor called this type of infinity C, the infinity of the continuum” (Maor 59). Mathematicians had to think the absurd (an infinite set of numbers can have a finite sum) or the impossible (the part is the same size as the whole) before mathematics could take these major steps forward.

Poets, too, make inspired, unexpected connections. To illustrate the richness of poetic metaphor based on analogy, we turn to a poem by Wallace Stevens in which mathematics provides the context for an analogy. This poem offers an opportunity to engage students in a cross-disciplinary discussion of how analogy is used in creative expression. Such a discussion would be enlivened by drawing on the students’ knowledge or preconceptions about poetry or mathematics.

VI from “Six Significant Landscapes”

Rationalists, wearing square hats,

Think, in square rooms,

Looking at the floor,

Looking at the ceiling.

They confine themselves

To right-angled triangles.

If they tried rhomboids,

Cones, waving lines, ellipses--

As, for example, the ellipse of the half-moon--

Rationalists would wear sombreros. (Stevens 17)

The central idea here establishes, by analogy, that “rationalists” dress, and live, and think in ways that are intellectually and personally confining. The poet invites the reader to make a leap with him and apply characteristics of geometric shapes to the ways certain human beings think and live their lives. We observe the repetitive “square-ness” of the rationalists’ existence, along with the limitations on their field of vision. We are led to believe the rationalists lack imagination, even joy, in their lives. The poem goes further, suggesting that rationalists, if they tried other, less ”squared” ways of being, again represented analogously by geometric figures that we are now encouraged to interpret as more flamboyant (rhomboids, cones, waving lines, ellipses), would “wear sombreros,” the hats of dance. In this relatively short poem, Stevens is not only eloquent about the privations rationalists inflict upon themselves, he also comments, indirectly, on the realm of formal learning, suggesting that there is more pleasure and creativity possible in forms or ideas that go beyond the “right angles” with which we are most familiar.

Encouraging students to see how analogy is used creatively in both mathematics and poetry challenges the students’ assumptions about the “hardheadedness” of one or the “utter ineffectualness” of the other and may provide students with increased insight and confidence for approaching both disciplines.

III. Using Analogy to Change Pedagogy

Pedagogical shifts occur when teachers experiment with different ways of teaching and learning. Current trends in pedagogy, such as the infusion of technology into the classroom and the use of collaborative learning, developed from considering teaching and learning in a new light. By making analogies between the classroom and the corporate boardroom (team work), between teaching and entertaining (multimedia), between teacher as autocrat and classroom as democracy (collaborative learning), and between the linear textbook and the non-linear brain (hyperlinking, morphing, interactivity of computers), we created the “new classroom” with its technological emphasis.

Technology is most useful and successful as an educational tool when it allows students to see or “imagine” differently or to make analogical leaps. When students are far away from museums, but can experience a virtual tour on the computer, examining artwork, comparing artists and holding “chats” with docents, then technology has been successfully integrated into the curriculum. Similarly, when technology allows us to see mathematics in new ways, by rotating a three-dimensional object on a computer screen, by changing one parameter in an equation while simultaneously viewing the changes to its graphical representation, or by creating animations to show change and growth, then we are creating paradigm shifts in understanding.

Technology can also make it easier for students to conduct poetic analysis and understand analogic references. For instance, a poem may contain obscure allusions, such as those found in “The Merman and the Seraph” by William Benjamin Smith, a poet and mathematician. JoAnne Growney, herself a poet and mathematician, offers one explanation, saying,

“[Smith] sings sadly of the separation between the Merman - perhaps a mathematician, isolated in his sterile world of thought, and separated from beauty, from feeling and desire - and an angel or Seraph, who represents the world of whatsoever is good” (Growney “Isolated or Integrated” 63-64).

The terms “Merman” and “Seraph” are probably unfamiliar to most undergraduate students, but a structured computer search could help define these terms, explain their similarities and differences, and lead to more information about their historical, societal, and psychological implications. Continued internet searching on William Benjamin Smith himself would yield information on the author’s background in philosophy, physics, and biblical scholarship, as well as the dates he lived. Collecting and analyzing a range of material, from the etymological to the biographical, would provide students with insights into the domains or contexts Smith draws upon to construct the “Merman” and “Seraph” analogies.

In addition to enhancing students’ ability to conduct research, the multimedia capabilities and interactive nature of technology also allow students an unprecedented level of involvement with writers and texts. Students may easily compare and contrast different versions of a manuscript; write and experiment within the parameters of a specific poetic form; make visual connections among rhyme schemes or figurative language by creating links with color and motion; add sound or animation to a poem; and listen to a poem being read aloud in a variety of voices.

Whether we use sophisticated searching or graphing methods, visual images, or sounds, technology has the potential to greatly enhance teaching in the disciplines of mathematics and poetry. Furthermore, these same elements of technology may make it easier for students to analyze and create analogies on their own, as they work on independent projects.

With the support of a Rochester Institute of Technology Provost’s Grant, we are currently developing an undergraduate course titled “Analogy, Mathematics, and Poetry.” We believe that by familiarizing our students with the uses of analogy in mathematics and poetry, we will broaden their understanding of content material from both areas and will encourage them to apply analogous thinking in their own fields of study. Beyond that, however, we hope that students’ intellectual curiosity will be challenged by seeing the connections, made by analogy, among disparate fields and across different historical periods. By encouraging students to see their college curriculum not merely as a series separate, discrete courses, but as a body of knowledge held together by recurring cognitive links, we are providing them with the tools, vocabulary, and conceptual framework essential for creative thinking.

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