Mixing Apples and Oranges: What Poetry and Applied Mathematics Have in Common1 JOEL E. COHEN Abby Rockefeller Mauzé Professor of Populations The Rockefeller University and Columbia University M Y GRADE-SCHOOL EDUCATION in mathematics in- cluded a strict prohibition against mixing apples and or- anges. As an adult buying fruit, I often find it convenient to mix the two. If they have the same price, the arithmetic works out well. The added thrill of doing something forbidden, like eating dessert first, comes free. In any case, the prohibition against combining apples and oranges falls away as soon as we care about what two subjects, differ- ent in some respects, have in common. I want to mix apples and oranges by insisting on the important features shared by poetry and applied mathematics. Poetry and ap- plied mathematics both mix apples and oranges by aspiring to com- bine multiple meanings and beauty using symbols. These symbols point to things outside themselves, and create internal structures that aim for beauty. In addition to meanings conveyed by patterned sym- bols, poetry and applied mathematics have in common both economy and mystery. A few symbols convey a great deal. The symbols’ full meanings and their effectiveness in creating meanings and beauty re- main inexhaustible. Consider the following examples, which involve a beautiful poem of A. E. Housman (1859–1936) and some applied mathematics from my own recent research. In August 1893, Housman wrote: 1 I thank Helen Vendler and Rosanna Warren for very helpful comments on prior drafts. This article is based on a talk I gave at the American Philosophical Society in Philadelphia on 26 April 2002. The late Alexander G. Bearn invited me to speak on any topic of my choice. A condensed version, skillfully edited by Robert Wilson and Jean Stipicevic, was published in The American Scholar under the title “A Mindful Beauty” in fall 2009. The late William T. Golden introduced me to Housman’s poem. Johannes H. B. Kemperman, our collaborator in this work, died 13 June 2011, shortly before publication of this article. PROCEEDINGS OF THE AMERICAN PHILOSOPHICAL SOCIETY VOL. 155, NO. 2, JUNE 2011 [ 189 ]

190 joel e. cohen With rue my heart is laden For golden friends I had, For many a rose-lipt maiden And many a lightfoot lad. By brooks too broad for leaping The lightfoot boys are laid; The rose-lipt girls are sleeping In fields where roses fade. The surface meaning is simple: I regret that my friends, once young, have died. At that level of sophistication, the surface meaning of The Odyssey is equally simple: Odysseus has trouble getting home. Below the surface of Housman’s poem, though, multiple meanings (social, personal, and allusive) interact. The poem’s social meanings arise from its time and place. The 63 poems in the collection A Shropshire Lad (of which this is num- ber 54) describe the nostalgia of a country boy who moved to the big city. The poems, published in 1896, resonated widely in English society, where the population was rapidly urbanizing. By 1900, England would Figure 1. Title page of the 1896 first edition of A Shropshire Lad. Courtesy of the Housman Society.

poetry and applied mathematics 191 Figure 2. A. E. Housman in 1910. Courtesy of the Housman Society. Photo by E. O. Hoppé (public domain). become the first country in the world to have most of its people living in cities. The poem also had personal meanings for Housman. The scholar Archie Burnett’s 2003 essay “Silence and Allusion in Housman” showed that many of his poems were “for Housman a means of finding a voice for the love that dare not speak its name, a way of breaking silence, a veil for disclosure, at once catering to reticence and facilitating expres- sion.” In May 1895, Oscar Wilde was sentenced for the crime of “gross indecency” (homosexuality but not buggery) to two years’ imprison- ment with hard labor. Housman’s Shropshire 54 seems benignly neutral about boys and girls, maidens and lads, and Housman went to great lengths from his youth onward to conceal his homosexuality. But his passionate objection to society’s treatment of homosexuals, including Wilde, is clear in several poems in A Shropshire Lad and in his later

