First published in Under the Bashō on 3 June 2016. Originally written in April of 1994, and tinkered with in 1995, 1999, 2007–2009, and 2014–2016. New here are the last half of the paragraph beginning with “As for the understanding of jokes” (“To give an example” to the end) and postscripts #6 and later, among a few other minor changes. In some ways, the leap or implication in the topological structure of haiku presented here relates to the Japanese concept of “ma” 間, or space. + + + + + +
“The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.”
—William James, Collected Essays and Reviews
“Mathematics and Poetry are . . . the utterance of the same power of imagination, only that in the one case it is addressed to the head, in the other, to the heart.” —Thomas Hill
“I think a lot of scientists are secret poets. The modes of exploration are quite similar, and the physical embodiment of discovery feels the same, too. That surprise and delight of discovery is something that poets and scientists share.” —Kate Greene +
“Science is not omniscient, nor is it omnipotent, and there will always be room enough for haiku.” —D. T. Suzuki
In 1988 Professor John Allen Paulos of Temple University published a best-selling book called Innumeracy (New York: Hill and Wang). His overriding concern was to draw attention to the fear and misunderstanding of numbers and mathematics in modern American culture—and to emphasize the unfortunate consequences brought on by mathematical illiteracy. He is probably best known for this book. In an earlier book, titled Mathematics and Humor (Chicago: University of Chicago Press, 1980; all page references to this edition), Paulos asserts a three-dimensional model for the structure, development, and understanding of jokes and humor. Numbers and things mathematical may seem far removed from the ken of haiku—except perhaps for the counting of syllables, if one is so inclined. Yet the structural development within many haiku bears a demonstrable resemblance to the mathematical model that Paulos employs to reveal the methodology of humor. Indeed, the structure of jokes and the realization of humor hold many similarities with the structure of haiku and the “aha” moment that releases this small poem’s reverberations. Haiku poets have realized this similarity on an intuitive level for many years, but through Mathematics and Humor it is possible to review the subject in significant detail. With a close study of Paulos’s book, haiku poets might readily recognize the parallels in this model to the moment of heightened awareness in haiku.
Critical Assessments of Humor
Before exploring this mathematical model as an aid to understanding haiku’s structure and development, it is helpful to know Paulos’s application of the model to humor. In prelude, however, Paulos reviews a short history of the various critical assessments and classifications of humor. Many of these observations also hold true of haiku. William Hazlitt, for example, stated in 1819 that “the essence of the laughable is the incongruous” (3). Even as far back as the ancient Greeks, Aristotle proposed that the mark of a poet was to see a connection between apparently incongruous things. Given haiku’s common technique of juxtaposition—which sometimes leads to incongruity—it is no wonder haiku shares similarities with humor. Hazlitt also described humor as “the disconnecting of one idea from another, or the jostling of one feeling against another” (3). He could easily have been talking about haiku. Other philosophers and critics have emphasized the surprise and the unexpectedness in the incongruity of humor—and again this is part of what makes haiku work. Specifically, in 1790, Kant proposed that “Laughter is an affectation arising from the sudden transformation of a strained expectation into nothing” (3). Is this, perchance, the nothingness of Zen? In 1818 Schopenhauer wrote of humor that it “often occurs [when] two or more real objects are thought through one concept; it then becomes strikingly apparent from the entire difference of the objects in other respects, that the concept was only applicable to them from a one-sided point of view” (3). While haiku usually seek to present a unity that does not come apart, certainly humor shares with haiku at least that initial unity. That two objects could be thought at first similar in humor is akin to the technique of internal comparison in haiku. Yet perspective is also essential to haiku, intrinsic to the poet’s voice. In humor that perspective or one-sided point of view is intentionally flawed or absurd, but in haiku it takes observation and juxtaposition beyond the merely playful to hint at profound wholeness. Herbert Spencer observed that laughter can occur “when consciousness is unawares transferred from great things to small” (4)—which is exactly the focus of haiku, enabling readers to notice small and intimate details. No wonder an appropriate response to the sudden awareness found through haiku is a knowing smile, a quick chuckle, or perhaps a laugh, even if the haiku itself is not intentionally “funny.”
