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M@h*(pOet)?ica-Louis Zukofsky's Integral


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#StorySaturday is a Guest Blog weekend experiment in which we invite people to write about science in a different, unusual format – fiction, science fiction, lablit, personal story, fable, fairy tale, poetry, or comic strip. We hope you like it.


Two fun poems to begin with--because after that the going will be tough: calculus, no less! The first is on a card I got from my friend Endwar.

ten x ten

ten x ten

The second is a poem from 60 or 70 years ago that's one of sixty-four collected in a small hardbound book called, Geo-Metric Verse, which was published in 1948. Its author, Gerald L. Kaufman, now deceased, was a distinguished architect--like later mathematical poet, Scott Helmes, architecture perhaps more than any other profession likely to produce people able to marry mathematics and poetry. Kaufman describes his sixty-four poems as “Poetry forms in mathematics/ Written mostly for fanatics.” Many of them were published in that fine old family culture magazine, the Saturday Review of Literature (in which John Ciardi provided me with the first intelligent writing about poetry I ever encountered).



This is actually more a visual poem than a mathematical one. Note in particular its subtle, albeit simply-produced, three-dimensionality. But it is also a poem about as traditional as possible. A specimen of light verse, in fact. Neither its rhyming so old-fashionedly nor its going after laughter lower its status for me, however.

Quite a bit more dignified is Louis Zukofsky's use of the following integral sign that most of the rest of this entry of M@h*(pOet)?ica will be about:



It appears in Zukofsky's famous long poem, “A” (first published in 1969), which I won't say much about--except that the fact that the quotation marks are part of the poem's title should suggest to you a little of its expressive concern with matters as seemingly minor as punctuation marks.

According to Jeffrey Twitchell-Waas, the webmaster of Z-site: A Companion to the Works of Louis Zukofsky at, who was kind enough to fill me in on what he knew of Zukofsky's background in mathematics, Zukofsky “often said that he should have been an engineer--and that wasn't just something he said later in life--recently I saw a letter from the early 1930s to (poet Carl) Rakosi (I think) where he says this. He went to the top math and science HS in NYC and later in the late 1920s he substitute taught there (presumably English) where he met Jerry Reisman as one of his students and who would become one of his best friends over the next decade plus.

Reisman was studying to be an electrical engineer and in the late 20s and early 30s LZ was reading some of his textbooks, such as on the new quantum physics. The mathematical formulas used in "A"-9, LZ always credited to Reisman. During the early 40s LZ worked for Reisman editing electrical technical manuals--the poem "It's hard to see but think of a sea..." comes out of that. I haven't run across much evidence that LZ read very deeply in mathematics and hard to say how much he picked up from Reisman. Obviously he felt he had an affinity with numbers and liked playing with formulaic possibilities, such as the tone row of "A"-20. But I doubt any of this is very sophisticated from the perspective of someone who knows a fair bit of math. But, also, I may simply be blind to what I don't know. Don't know if John Taggart or his thesis on LZ from the early 1970s would have/know more on this.”

I can't quote the passage containing the integral sign because I'd have to sell my cat, Spike, to be able to afford the fee Zukofsky's son, Paul, would charge me, and I don't want to have to go to court to defend my use of it as scholar's privilege, which I do believe it would be. I also sympathize with Paul's reasons for charging what he does for any use of his father's work. The core of his belief, as I understand it, is that academics quote famous writers' work for material profit (not necessarily commercial profit but profit, nonetheless, via the tenure, increased salaries, etc., that they gain); hence, they should have to pay more than nominal rent on what they quote. If I didn't think I could get by with a sort of paraphrase, though, I'd fight PZ to the Supreme Court!

Here's the integral sign again. In Zukofsky's poem, it has the word, “music,” just above its top, and the word, “speech,” just below its bottom.



According to the poem, this represents Zukofsky's concept of poetry. The poem explains it as basically a continuum containing speech at one end and music at the other. More precisely, if much more clumsily, it means in calculus “the sum of all the infinitesimally-changing values of vocalization ascending from speech (at the lower limit) up to where it becomes music (at the upper limit).” Hence, it represents, “poetry as a patterned integration of words, beginning in speech and ending in music,” as my friend Sabrina Feldman (a, ahem, JPL source of help of mine) put it. (I also need to thank David Webb, a Dartmouth Math professor, who also gave me help.)

I feel it obligatory to discuss Zukofsky's poem here for several reasons. The first is because of its historical importance (however little the poetry establishment has yet made of it). While mathematics has for a long time occasionally gotten into poetry (notably in the work of some of the 17th Century English metaphysical poets), so far as I know it has done so only as subject, not as actual expressive component--until Zukofsky's poem. I therefore consider him the inventor of mathexpressive poetry. This, remember, I define as “poetry in which a mathematical operation (or a series of them) is carried out on non-mathematical terms, in the process adding significantly to the poem's aesthetic value.”