192 joel e. cohen writings, as the critic and scholar Christopher Ricks demonstrated in his essay “A. E. Housman and ‘the colour of his hair’ ” in 1997. Among the personal meanings of “With rue my heart is laden” is what Hous- man dared not say. This poem also has allusive meanings for those who read it with the literary background that Housman brought to writing it. In Cym- beline (act 4, scene 2), Shakespeare wrote a beautiful song of mourning for a boy, Fidele, who was thought to have died (but was in fact only drugged into a deep sleep): Golden lads and girls all must, As chimney-sweepers, come to dust. Here are Housman’s “golden” “lads” and “girls.” John Sparrow in 1934 noted echoes of Shakespeare’s dirge in this and two other poems of Housman’s. Beyond the specific words, Housman echoes Shakespeare’s point that mortality masters all. But there is more to this allusion, as the poet and critic Rosanna Warren has pointed out. Fidele is in fact a young woman, Imogen, in boys’ clothing. In Shakespeare’s time, the fe- male role of Imogen would have been played by a boy or young man, giving the audience a male actor playing a female (Imogen) pretending to be a male (Fidele). Given what is now clear about Housman’s sexual orientation, it seems plausible that Housman, consciously or not, iden- tified with the doubly cross-dressing Imogen/Fidele, and nurtured a hope that his poems, if not he, would live. The economy of the poem is evident in the many images compressed into eight lines and in the questions left unanswered. As the poem opens, the narrator speaks of his rue-laden heart, raising the question, why is he sad? The second line explains why: his friends are gone. Where have they gone? Wait till the second stanza. Who were the rose- lipt maidens (echo of Othello, act 4, scene 2, as noted in Archie Bur- nett’s 1997 edition of Housman’s poems) and lightfoot lads in lines 3 and 4, and what were his relations with them? What happened in those friendships? The narrator never says. Instead he speaks of brooks too broad for leaping, evoking not slender streams easily leaped but a broader, slower descent to the sea in the fullness of time. Only in the last three lines, where the boys and girls are laid in death, in fields where roses fade, do we finally learn that his friends are gone not only as a result of migration (possibly) but also as a result of mortality. The Oxford English Dictionary does not support a sexual interpretation of “laid” at the end of line 6, as its earliest quoted sexual use of “lay” or “laid” dates from 1932, but it is not clear that the OED would capture a slang usage immediately. Additional questions remain. Why is the narrator’s mourning plural and anonymous?

poetry and applied mathematics 193 Housman sets up a botanical contrast in opening the poem, but re- veals it only at the end. As computer scientist and attorney Lewis Stiller pointed out, “rue” means both “regret” and a very bitter evergreen herb (genus Ruta) with a toxic oil. The rue in line 1 contrasts with the sweet roses that fade in line 8. Turning from meanings to patterns, we face another mystery. How do the combined patterns of the symbols, on the page or spoken, evoke so much beauty? The patterns in these eight lines interweave meter, rhyme, ending accent, internal repetition, play on the letters r and l, alliteration, du-bi-du consonants, and two layers of chiasmus, within two symmetrical stanzas. Each line is written in iambic trimeter, and each set of four lines constitutes a quatrain. With rue | my heart | is laden For gold | en friends | I had, For man | y a rose | -lipt maiden And man | y a light | foot lad. By brooks | too broad | for leaping The light | foot boys | are laid; The rose | -lipt girls | are sleeping In fields | where ros | es fade. The irregularity of the anapests in lines 3 and 4 relieves the repeti- tious symmetry of the other lines. The rhyme scheme is equally simple: abab. The ending accent alternates feminine (“laden,” “leaping”) and masculine (“had,” “fade”). There is an extraordinary amount of internal repetition. The first syllable of “laden” reappears in “laid” and, with a slight change in vowel, in “lad.” The second syllable of “laden” reappears in “golden” and “maiden.” Lines 3 and 4 repeat “many a” exactly. The phoneme “r¯ in “rose-lipt” and “roses” appears in lines 3, 7, and 8. oz” “Lightfoot” appears in lines 4 and 6. Every even-numbered line ends with “d” (the initial consonant of “death,” a word that does not appear in the poem), preceded by one or another variant of the vowel-sounds that the letter “a” can exhibit. The poem uses two liquid consonants: r 10 times and l 12 times. Alliteration crosses lines: “many,” “maiden,” “many”; “lipt,” “lightfoot,” “lad,” “leaping,” “lightfoot,” “laid,” “lipt,” “(s)leeping”; “friends,” “fields,” “fade”; “brooks,” “broad,” “boys.” An- other pattern, for which I do not know a technical name, I have called du-bi-du consonants. It is the pattern illustrated by the consonants in its name, du-bi-du: “lightfoot lad” (l-f-l), “brooks too broad” (b-t-b), “lightfoot boys are laid” (l-b-l), “fields where roses fade” (f-r-f ). The pattern of chiasmus is central to this example of the connec- tion I want to make between poetry and applied mathematics. In po- etry, chiasmus refers to the statement of two words or ideas and then