The “incongruity theory” of humor has been emphasized by critics into the nineteenth and twentieth centuries. Baudelaire expanded on the idea by saying that laughter “is at once a token of an infinite grandeur and an infinite misery,” going on to say that “it is from the perpetual collision of these two infinities that laughter is struck” (5). Indeed, haiku makes an effective “approach to infinity”—the infinite now—by focusing on small objects or brief events.
Central to all humor, of course, is an innate playfulness. Here again humor shares a similarity with haiku, for even the word “haiku” in Japanese means “playful verse.” The writer Max Eastman emphasized this idea of playfulness in developing his “derailment theory” of humor in 1936. He suggested that some humor takes unpleasant topics and intentionally derails our negative emotions toward them by making light of them or observing the humor within them. In January of 1994 the Los Angeles area suffered a devastating 6.8-magnitude earthquake that killed 61 people and caused over $20 billion in damage. That same month, Midwestern Americans were suffering through one of the most consistently cold and snowy winters on record. Lee Gurga wrote to me at that time and shared an amusing “catastrophe” theory: He said that the L.A. quake was caused by all the Midwesterners shivering. Furthermore, one can judge the depth of emotion in response to a tragedy by noting how long it takes for jokes about it to begin circulating—if at all. For example, how long did it take for jokes about the death of Princess Diana to appear? Likewise, it took a very long time for jokes about September 11 to surface, and they still make many people feel uncomfortable. Rather than belittle the tragedy, such jokes derail our negative emotions and help us recover emotionally.
By the means of derailment, humor helps to disengage readers from what is unpleasant. In contrast, haiku usually seeks to engage rather than detach the reader from life (see Paul O. Williams’ article, “Engagement and Detachment in Haiku and Senryu” in Woodnotes #19, Winter, 1993, reprinted in The Nick of Time: Essays on Haiku Aesthetics, Foster City, California: Press Here, 2001). Yet the engagement in haiku is similar to the disengagement in the derailment theory of humor, in that the means to both ends is playful—that is, creative—and that the net aim is to create shared understanding. Humor of the “derailment” kind may disengage the reader from something negative, but it does still engage the reader in something positive to take its place—humor.
The contextual nature of the derailment theory of humor—where one must know that an earthquake just happened or a tornado just swept through the nearest trailer-park, or know the values or morals of some intended victim—is one recent understanding of humor. D. H. Munro wrote in 1951 that a sense of humor takes “delight in what is new and fresh” (7)—a comment that applies equally well to haiku. Having a “sense of haiku” is exactly what is meant by the haiku spirit and living the haiku life. In fact, Arthur Koestler commented very directly on the similarity between the “creative insights” of both humor and poetry. In 1964 he wrote that “the logical pattern of the creative process is the same in [humor and poetry]: it consists in the discovery of hidden similarities” (7). Indeed, T. S. Eliot’s notion of the objective correlative—whereby an external object of nature or the world around us may be found that intrinsically embodies and conveys an internal emotion without the stating of that emotion—is central to successful nature haiku, and relies heavily on the observation of exactly those “hidden similarities.” Koestler notes a difference in the emotional climate between humor and poetry, yet calls attention to the similar structure or pattern. Of larger significance, however, is the fact that Paulos follows “Koestler’s principle that creative insights in all fields . . . share the same logical pattern” (8). Paulos summarizes this notion by saying that, “for something to be funny, some unusual, inappropriate, or odd aspects [of something] must be perceived together and compared” (9). While haiku typically embraces and finds virtue in the usual, appropriate, and ordinary, the way haiku works—in the perception and comparison of sometimes disparate objects or events—is similar to the way humor works. Whereas the humor of a joke lies in the resolution of ambiguity, so too the understanding of a haiku lies in the similar resolution of a carefully crafted ambiguity—what I like to call a “perpetrated ambiguity,” wherein two or more ambiguities are intentionally suggested. Whether ordinary or odd, the revelations of suchness in both haiku and humor are strikingly similar in pattern, structure, and even response.