Zukofsky's poem (poem-within-a-poem, to be precise) is mathematically expressive due to the integral calculus operation a reader must carry out in order to understand it--or solve it, since Zukofsky makes it part of an equation by telling us the integral is what poetry equals. The derivative, “d(vocalization),” that should be included in the equation is implied, I should add, since it is infinitesimal values of vocalization's non-semantic content that are integrated from zero, presumably to some much higher value, a hundred, say. That is, the continuum begins with highly verbal unmusical speech and ascends (and Zukofsky does want us to believe poetry is at its best when most musical) through less and less verbal vocalizations until it is minimally verbal, and maximally musical).

Another reason for my discussion of the poem is that it was the poem that introduced me personally to mathexpressive poetry. I found it in the eighties in A.L. Nielson's “newsletter in poetics,” lower limit speech (a title Nielson took from Zukofsky's poem). I was bowled over by it. Within a few days, I'd come up with my first two mathexpressive poems:

Mathemaku No. 1

Mathemaku No. 1

Mathemaku No.2

Mathemaku No.2

To “solve” these, just do the simple algebra in each, although in the second you will need to consider what punctuation marks do. Complete answers will appear at the bottom of this entry.

My third reason for dealing with LZ's passage is simply that it seems to me a terrific poem. For one thing, I think no one can deny its having two of effective poetry's cardinal features, compactness and freshness. It could hardly be expressed in fewer words, if you take it as an equation with the integral on one side and “poetry” on the other. Its use of mathematics is so fresh even now, over forty years since it was first published, that it will strike most people as bizarre.

And I claim that nothing is more important for a poet than finding new ways to surprise people with the familiar. For one thing, it slows a person encountering a poem into a way of apprehension he's never employed in such a situation before, making it hard for him simply to register the surface of its words rather than their connotative undermeanings, their sound, and the sensual impact of what they signify--which is an important though rarely realized function of poetry. But it is also briefly annoying, which enhances the pleasure felt when he gets past it to the familiar matter at almost every poem's core.

But this poem provides a reader with something not attainable without mathematics: an expression of a flowing ascent of speech to music, rather than a step by step one, since each successive value of poetry in the continuum from speech to music shown is only infinitesimally larger than the previous value, due to the nature of the integral. So--for me, at any rate--the integral sign seems another machine like the “dividend shed,” the thing divided into I wrote about in my last entry: it takes two varieties (or “values”) of something and--with the speed of a machine, an electronic machine--it expands them to a river of language falling away from someone's speech, leaving song behind.

Hey, maybe I have some sort of synaesthesia--instead of hearing visual stimuli or seeing sounds, I experience certain symbols as operating machines! Not really, but surely a person can identify with a machine--a lawnmower making paths through a lawn, say. . . .

That Zukofsky's passage, to get back to that, is challenging most people will surely agree. But some will not agree that being challenging is a virtue. All I can say is that a problem like the one concerning what this poem is up to can be fun, and solving it as pleasurable as anything else one can get from a work of art.

That's something I emphasized in my last entry, as is the value of the tension between earth and sky, or the concrete and the abstract any kind of mathematical poem will have. Finally, Zukofsky's poem should deliver those able to follow what it does to the Manywhere-at-Once I hold to be the place the best poems end in. For, lo, there such oppositions as speech and song, or mathematics and poetry become one. And the ultimate wholeness of the cosmos is once again verified. Or so, I say unto you, it seems to me when I'm in the proper mood.

To finish this session with Zukofsky's poem, here's a variation on it by Anna Everettt that was in A.L. Nielson's newsletter:

Everett Variation

Everett Variation

Where Zukofsky defines a poem as the sum of all possible forms of speech up to speech-become pure-music, Everett's variation extends its range to speech-become-pure-music . . . becoming-pure-ethereality!

Now, to finish off this installment of my blog, here's “Derivation of Wisdom,” by Connie Tettenborn, another mix of poetry and calculus:

Derivation of Wisdom

Derivation of Wisdom

If you carry out the differentiation shown in this, you will get (2)(½)(experience) + knowledge. This, the poet goes on to claim, equals, or yields, Insight plus Knowledge, the two together adding up to Wisdom. In just the right graphic scene.

Now for the explanations of my two “mathemaku” (called “mathemaku” because they are intended to be mathematical haiku, so contain for those familiar with that form of poetry, a kind of serenity, feelings of reverence for nature--especially the change of seasons--and intimations of “the ultimate wholeness of the cosmos” most haiku contain, or try to contain). I'm not sure whether my image of the different sounds of rain as a mapping process works, but “cold rain” by itself should be enough to evoke late fall or winter--the dying of the year in contrast with the couple's becoming one, according to the poem's exponent, and--we know from biology--as they symbolically take the necessary steps that will lead to the rebirth of the year. The poem, then, represents a synthesis of final, archetypal opposites. (It is also, I like to boast, the world's only pornographic mathematical poem.)

The other mathemaku requires one to remember how to carry out the multiplication of simple fractions, such as the multiplication here of “meadows.” times “:/.”--by letting the period of “meadows.” cancel out the period of “:/.” to get “meadows:” All that is then necessary is to remember that a period stops a text, and colon indicates “more to come.” The poem thus states that March turns winter into spring. A truism freshly expressed. What I like best about it is the idea that the multiplication, generally instantaneous, somehow takes place slowly.

Previously in this series:


M@h*(pOet)?ica: Summerthings

The views expressed are those of the author and are not necessarily those of Scientific American.

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