194 joel e. cohen their restatement in reverse order. A subtle example of chiasmus is the appearance of maiden and lad (female, male in elevated language) in lines 3 and 4 followed by boys and girls (male, female in demotic lan- guage) in lines 6 and 7. The change in language from elevated to demotic suggests that even the highborn are brought to earth, and the reversal of order suggests that any precedence in life (“ladies first,” maidens be- fore lads) may be undone in death. The second stanza gives another ex- ample of chiasmus. Line 5 tells where (by brooks too broad for leap- ing), and line 6 tells who (the lightfoot boys). Line 7 tells who (the rose-lipt girls) followed by line 8, which tells where (in fields where roses fade). Again the pattern conveys a meaning: the brooks (line 5) and fields (line 8) enclose the boys and girls (lines 6 and 7) as a coffin contains its cadaver. The patterns of the symbols and the messages of the poem are inextricable. Chiasmus occurs in a famous song from The Princess by Alfred Lord Tennyson (1809–1892): Fresh as the first beam glittering on a sail, That brings our friends up from the underworld, Sad as the last which reddens over one That sinks with all we love below the verge; So sad, so fresh, the days that are no more. Here “fresh” (line 1) and “sad” (line 3) are repeated in reverse order (line 5). The patterning of the symbols and the meaning of the verse are interwoven. Going forward from line 1 to line 4, “fresh” is followed by “sad” only after one sail has brought our friends up from the under- world and, later, another sail has carried them away. Much happiness intervenes between “fresh” and “sad.” In retrospect (that is, from the perspective of line 5), by contrast, time is compressed and “sad” and “fresh” are nearly contiguous. In “An Irish Airman Foresees His Death,” William Butler Yeats (1865–1939) uses chiasmus to express the waste of years past and future: I balanced all, brought all to mind, The years to come seemed waste of breath, A waste of breath the years behind In balance with this life, this death. Helen Vendler, in Our Secret Discipline: Yeats and Lyric Form (2007), pointed out that these four lines contain a triple chiasmus (bal- ance, years, waste of breath; waste of breath, years, balance), and that this balanced, symmetrical pattern conveys “an intellectually meditated decision,” a deliberate choice of “this life, this death.”

poetry and applied mathematics 195 The term “chiasmus” is a nineteenth-century Latinized English word from the Greek chiasmos, “placing crosswise.” The closely related term “chiasma” was first used in nineteenth-century neuroanatomy to describe the crossing or intersection of optic nerves on the ventral sur- face of the brain. “Chiasma” was used at the beginning of the twentieth century in genetics and cytology to describe a crossing-over between two chromosomes. Both crossings make a picture that looks like the Greek letter χ (chi). The term “chiasma” comes from the Greek chi- asma, “two lines crossed,” which comes from chiazein, “to mark with χ.” The poetic and the scientific terms share etymological roots in χ, mixing apples and oranges philologically through Latin and Greek. As a master classicist, Housman would have known the uses of chiasmus in Latin and Greek literatures well. Though he went to great pains to separate his scholarly life as a classicist from his poetical life, his use of chiasmus crossed the two lines of his work. In mathematics (pure or applied), a strict definition would limit “chiasmus” to the exact repetition of two symbols (a, b) in reverse or- der (b, a). A more inclusive definition includes any repetition of a se- quence of symbols in a permuted order. We soon encounter an example that illustrates both possibilities. Like poetry, applied mathematics combines multiple meanings, economy, pattern, and mystery. In its scientific or practical applications, applied mathematics points to something external. It also alludes to prior mathematics. Its few symbols convey a lot. Its use of symbols of- ten involves internal repetition, symmetry, and chiasmus. It is replete with unexpected truths, unexpected applications, and diverse proofs that illuminate different aspects of a single truth. My example from applied mathematics comes from work I did with two outstanding colleagues, Johannes H. B. Kemperman, retired from Rutgers University, and Gheorghe Zb˘ ganu, University of Bucha- a rest. In 2000, Zb˘ ganu published a fact new to mathematics: a If n is a positive integer (a counting number like 1, 2, 3, . . . ), and a1, a2, . . . , an and b1, b2, . . . , bn are any nonnegative real numbers (any fractional or whole number larger than or equal to zero, such as 17 or 0.333333 . . . or 3.14159 . . . ), then Σ min ((ai × aj),(bi × bj)) ≤ Σ min ((ai × bj),(bi × aj)). i, j i, j No understanding of the meanings of this beautiful formula is nec- essary to appreciate that it is an intricately patterned array of symbols. Whatever it means, the formula has a left lobe Σi,j min ((ai × aj),(bi × bj)) and a right lobe Σi,j min ((ai × bj),(bi × aj)) mediated by ≤. The symbols in the left lobe are exactly the same as the symbols in the right lobe, but