For humor to work, however, Paulos stresses that the “emotional climate” must be appropriate (10). As an example, earthquake jokes are hardly appropriate while one is excavating the debris of a fallen building looking for bodies—or survivors. As already mentioned, the severity of a disaster may be measured by the delay after which jokes about it begin to surface. Even the common preface “Have you heard the joke about . . .” serves a useful purpose because it alerts the listener to the onslaught of a joke. Thus, despite all its cloying, it helps to create an appropriate emotional climate. By this emotional climate Paulos means to acknowledge the contexts and psychological aspects of humor. These can become quite involved, and if one wants to delve into the writings of Freud and others batting about this topic, one may do so, but it is not directly relevant to the study of the structural patterns of humor—or haiku. Of relevance to the subject of haiku, though, is the analogy that, as with humor, perhaps the “emotional climate” must also be appropriate for haiku. I fear this topic is too large to explore adequately here, and would only diverge from the topic at hand. But it would be worthy of exploration. Suffice it to say that haiku and humor both require an appropriate emotional climate—and this similarity again strengthens the idea that haiku and humor have much in common.
Affinities Between Mathematics and Humor—and Haiku
For his purposes, Paulos notes numerous affinities between mathematics and humor. Many are also surprisingly true of haiku. Not only does he state that “both mathematics and humor are forms of intellectual play” (10), but that they are both “economical and explicit” (11). What could be more true of haiku? The elegance of brevity is no more refined than it is in haiku, and explicitness may also be found in haiku’s directness and objectivity. I cannot vouch for most mathematical proofs, but a long-winded or awkwardly told joke seldom makes one laugh.
Paulos also suggests that “combinatorial ingenuity” (14) is strong in both humor and mathematics (the cleverness of combining unexpected things in the right way). This is emphatically true for haiku, where the choice of which elements to juxtapose is central to the vibrancy of the poem. More subtle, but vitally important for haiku, is the similarity in logic. For all the absurdity in humor, it is always held together by some internal logic or the joke would not work—it makes sense on some level. Mathematical proofs also depend on logic, of course—and so do haiku. Paulos indicates that humor requires “a perceived incongruity with a point and an appropriate emotional climate” (10, emphasis added). Just as there must be some “point” to a joke, a haiku flounders like a badly told gag if it does not have some sort of goal such as showing the unity or internal comparison of two carefully selected objects or events in juxtaposition. By “goal” in haiku, I do not mean just the poet’s intent, for the poet’s desire for his or her poem can be misguided or the experience may be inadequately told in words. Nor do I mean any overarching moral or manipulative message—polemical agendas that seldom have a place in haiku anyway, because the poem seeks simply to reveal the suchness of things without agenda or ego assertion. Rather, I mean that the goal of a haiku is to have whatever element makes a haiku work, if it works. In a sense, this elusive goal is larger than the poet. Perhaps what I’m saying can be made clearer by stating that a jokester’s “intent” may be to make another person laugh or to prick the conscience through satire, but the method of how the structure of a joke brings about that laughter or realization of satire is quite distinct from what he or she says or may have intended by the joke. In other words, my focus on humor and haiku is objective rather than subjective, a focus not so much on what is said or intended, but how it is said. The similarity to the logic of humor and mathematics is that for the haiku to work, that “how” must, of necessity, be internally logical.
Paulos also notes an assertive tone in most humor. This assertiveness, the surety of what is being said (“this is so”), is also a clear trait of successful haiku. Its apparent assertiveness arises, I think, from the poem’s clarity, concision, and directness. In a good haiku, as with a good joke or refined mathematical proof, no redundancies exist to suggest doubt, no woolly phrases or flawed syntax, just the clearest verbal path, the most elegant means to an end (one could say that they have the ideal organic form, to use Denise Levertov’s term). The implication for beginning haiku poets is that their work might improve if they write with greater assertiveness—not naively, of course, but to strive after the substance behind the poem that intrinsically gives rise to polite assertiveness.
