Held at Philadelphia for Promoting Useful Knowledge volume 155 • number 2 • june 2011 THE AMERICAN PHILOSOPHICAL SOCIETY independence square: philadelphia 2011 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 fi nd 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 fi rst, 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 fi elds 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 fi rst 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 fi rst 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 fi nding 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 specifi c 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- tifi ed 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 fi elds where roses fade, do we fi nally 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 fi elds | 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 fi rst 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 “ro¯z” in “rose-lipt” and “roses” appears in lines 3, 7, and 8. “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,” “fi elds,” “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), “fi elds 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 fi rst,” 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 fi elds where roses fade). Again the pattern conveys a meaning: the brooks (line 5) and fi elds (line 8) enclose the boys and girls (lines 6 and 7) as a coffi n 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 fi rst 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 fi rst 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 scientifi c 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 defi nition would limit “chiasmus” to the exact repetition of two symbols (a, b) in reverse or- der (b, a). A more inclusive defi nition 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 scientifi c 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 Zba˘ganu, University of Bucha- rest. In 2000, Zba˘ganu published a fact new to mathematics: If n is a positive integer (a counting number like 1, 2, 3, . . . ), and a , 1 a , . . . , a and b , b , . . . , b are any nonnegative real numbers (any 2 n 1 2 n fractional or whole number larger than or equal to zero, such as 17 or 0.333333 . . . or 3.14159 . . . ), then Σ min ((a × a),(b × b)) ≤ Σ min ((a × b),(b × a)). i j i j i j i j 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 Σ min ((a × a),(b × b)) i,j i j i j and a right lobe Σ min ((a × b),(b × a)) mediated by ≤. The symbols i,j i j i j 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 Zba˘ganu in 2001. Photograph © 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 Σ means sum (add) for all pairs i,j i, j, where i and j are positive whole numbers from 1 to n. Finally, min((a × b),(b × a)) means the minimum (smaller) of a × b and b × i j i j i j i a, and similarly for min((a × a),(b × b)). j i j i j The words “If n is a positive integer, and a , a , . . . , a and b , b , . . . , 1 2 n 1 2 b are any nonnegative real numbers, then,” which precede the formula, n declare that the inequality holds for any such numbers. Therein lies the immense and surprising power of Zba˘ganu’s inequality. A numeri- cal example illustrates its economy of expression. If n = 2 and if a = 2, 1 poetry and applied mathematics 197 a = 5, b = 4, b = 3, then the expression on the left side of the in- 2 1 2 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. Zba˘ganu’s inequality asserts that, no matter what natural number n you may pick, and no matter what nonnegative real numbers a , a , . . . , a and b , b , . . . , b you may pick, the value of the 1 2 n 1 2 n left side will be less than or equal to the value of the right side. Zba˘ganu’s inequality has social or referential meanings because it answers a question Zba˘ganu was considering in the mathematical the- 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 fi rst message is sent as AA and the second message as BB, or if the fi rst mes- sage is sent as AB and the second message as BA? The left lobe of Zba˘ganu’s inequality represents the probability of an error in transmission if the fi rst 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 Zba˘ganu’s inequal- 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 b (j = 1, 2, . . . , n) represents the j probability that the input symbol B is received as the output symbol j, and if the channel lacks memory, then (a × a) is the probability of output symbols (i, j) when the input symbols are i j AA, (b × b) is the probability of output symbols (i, j) when the input symbols are BB, (a × i j i b) is the probability of output symbols (i, j) when the input symbols are AB, and (b × a) is j i j 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 (a × a) and that with elements (b × b), because each summation equals 1 if the two matrices i j i j 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 (a × b) and that with elements (b × a). The inequality asserts that the i j i j 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.