196 joel e. cohen Figure 3. Johannes Kemperman and Gheorghe Zb˘ ganu in 2001. Photograph a © 2009 by Joel E. Cohen. the letters a and b appear in different order; this is chiasmus in the broad sense. In the right lobe, (a, b, b, a) is an example of chiasmus in the strict sense as the sequence (a, b) is repeated in reverse order (b, a). That is about as far as one can go without having any idea of the mean- ings of the symbols. Understanding the formula’s meanings only enhances one’s sense of its beauty, economy, and mystery. The connective ≤ between the two lobes means that the quantity on the left is less than or equal to the quantity on the right. The expression Σi,j means sum (add) for all pairs i, j, where i and j are positive whole numbers from 1 to n. Finally, min((ai × bj),(bi × aj)) means the minimum (smaller) of ai × bj and bi × aj, and similarly for min((ai × aj),(bi × bj)). The words “If n is a positive integer, and a1, a2, . . . , an and b1, b2, . . . , bn are any nonnegative real numbers, then,” which precede the formula, declare that the inequality holds for any such numbers. Therein lies the immense and surprising power of Zb˘ ganu’s inequality. A numeri- a cal example illustrates its economy of expression. If n = 2 and if a1 = 2,

poetry and applied mathematics 197 a2 = 5, b1 = 4, b2 = 3, then the expression on the left side of the in- equality equals min(2 × 2, 4 × 4) + min(2 × 5, 4 × 3) + min(5 × 2, 3 × 4) + min(5 × 5, 3 × 3) = 4 + 10 + 10 + 9 = 33, while the ex- pression on the right side equals min(2 × 4, 4 × 2) + min(2 × 3, 4 × 5) + min(5 × 4, 3 × 2) + min(5 × 3, 3 × 5) = 8 + 6 + 6 + 15 = 35. As claimed, 33 < 35. Zb˘ ganu’s inequality asserts that, no matter what a natural number n you may pick, and no matter what nonnegative real numbers a1, a2, . . . , an and b1, b2, . . . , bn you may pick, the value of the left side will be less than or equal to the value of the right side. Zb˘ ganu’s inequality has social or referential meanings because it a answers a question Zb˘ ganu was considering in the mathematical the- a ory of information systems: If one of two messages must be sent over a channel with only two input symbols, A and B, and with n output sym- bols, 1, . . . , n, is the chance of error in transmission smaller if the first message is sent as AA and the second message as BB, or if the first mes- sage is sent as AB and the second message as BA? The left lobe of Zb˘ ganu’s inequality represents the probability of a an error in transmission if the first message sent is AA and the second is BB, while the right lobe represents the probability of error in the alter- native.2 Hence the inequality says that coding the two messages by AA and BB gives a lower risk that the wrong message will be received than coding by AB and BA. (The tongue in cheek message for teachers might be, “If you’re trying to teach your students one of two messages, it’s better to convey the message twice in the same way than to convey it once in each of two different ways.” But don’t take this interpretation too seriously; some students have memories, unlike the communication channels in this theory.) Nothing prevents us from playing formally with Zb˘ ganu’s inequal- a ity as long as we remember that such formal play yields only questions, 2 For those interested in the technical details, if a (i = 1, 2, . . . , n) represents the probabil- i ity that input symbol A is received as output symbol i and bj (j = 1, 2, . . . , n) represents the probability that the input symbol B is received as the output symbol j, and if the channel lacks memory, then (ai × aj) is the probability of output symbols (i, j) when the input symbols are AA, (bi × bj) is the probability of output symbols (i, j) when the input symbols are BB, (ai × bj) is the probability of output symbols (i, j) when the input symbols are AB, and (bi × aj) is the probability of output symbols (i, j) when the input symbols are BA. The sum of the minima on the left side of the inequality measures the similarity between the matrix with elements (ai × aj) and that with elements (bi × bj), because each summation equals 1 if the two matrices being compared are identical, and is 0 if the probability distributions have disjoint support. In the same way, the sum of the minima on the right measures the similarity between the ma- trix with elements (ai × bj) and that with elements (bi × aj). The inequality asserts that the probability distributions being compared on the left are less similar than those being com- pared on the right, that is, that sending AA or alternatively BB produces symbols (i, j) that are easier to distinguish than sending AB or alternatively BA.