A final trait of both mathematics and humor that also applies to haiku involves the necessity of deduction. Just as one may deduce the conclusion of a syllogism through logic, and just as humor works by offering surprise deductions, haiku has its life-blood in the deductions of both writer and reader—little leaps of understanding. The writer makes the first deduction, of course, keenly perceiving one or more elements of nature or human nature in a moment of heightened awareness, deducing an internal comparison or objective correlative or using some other haiku-writing technique to convey meaning. But, because haiku is a poetry of implication, it is also up to the reader to deduce those implications. This is why haiku seem unfinished, like “a hand beckoning, a door half-opened,” as R. H. Blyth has said (Haiku, Volume I: Eastern Culture, Tokyo: The Hokuseido Press, 1949; 272). This is what sets haiku apart from most other poetry, in that the reader has to be engaged to complete the poem (and perhaps why many haiku poets read each of their haiku twice at public readings). And as with humor, where the listener reaps the reward of a joke by “figuring out” the punch line and making a leap of understanding without the joke being explained, so too the haiku works best if it avoids explanation. If the poet knows when to stop and lets the reader figure out what makes the poem work, then that is where the haiku begins to reverberate and reward the reader. For example, consider this poem of mine (from Frogpond 14:3, Autumn 1991, and second-place winner in the Haiku Society of America’s 1991 Henderson haiku contest):
an old woolen sweater
taken yarn by yarn
from the snowbank
I’ve shared this poem in workshops and seen the light go off in the eyes of students when they realize that this poem is about a bird building a nest. Yet it takes a risk in hinting at that meaning, and not all readers get it. The light doesn’t always click on. But I believe the poem creates a greater reward by not explaining itself or by avoiding excess information that would make the poem easier for a less sensitive reader. As Toni Morrison once said, “All art is knowing when to stop.”
Nevertheless, too many haiku, especially by beginners, fail to let the reader figure anything out. For example, a statement of emotion, or an explanation or intellectualization about the relationship between certain natural elements chokes all hope from the poem with these shortcomings. Even concrete poetry must be put together or reconstructed and thus figured out. It would be absurd to explain a successful concrete poem; such an explanation would kill it just as surely as explanations kill the humor of jokes.
Incidentally, this is one reason why I consider Cor van den Heuvel’s controversial “tundra” poem to be a haiku, though some poets and critics do not. In this poem, the reader figures out that the white space of the page is snow surrounding the lone “rock” of the word “tundra”—or that the expanse of white space is a concrete representation of the tundra itself. This “putting together” of the poem occurs in a spark of understanding, a leap of enlightenment. Indeed, the very “haikuness” of this poem takes place in that realization, rather than in the verbalization or the space of the poem. Meaning has been put in a different place, and it’s up to the reader to figure that out. Because the verbal (surface) structure is so reduced in Cor’s poem, having only one word, it leaves little for poets to understand as “haiku” unless they grasp the metalevel (deep) structure—the relationship of this particular word to the expanse of space around it. To say that any word could be tossed onto a page by itself and called a haiku is a grave mistake that ignores the spark of deep-structure realization that defines not only this minimalist haiku but also more conventional haiku. For me it is this spark, this vital leap of understanding so perfectly crystallized in haiku, that defines haiku and its spirit more than any other characteristic. What seemingly confuses some poets into disallowing “tundra” as a haiku (albeit experimental and on the fringe), I suspect, is too great a focus on the object level or surface structure of the poem, and inadequate focus (even a total lack of awareness thereof) on the metalevel or deep structure of this poem—the level at which I feel haiku is truly defined. I do not mean to suggest, however, that more haiku should be like “tundra.” Quite the opposite, in fact. It’s been done, and because it’s so short, most poems like it would be overly derivative (John Stevenson’s “core” being one brilliant exception). Rather, more haiku poets could benefit from recognizing this poem’s deep structure and how it serves to create a moment of realization. Perhaps I am being generous, but it’s that similarity of deep structure and the emotional effect that “tundra” has on the reader that makes it akin to more traditional haiku, and thus makes it, in my estimation, a haiku.
Levels and the deep structures of transformational grammar will be returned to shortly. For now, however, note that the understanding of a joke is different from its humor. Similarly, the understanding of a haiku is different from its implied emotion. The problem is simply that some haiku are too often explanatory, thus committing a sin equivalent to the explaining of a joke.
The traits of deduction and other similarities between humor and haiku temporarily aside, much of this discussion might be abetted with some examples of haiku. The best examples, however, show the failings of specific haiku, and I would rather not discuss any individual failed poems. In general, though, consider the example of a haiku that explains too much. In such a case, the explanation removes the element of deduction, the very “playing with the poet” that makes the game enjoyable—worth playing, even. With too much explanation in a haiku (directly stating one’s intellectualization or emotion rather than freshly describing an object or event, for example), the poet is saying, in effect, “watch me play.” This completely misses two essential aspects of haiku, those of suppressing the ego, and of actively engaging the reader, getting him or her off the bench to join the game and help “finish” the poem. A game of tennis is no fun—indeed, not even possible—without a partner.
