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This entry will be my laziest ever. In fact, it may be as lazy an entry as it’s possible even for me to make. That doesn’t mean it won’t be terrific! It begins with:

This is from Number Poems, a book by Irving Weiss I used my one-man outfit, the Runaway Spoon Press, to publish in 1997. I liked it so much, I had it on the cover, title-page and as the first poem in the book!

As those who have been reading this blog for a while will know, I take its mandate Very Seriousfully to be a concern with Real Mathematics, preferably mathematics in operation, if on non-mathematical material. That which is mathematics, as opposed to that which is about mathematics. Not that I have wholly neglected poetry about mathematics. Indeed, I hope to have more of that here in the future, for I do consider it as valuable as poetry that is mathematics (in my view), just different from it.

Which brings me to Irving’s book. The works in it are not real mathematics by any stretch of the imagination. (Some of them aren’t poems, either!) But I really like them. What’s more, he’s a good friend of mine—not to mention over ninety now which means that featuring him will win me a bunch of anti-gerontophobia brownie points. Ergo, I had to find a way to get them into this entry. Finally (after many weeks of thought, I assure you), I came up with the concept of something I termed “matheconceptuality.” This I defined as mathematical thinking about non-mathematical subjects. Or: a way of treating reality that is not, strictly speaking, mathematical but is more mathematical than anything else.

To clarify, my impression is that strictly mathematical thinking is used in just two ways: to measure (i.e., applied math), and to perform number theory (i.e., pure math). What Irving’s matheconceptuality results in is not quite either of those, nor is it a kind of discussion of mathematics (like Rita Dove’s poem in my previous entry about the joy of geometry). It’s closest to the use of simple analogies like equating a boulder’s height to some friend’s. The latter, however, seems to me more the use of a kind of “ur-math,” or the application of math at its most primitive.

Perhaps the best way to get near what Irving’s matheconceptuality is, is to work out what his “Invasion” does . . . if I can. If only intuitively, and confusingly . . .

First impression: the piece is minimally “polluted” by sensory reality. As number theory is.

No, it is emphatically involved with the real—with real numeral ones, with the more real hand-printed numerals (because conveying a sense of the person who made them) . . . the color black. But these material elements are employed primarily in a mathematical manner . . .

The apeiron!1 It just occurred to me that a word I got from a piece by Irving coming up may be just what I need here. It is the Greek philosopher Anaximander’s term for what “space” used to mean before Einstein: the essential nothingness everything in reality is “in.” I suggest that—at least as a start—we take what Irving has done in “Invasion” as apeironian number theory.

The premise for the piece (make that my premise for it) is that the universe was once nothing but ones. This, for me, is the first bit of its mathaesthetic magic, for it presents an arresting archetypal locus that is almost minimally complex, yet capable of dropping someone into fascinating questions. Like how a universe can be all ones. It could not be! Or so I was convinced by my belief in the Eternal Dichotomization of Reality (i.e., that nothing exists that lacks an opposite2). I’d love to be hit with any sane, or even insane, argument against that.

In any case, as I see it, Irving’s poem begins in a pre-mathematical apeiron. The introduction of his ones did not begin an exercise in number theory, but began something prior to the possibility of number theory. What followed was a depiction of an “invasion” by hand-printed numerals—commanded by a zero, it would seem--of the pre-math of a universe of nothing but ones and absences of ones. A binary universe, actually. The hand-printing is not mathematically important but poetically important, for it suggests that the white numerals are giving this too-purely symbolical binary universe life—even personality.

Note, too, the size of the mouthful of invaded universe, and the distance between its numerals compared with the distance between the invaded universe’s ones. That the complicating elements of the invading forces are white is interesting, as well. Light climbing into numbers and leaving nothing behind for the numeral ones to contrast into meaningfulness against. . .

What seems to me the most important result depicted by the poem is that the numerals needed for fullest, uncumbersome number theory—i.e., real mathematics—become available. The poem, a mathematical way of dealing with the universe just short of doing math, has carried pre-math up to but not into math. Numbers not doing anything, but ready to.3 Or so my thoughts about it go. That the work can make such thoughts possible, regardless of how loony they may be, is—I contend--what gives it the high aesthetic—mathaesthetic--value it seems to me vibrantly to have.4

“Invasion” leads smoothly into the next two pieces from Irving’s Number Poems that I’ve chosen. “Ones,” is the name I’ve given to the first of these:

This puts us back before the time of “Invasion.” To what may be a contradiction of the universe depicted in it, in a way, because it strongly suggests that ones actually are all that are needed for maximal explanatory complexity—just a final simplicity of ones, black or white, can represent any universe, however complex. A pre-mathematical picture of a binary universe. Also a fascinating maze/wilderness (in spite of being about as linear as possible) to wander in . . .

“One More Is One Less” may be a different part of the universe:

The idea behind it, I think, is that each another of the dots enters the word, “ONE,” the less ones over-all there are. We are leaving the chaos of a universe of individuals for a final deadness of the single ONE. Entropy.

From this point on Irving’s pieces here are less clearly matheconceptual, but continuing to remind us how inescapable present mathematics is in everything. I think Irving himself best describes them in the preface he wrote for Number Poems, which begins: “Numbering is inseparable from living. The numbers are there waiting for us in our bodies and in the world of other people and things-the alphabet and ideographic numerals may belong to the educated but everyone can mark and count.-

“The numbers are in the centrality of nose, navel, penis, and vagina; in the symmetry of eyes, ears, hands, and feet; and in all further combinations of odd and even. The decimal system lies in our fingers and toes. And, all peoples invent some kind of calendar to manage the cosmos with.”

An excellent example of how the above translated into texts is Irving’s following musing from his book. It seems mathematical to me only in that it consists of five sets of observations and is placed in a section of Number Poems called “Fives.” But how entertainingly unusual its wry slants into what it is to be a human being!

“Number Poems,” Irving goes on to write, “is a sequence of visual poems and word-only poems based on our familiar numbering system. Since numbers are an integral part of symbolic thinking, everyone everywhere has some if not many relations to numbers long before a few among us become tabulators, accountants, mathematicians, or numerologists. I, myself, can't do more than ordinary arithmetic, but my ignorance doesn't prevent me from having feelings about and attachments to numbers and even from thinking about their mathematical, historical, mystic, mythic, and familiar uses or accidental significance.

“Number Poems originates in my own sense of number experience. The number words used as the page-titles of my poems represent various ways in which a number came into mind as an emblem or marked an occasion or turned up by chance and became fixed without reason in my memory: I construct the poems as imaginative forms or as abstract ideas.”

“Number experience,” as Irving seems to have undergone it, may be quite close to an engagement with apeironian number theory. My thinking here is definitely fuzzy, but I stand by my main point, which is that in his “number poems,” Irving is generally working out of his mathematical brain as significantly as he is working out of his verbal brain, and that in the process, he provides us with aesthetic pleasure neither alone is capable of providing.

Instant proof of this is one of Irving’s cartoons, with a genuine mathematical poem in it. Unless it’s a mathematical joke. But surely it can be both.

It is from a larger work:

I think a full philosophy could by developed from this diagram of what may be the interior that consciousness is inside the exterior that feeds it. Or of a quest for knowledge sleeping into the essential understandings of reality both maximally and minimally mundane. Or simply of “where” as both place and question . . . Or even a satire on people who would react to it as I now am.

Okay, you’re on your own, now, mine readers. I hope those of my bumbles into Irving’s works I’ve tried to describe will help you to your own at least partially profitable bumbling into the three pieces that follow--which I say have to do with geometry! I hope, that you will later return for new bumbles into the previous ones, too!

* * * *

1 Apeiron: in the philosophy of Anaximander (610-c.547 B.C.), a limitless, incommensurable, and boundless primary substance; that which precedes everything else.

Here’s more, which is translated from a 1973 Soviet Encyclopedia of all things: “Anaximander’s concept was a step forward in developing the concept of matter compared to the theories prevailing at that time. These were propounded by Thales and Anaximenes, who held that the primary substance was one ‘element’ (water or air). Anaximander’s concept was interpreted by Alexander of Aphrodisias as something between water and air or between fire and water or air. The Pythagoreans viewed it as the limitless, formless principle, which with its opposing ‘limit’ constituted the basic grounds of being.”

2 Or that the structure of our brains makes it impossible for us to imagine anything without an opposite.

3 There is a nice suggestion of Roman Numerals changing to Arabic Numerals, too.

4 I really do try to make sense of the works I write about. Here I only hope I’ve made interesting half-sense.

**Previously in this series:**

M@h*(pOet)?ica–Louis Zukofsky’s Integral

M@h*(pOet)?ica—of Pi and the Circle, Part 1

M@h*(pOet)?ica – Happy Holidays!

M@h*(pOet)?ica—Circles, Part 3

M@h*(pOet)?ica – Mathematics and Love