198 joel e. cohen not answers. For example, if we exchange multiplication and addition, we get another formula: Π (min [(ai + aj),(bi + bj)]) ≤ Π (min [(ai + bj),(bi + aj)]). i, j i, j Is this formula always true if n is a positive integer, and a1, a2, . . . , an and b1, b2, . . . , bn are any nonnegative real numbers? Pushing the same idea further, we produced 64 possible generaliza- tions of Zb˘ ganu’s inequality by replacing each occurrence of addition a or summation, minimum, and multiplication by each of four opera- tions: addition, multiplication, minimum, and maximum, then looking for a direction of the inequality ≤ or ≥ that would make the statement true whenever n is a positive integer and a1, a2, . . . , an and b1, b2, . . . , bn are any nonnegative real numbers. Zb˘ ganu’s inequality is one of a these 64. We produced another 32 formal generalizations of Zb˘ ganu’sa inequality by swapping in other mathematical functions (for cogno- scenti, the spectral radius of a square nonnegative matrix, and qua- dratic forms).3 The mathematical question for us became, are these other 95 inequalities besides Zb˘ ganu’s inequality true always or under a some interesting conditions? Here are a few examples of the resulting generalizations of Zb˘ ganu’s a inequality (which we proved to be true under appropriate conditions): Σ max ((ai + aj),(bi + bj)) ≥ Σ max ((ai + bj),(bi + aj)), i, j i, j Π min (max (ai,aj),max (bi,bj)) ≤ Π min (max (ai,bj),max (bi,aj)), i, j i, j Σ min ((aiaj),(bibj))xixj ≤ Σ min ((aibj),(biaj))xixj, i, j i, j ρ(min ((aiaj),(bibj))) ≤ ρ(min ((aibj),(biaj))), ∫∫ log [(f(x) + f(y))(g(x) + g(y))]dμ(x)dμ(y) ≤ ∫∫ log [(f(x) + g(y))(g(x) + f(y))]dμ(x)dμ(y). You are supposed to be saying, or at least thinking, ooooooh, aaaaaaaaah, how beautiful these formulas are. Every mathematician knows, and every student of elementary mathematics has to learn, that swapping the operations of addition, 3 Specifically, we replaced Σ by the spectral radius of a square nonnegative matrix. Alter- natively, we took each summand as a coefficient in a quadratic form. Then, as before, we re- placed each remaining occurrence of minimum and multiplication by each of four operations: addition, multiplication, minimum, and maximum, and looked for a direction of the inequal- ity that would make the resulting statement true always.

poetry and applied mathematics 199 multiplication, minimum, and maximum does not generally produce true formulas.4 But after a few years of hard, exciting work, Zb˘ ganu, a Kemperman, and I were astonished to find that when n = 2, all 96 for- mulas we had invented by purely formal manipulations are true. When n > 2, 62 of our first 64 generalizations are true but two are false in general. Some of the inequalities involving the spectral radius and qua- dratic forms are false in general; some are true. We were able to distill several of our inequalities into a more abstract so-called matrix-norm inequality that is valid for any one of the four operations of addition, multiplication, maximum, and minimum—the kind of mathematical discovery that brings joy to the hearts of applied mathematicians (or at least to the hearts of the three of us).5 Our results are not the only inequalities in mathematics where key elements reappear in permuted order. For example, in 1979, Ralph Boas (1912–1992) published these lovely inequalities: sin−1 (sinh [x/2]) ≤ sinh ([sin−1 x]/2), 0 ≤ x ≤ 1. sin−1 (xa) ≤ (sin−1 x)a, a > 1, 0 ≤ x ≤ 1. Our inequalities have meanings both referential and allusive. I de- scribed earlier Zb˘ ganu’s construction of his inequality in the con- a text of information theory. Among our other inequalities, several have interpretations in operations research with potential applications to scheduling of transportation and manufacturing. In addition, our in- equalities allude by similarity of form to a beautiful inequality by Augustin Louis Cauchy (1789–1857): If n is a positive integer, and a1, a2, . . . , an and b1, b2, . . . , bn are any real numbers, then [a12 + . . . + an2][b12 + . . . + bn2] ≥ [a1b1 + . . . + anbn]2. Equality holds if and only if for some real constants λ and μ with λ2 + μ2 > 0 we have λaj + μbj = 0 for j = 1, . . . , n. In 2004 Michael Steele called Cauchy’s inequality “one of the most widely used and most important inequalities in all of mathemat- ics.” The opening words and symbols are identical with those of 4 For example, the distributive law tells us that for any three real numbers a, b, c, it is true that a × (b + c) = (a × b) + (a × c). For example, 2 × (3 + 4) = 14 = (2 × 3) + (2 × 4). But does swapping + and × give a formula that is true for all the numbers a, b, c, namely, a + (b × c) ? = ? (a + b) × (a + c)? If we test the formula using the same example, we find that 2 + (3 × 4) = 14 < 30 = (2 + 3) × (2 + 4), so our would-be equality is not generally true. * be one of the four operations +, ×, min and max on an appropriate in- 5 Specifically, let terval J of the real numbers. Let a, b ∈ Jn where n is a positive integer. Denote by a*a the n×n matrix ai,j = ai*aj. Then the matrix a*a is more different from b*b than a*b is from b*a. Pre- cisely, if ʈAʈ = Σ1≤i,j≤n ͉ ai,j ͉, then ʈ a*a − b*b ʈ ≥ ʈ a*b − b*a ʈ.

200 joel e. cohen Zb˘ ganu’s original inequality except that Zb˘ ganu’s restriction to non- a a negative numbers is absent from Cauchy’s inequality. Here the letters a and b, which appear in the left lobe in the order (a, a, b, b), appear in the right lobe in the order (a, b, a, b), whereas in Zb˘ ganu’s inequality a they appear in the right lobe in the order (a, b, b, a). For readers with the requisite mathematical background, Zb˘ ganu’s inequality alludes a to Cauchy’s inequality no less clearly than Housman’s verse alludes to Shakespeare’s. A deeper and more substantial connection lies beneath the similar- ity in the pattern of the symbols. After Zb˘ ganu’s inequality appeared in a 2000, Ravi Boppana of New York University proved it in a new way. In 2006, Titu Andreescu of the University of Texas at Dallas and Gabriel Dospinescu of the Lycée Louis-le-Grand, in Paris, found new proofs that strengthen Boppana’s intermediate result on the path to Zb˘ ganu’s a inequality. Some of these proofs depend on the Cauchy-Schwarz in- equality, a generalization of Cauchy’s inequality. So among the lines of logical genealogy that lead to Zb˘ ganu’s inequality are lines that pass a through Cauchy’s inequality. In applied mathematics, as in poetry, at the end of the analysis, things not yet understood remain. Why are so many of our formulas true? They were generated by a formal process that any mathematician, pure or applied, regards with total disbelief. In the first group of 64 pu- tative inequalities, why are the two particular failed inequalities false when n > 2? Why are the proofs of the true inequalities so extraordi- narily diverse? Why can’t we find a unifying approach that separates the sheep from the goats, the true from the false? Perhaps most myste- riously, why do several of our formulas have meanings for potential practical applications? The critic and scholar Helen Vendler showed me a precedent for the mixing of poetry and applied mathematics in Seven Types of Ambi- guity, written by William Empson (1906–1984) at the age of twenty- two and published when he was twenty-four. At Cambridge, Empson won firsts in mathematics and English. His book repeatedly cites the commonalities of poetry and math. For example, Empson quotes from George Herbert’s book The Temple (1633) the eight-line poem “Hope,” which alternates lines of iambic pentameter and iambic trimeter. Emp- son wrote, One can accept the poem without plunging deeply into its meaning, because of the bump with which the short lines, giving the flat, poor, surprising answer of reality, break the momentum of the long hopeful lines in which a new effort has been made; the movement is so impec- cable as to be almost independent of the meaning of the symbols.

poetry and applied mathematics 201 And, indeed, the symbols themselves seem almost to be used in a way familiar to the mathematician; as when a set of letters may stand for any numbers of a certain sort, and you are not curious to know which numbers are meant because you are only interested in the rela- tions between them. As apples do differ from oranges, poetry does differ from applied mathematics, despite their commonalities. For example, in poetry, but not mathematics, sound and “mouthfeel” (Galway Kinnell) matter. In applied mathematics, unlike poetry, calculation and shared scientific con- cepts and data, rather than intimate experience, lend conviction. Exam- ples of differences could be multiplied. They do not undermine the sig- nificance of the similarities. Poetry and applied mathematics, with mysterious success, both use symbols for beautiful, economical pointing and patterning. Pointing es- tablishes a relation between symbols and a world beyond the domain of symbols. Patterning establishes a relation between symbols and other symbols in the same domain. Poetry and applied mathematics fall along a continuum between pointing and patterning. The pointing of symbols to something else is most important. Formal prose Numerical data collection Poetry Engineering Songs with words, program music Applied mathematics Abstract music Pure mathematics The patterning of symbols themselves is most important. The same continuum runs in the visual arts from journalistic pho- tography at the extreme of pointing to purely abstract art at the ex- treme of patterning. Between those extremes lies most of the world of art, mixing apples and oranges, mixing meanings and patterns, along with poetry and applied mathematics. The differences between poetry and applied mathematics coexist with shared strategies for symbolizing experiences. Understanding those commonalities makes poetry a point of entry into understanding the heart of applied mathematics, and makes applied mathematics a point of entry into understanding the heart of poetry. With this understanding, both poetry and applied mathematics become points of entry into under- standing others and ourselves as animals who make and use symbols. References Andreescu, Titu, and Gabriel Dospinescu. 2006. On some elementary inequalities. Mathematical Reflections issue 1, http://reflections.awesomemath.org/2006_1. html, accessed 29 August 2008.

202 joel e. cohen Boas, Ralph P. 1979. Inequalities for a collection. Mathematics Magazine 52.1:28–31, January 1979. Burnett, Archie, ed. 1997. The Poems of A. E. Housman. Oxford: Clarendon Press. ———. 2003. Silence and allusion in Housman. Essays in Criticism: A Quarterly Jour- nal of Literary Criticism 53.2:151–73. Cohen, J. E., J.H.B. Kemperman, and Gheorghe Zb˘ ganu. 2004. Elementary inequali- a ties that involve two nonnegative vectors or functions. Proceedings of the National Academy of Sciences USA 101.42:15018–22, 19 October. ———. 2005. A family of inequalities originating from coding of messages. Linear Algebra and Its Applications 395:1–82, 15 January. Empson, William. 1970. Seven Types of Ambiguity. Norfolk, Conn.: New Directions, 3rd ed. rev., 118–19. Full text online http://www.archive.org/details/seventypeso fambi030525mbp based on 2nd ed., London: Chatto & Windus, 1949, accessed 11 October 2008. Gray, Allan. 1970. A Shakespearian allusion in Housman, Last Poems XXIX. English Language Notes 8:36–39. Kinnell, Galway. 1993. 3 Books: Body Rags; Mortal Acts, Mortal Words; The Past. New York: Houghton Mifflin. Ricks, Christopher, ed. 1968. A. E. Housman; a collection of critical essays. Englewood Cliffs, N.J.: Prentice-Hall. ———. 1995. The force of poetry. Oxford University Press, 453 pages. According to Ricks at p. 176, John Crowe Ransom (Southern Review 6 [1940]: 6–8) attacks “With rue my heart is laden” “partly on the grounds that one of its images ‘runs counter to the ironic intention.’ ” ———. 1997. A. E. Housman and “the colour of his hair.” Essays in Criticism 47.3:240– 55. Reprinted in Allusions to the Poets, 282–94. Oxford and New York: Oxford University Press. Sparrow, John. 1934. Echoes in the poetry of A. E. Housman. Nineteenth Century and After 115:243–56, February. Steele, J. Michael. 2004. The Cauchy-Schwarz Master Class. Cambridge: Cambridge University Press. Vendler, Helen. 2007. Our Secret Discipline: Yeats and Lyric Form. Cambridge, Mass.: Belknap Press of Harvard University Press. Wysong, J. N. 1968. The influence of Shakespeare’s songs on the poetry of A. E. Hous- man. Shakespeare Quarterly 19.4:333–39.

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