As for the understanding of jokes, Paulos asserts that they are like mathematics because they “depend for their humor on an implicit understanding of the axiomatic method” (19). He summarizes this method as the selection of “certain self-evident statements as axioms and deducing from them, by logic alone, other statements, which often are not so self-evident” (19). The corollary to haiku is extremely compelling: The haiku poet selects certain self-evident (that is, universal) objects or events as axioms from which certain intrinsic, non-self-evident conclusions may be deduced. It’s where we get the haiku poem’s leap of understanding, indeed, the aha moment. This is the heart of how haiku works. To give an example, a window and a rock might be selected for a haiku as axiomatic objects from everyday life. With the appearance of those two objects in a poem, we can leap to the conclusion of what the rock might do to the window. This may be an easy conclusion to make, but it’s not intrinsically evident in the “self” of the window and the rock; rather, it becomes evident in the reader’s interpretation. In this manner, one plus one becomes three.
I note with more than passing interest, incidentally, that the logical deductions that may be made from the so-called “axioms” in a haiku are sometimes confused with a mystical sort of intuition. Some readers think they “intuitively” understand what a poem is about. That may be so, but quite likely the truth in the way they understand a poem is less subjective: they may have simply deduced a meaning by objective logic, thanks to the axiomatic nature of the primarily noun-based elements of haiku. Indeed, Paulos suggests that axioms may be thought of “as being clues to a mystery and the different possible scenarios for the crime as being models of these axioms” (22). Likewise, the elements of a haiku are indeed clues to a mystery. It may not be a dastardly crime, however, when the poet fails to understand the poem’s intrinsic structure.
Object Levels and Metalevels in Humor and Haiku
Another vital distinction to do with mathematics and humor—and also haiku—is between “object level” and “metalevel.” In terms of haiku, what a haiku describes is the object level—a rock, a tree, a passing glance. But what the poem is about, that is, what it implies, is the metalevel—old age, loneliness, love, or whatever, and the emotions related to the implication. I have at times described haiku as “a brief poem using objective words to convey heightened subjective feeling about nature and human nature.” Here the objective/subjective dichotomy maps neatly to the object level/metalevel distinction—and to surface and deep structures in terms of transformational grammar. I feel that many beginning poets confuse these levels, and doom their poems to failure by trying to directly state a subjective feeling within what is best left as an objective domain. Yet often the very subject of a poem is best removed so that the reader can be left alone to figure it out and make that leap from the objective to the subjective (hopefully, the same subjective intent as the poet). It is in the metalevels of meaning where haiku begin to reverberate. Paulos emphasizes the need to rise to the metalevel for humor to succeed: “to get (i.e., understand) a joke . . . one must ascend, so to speak, to the metalevel at which both interpretations, the familiar and the incongruous, can be imagined and compared” (26). The same is also true with haiku, yet again this “ascent” to the metalevel, the level of meaning, need not be considered mystical.
I find another intriguing parallel between humor and haiku in Paulos’s assertion that certain basic paradigms must be understood by a common body of people for humor to work (among those people at least). Perhaps, in the widest way, in Jungian terms, this is nothing less than the collective unconscious, which asserts that we all know, feel, or understand certain paradigms and archetypes. In haiku terms, these basic paradigms include the so-called “universal” in the particular that haiku so often espouses, and also include the common understanding we have of most seasonal references. If a poet describes an obscure local custom or insect or place, that poem’s meaning is necessarily limited even among a body of knowledgeable haiku readers. But if one describes what is more universally known, or archetypal, as in the common name for a certain genus of flowers (rather than, say, Helianthus annuus), the chances of successful communication are vastly improved—first on the obvious object level, but also on the more important metalevel. Actually, if there is no possibility to ascend to any metalevel, that is, if the poem operates only on the object level, then no reverberations are possible and the poem will be unremarkable and forgettable—a “so what?” poem. Paulos puts it this way: “With no feeling for what is correct, congruous, or natural, there can be no perception of what is incorrect, incongruous, or unnatural” (27–28). It is not completely accurate to call the metalevels of haiku “incorrect or unnatural,” necessarily, but certainly something unexpected or incongruous is the domain of the meaning level in haiku. Former Haiku Society of America president Virginia Brady Young has described haiku as “leaping the chasm” (Raking Sand, Foster City, California: Press Here, 1993; 28). The point regarding humor and haiku paradigms is that haiku poets must first stand on an identifiable, common cliff edge before they can go about leaping chasms. If poets begin with carefully chosen commonality, they will more readily engage the reader with their leaps to what is uncommon. This is an assumption inherent to humor and haiku that is too often overlooked.
In contrast to the ideas of object level, metalevel, and shared paradigms, Paulos reminds his readers that Zen philosophers have suggested that “[dualistic] notions like truth and falsity, subject and object, external and internal, while essential in everyday life as well as in scientific thought, nevertheless prevent one from attaining a mystic, oceanic union with the universe” (52). Because some jokes, especially paradoxes, intentionally confuse and then resolve levels, and challenge the notions of truth and falsity, they “might thus be taken as a reminder of this essential is-ness of the universe—a reminder that these distinctions, in some fundamental sense, are unimportant” (52). How true this is of haiku. Certainly the ascent to the meaning level/metalevel in haiku, while not necessarily confused and then resolved intentionally, does remind one of the is-ness and suchness of our surroundings. Therein, it would seem, lies the fundamental brilliance of haiku: By using objective words to convey—that is, imply—heightened subjective feeling about nature or human nature, haiku lays an object-level groundwork from which the reader may make the gestalt leap from the subjective metalevel, and, in the process, unifies the objective/subjective dichotomy into the holistic suchness of Zen’s eternal now. That’s a mouthful, I realize. The point is that, if we trust it, the right objective imagery, carefully juxtaposed, can have magical poetic and emotional effects. The end feeling can seem in many ways mystical or euphoric, like a taste of enlightenment, but here again it is a result, not the full process. And it is, in effect, a meta-metalevel. This figure/ground wholeness might be thought of, if you will, as the grand unifying theory of haiku. At its very base it shows that haiku recognizes that the universe simply is. Just as one laughs at the point when one has resolved a joke’s incongruity, one also breaks into a knowing smile at the “aha” moment of resolving, when it occurs, a haiku’s apparent incongruity—of figuring out what one part of the haiku has to do with the other, for example. At the instant of resolution in haiku, the levels seem to dissolve, the objective and subjective are superseded by the meta-metalevel, and one returns—leaps—to the ultimate yet intimate suchness of life, a mystical unity that is as Zen as Zen can be.
Catastrophe Theory: A Topological Model for the Structure of Humor
Paulos proposes a three-dimensional topological model, based on René Thom’s catastrophe theory, for the unifying moment central to humor. It also fits the structure of haiku, as I hope the reader will see. But before introducing the model, Paulos discusses types of humor, transformational grammar, deep and surface structures, and larger aspects of philosophy. He explains, for example, that “a pun forces one to perceive in quick succession two incongruous and unrelated sets of ideas” (61). This much is already understood, but then he immediately goes on to say that “the suddenness is, as in much humor, very important” (61, emphasis added). He means the suddenness of one’s realization of a joke, and here again is a parallel to haiku—its subject is here and now, but the sudden realization of its metalevel is very sharply here and now also. In presenting a mathematical model for humor, Paulos emphasizes “the idea of an abrupt switch or reversal of interpretation in the sudden perception of some situation, statement, or person in a different and incongruous way” (75). He might just as well say this of haiku. This abrupt switch or discontinuity is akin to René Thom’s topological mapping known as catastrophe theory (and here “catastrophe” is meant more in its primary definition of “the culminating event of a drama . . . by which the plot is resolved,” rather than any natural or personal disaster). As Paulos explains, the model “provides a sort of mathematical metaphor for the structure of humor [and helps observers] to visualize that structure more clearly” (75–76). When I first noticed that this model applies rather neatly to haiku as well as humor, it was not without a good deal of intellectual excitement.
This topological model presupposes the understanding of a three-dimensional coordinate system, where points are located by their x, y, and z coordinates. Much of the mathematics and topology is perhaps best avoided here, but essentially, given any response to a joke that depends on two factors (that is, an incongruity and its resolution), and given that it is discontinuous and satisfies mild general conditions, a consequence of Thom’s main theorem is that a graph in three-dimensional space of empirically measured results gives rise to the following surface shape: