Is Mathematical Poetry A Subset of Visual Poetry?

This is some of the comments to one of Geof Huth’s blog post reviewing Bob Grumman’s new book, really a chapbook, entitled A Preliminary Taxonomy of Poetry

Geof said, “Mathematical poems add mathematical features that visualize the poetry, so I consider them visual poems, and to have a category for flowchart poetry assumes that process symbols are textual and thus not visual. I'd argue, again, that they are not orthodox text, so these poems are also visual poems.

Also, Bob's definition remains indefensible: "poetry that uses mathematical symbols that actually carry out mathematical operations." These mathematical operations are not actual; they are apparent. That is a big different. Duck cannot be divided by yellow in any mathematical way, though it could in a metaphoric way that has nothing to do with math directly.”

Kaz said:
Gee Geof,
I am going to have to take exception to both of you on a couple of things. First I will start with you and the top paragraph. Unfortunately I have never seen a definition of Visual Poetry that everyone agrees upon. Yet I will have to say that I like what I understand to be Karl Kempton and Karl Young’s definition of: “Visual Poetry is a Poetry that has to be seen” This is such a simple yet powerful definition that seems to me to be true in every case of vizpo that I have seen. With that being said, There are what I would consider pure mathematical poems whereby they can be understood by reading them alone. An example would be, “Love is equal to the limit of 1 over ‘x’ as ‘x’ approaches zero”. This mathematical poem can be understood perfectly without seeing it therefore it would not be visual poetry.
In the next paragraph above Bob states that, " These mathematical operations are not actual; they are apparent. That is a big different.”
I will argue that these operations are actual and they work the same as any equation in applied mathematics. The ‘variable’ or we can say ‘concept’ or ‘word’ in any mathematical poem can be substituted with a number that represents the value of the variable/concept/word/term. The ‘word’ can be substituted with a multitude of numbers just like in the equation ‘x’ equals ‘y’ squared whereby x can equal anything and y will equal whatever x is squared. The thing to focus on is that the words have value or magnitude and they have mathematical relationship to each other. This means the words in a mathematical poem can be substituted with a number and the words or concepts along with their mathematical syntax within the equation provides the units or “unit meaning”. To make this clear let’s look at the equation from physics d=vt or distance is equal to the velocity multiplied by time. If you look at velocity you get units of miles per hour. If you look at time you get the units hours and when you divide the unit ‘miles per hour’ by ‘hour’ you simple get the unit ‘miles’. And ‘miles’ is the unit for distance. Notice we did not talk a bit about numbers, yet, those variables can all be replaced with numbers and it is important to note, the units will remain. Mathematical poetry is the same however the units are created within the poem itself. Unfortunately all the mathematical poets I know are not addressing this issue and thus are missing the boat by thinking that mathematical poems don’t do math.
In your next example where Duck is divided by yellow you say that you cannot divide it in any mathematical way. This is not true you can divide it, however, it is pretty much meaningless gibberish at worse and a wild metaphor at best. The bottom line is that Duck divided by yellow is not anymore incoherent than much of Gertrude Stein’s work.

Endwar (Andrew Russ) wrote:
On mathematical poetry and mathematics: I’m not sure I agree completely with anyone here. It seems to me that in a mathematical poem one sees a mathematical operation with words (usually) operating in a metaphorical way (thus the poetry enters). That said, the mathematical operations involved are usually well-defined for numbers, but not for various words and concepts. “3+1=2” is something everyone (is taught to) agrees on in a literal way, and it follows from the definitions of each number and the signs “+” and “=”. The statement "candy cane + child = happiness" is also probably pretty generally understood, but not with the same level of definiteness (or definition, as per the previous sentence) as the numerical example earlier. You could write "candy cane + child = obesity", which would probably also be understood, but because of the metaphorical nature of the math, you can’t conclude (via the law of substitution) that “happiness = obesity” (though some may point out the phrase “fat, dumb, and happy”, which could then lead us to conclude “happiness = obesity = stupidity” . . . You can see, then where the multiple meanings of words (bifurcations of meaning, to throw in another mathematical metaphor popular in some at one time trendy lit-crit circles)) can lead.)
I would argue that a mathematical poem is a statement that represents a mathematical operation on the words involved, but which isn’t necessarily one that can be checked the way mathematical statements with numbers can be. I will even go one step further and assert that one can create a mathematical poem that is mathematically wrong but which still makes a metaphorical point. I have done this using matrix multiplication – a 2x2 matrix times a 2x1 vector is set equal to a 3x1 vector. That’s not something you can do with real number (or even imaginary number) math, but I think it works as a poem.
Written mathematics is inherently visual, not verbal: I can grant Bob’s point that “3-1=2” is visually not interesting, and furthermore it hardly matters what font is used. It does matter a bit what numbers are used – roman numerals will say “III-I=II”, and binary says “11-1=10”, and ternary says “10-1=2”, which are all the same numerically. But it becomes evident for large numbers that roman numerals are unwieldy for calculating, and we are used to the decimal number system, so the non-decimal numbers need cumbersome subscripts or context to be read as intended. I would argue, though, that the real test of whether we have something verbal versus something visual is whether the statement can be read aloud. Again “Three minus one equals two,” is pretty straightforward, but that is merely because of the simplicity of the expression. Try reading, say, a passage out of the middle of J.D. Jackson’s Classical Electrodynamics or any other graduate physics or mathematics text, and it will be immediately obvious why these equations aren’t written out in words and why mathematicians and scientists do nearly all their professional discussions with slides or in the presence of a blackboard. And even if one does manage to put the text purely into words read aloud, you will find nobody in the audience who will understand what has been said who hasn’t at least written down some equations or a drawing as a guide. One of the most tedious reading experiences I had was a few pages out of an algebra text written by Leonhard Euler, who felt it was necessary to write down an equation and then repeat the equation in words, such as:
“E=mv ²/2
The kinetic energy is equal to half the product of the mass and the square of the velocity.” This continues for page after page.
If you’re still not convinced, show me how to do read calculus aloud and make it intelligible. Two pages minimum.
Because the visual representation is integral to the intelligible communication of all but the simplest mathematics, I would argue that mathematics is inherently visual language, and that by extension, mathematical poetry is also inherently visual poetry. The visual poem may still not depend on which font is used (though I have examples where that is the case as well), but it still can’t be read aloud and have the same meaning, because it will not then register as mathematical.

Kaz wrote in response to Endwar:

That is an interesting argument however, you seem to be making a distinction between the existence of a math equation which doesn’t have to be seen (like your Euler example) and then the distinction of performing the mathematical operations which have to been seen. (or at least I will agree that I would have extreme difficulty working out equations with out seeing them). Yet, since you can have math equations in verbal form (you just can’t work them out) it seems that math does not have to be in visual form and therefore not necessarily ‘exclusively’ visual. Or this begs the question what is math? Is it the performing of mathematical expressions or is it the expression itself? Or a mathematical Platonist would claim that math is an inherent object in nature … Gee why did I have to drag the Platonists into this – go ahead and slap me and forget that I said that.

Yours,
Kaz

Bob Grumman wrote:
Thanks for all the comments, endwar. I’ll get to all of them, I hope. Right now, just some thoughts in response to your comments about mathematical poetry.
I don’t care whether a poem can be read aloud or not. Mathematics is written in text just as ordinary verbal material is. Text printed standardly is effectively not visual, as far as I’m concerned: it’s symbolic. So a purely mathematical poem, in my definition, would be expressed in verbal and mathematical symbols.
On further thought, it seems to me all mathematics can be read out loud. So what if one needs to see it on the page to understand it? That would be true of many linguexclusive poems, too. Even relatively simple ones. I’ve almost never understood poems I was unfamiliar with when read at poetry readings.
As for the child and candy cane, I like your reasoning, but it now seems to me you have simple shown that “candy cane + child = happiness” and “candy cane + child = obesity” are both incorrect! They should be “candy cane + child = happiness + X” and “candy cane + child = obesity +Y.” And “happiness – obesity + X – Y.”
* * * * * * *
.
By the way, I love this discussion of mathematical poetry. I suddenly wondered, though, if there’s a subject fewer people in the world would be interested in.
One futher note: even if we admitted that difficult math must be seen to be understood, that would not make “candy cane + child – X = happiness” a visual poem since that particular poem would not have to be seen to be understood. That said, I can’t wait for the first mathematical poem based on mathematics you have to see on the page to understand.
–Bob

Kaz wrote:
As far as this Candy Cane analogy goes. I think that in both cases multiplication works better than addition. That said, I would imagine that people would relate to the following best.

Candy cane + childhood = happiness

Candy Cane x childhood = obesity

I am going to ignore the two equations above and rewrite them as multiplication problems with coefficients. The bottom-line is asking what numerical values you assign to these variables or words:

1(Candy Cane) multiplied by 100000(Childhood) equals 1(happiness)

Yet,

1000(Candy Cane) multiplied by 1(Childhood) equal 1(Obesity)
Kaz wrote:
Bob said, “Text printed standardly is effectively not visual, as far as I'm concerned: it's symbolic”

Gee Bob, if symbols are not visual then what are they? … verbal descriptions of symbols are just that ‘descriptions’ they are not the symbol.

Here you make an excellent point that language is just as difficult to understand when listened to as large mathematical equations Thus making a stronger case that pure mathematical poetry is not visual poetry or possibly making the case that all poetry is visual:

“On further thought, it seems to me all mathematics can be read out loud. So what if one needs to see it on the page to understand it? That would be true of many linguexclusive poems, too. Even relatively simple ones. I've almost never understood poems I was unfamiliar with when read at poetry readings.”

Instead of the definition of Visual poetry being – Poetry that has to be seen then state it as such: “Visual poetry is poetry that cannot be verbalized.”

Kaz wrote:
Bob said on his blog:
This is, I believe, the first time I’ve accepted that the operations are metaphorical, as Gregory St. Thomasino tried to convince me six months or so ago. My trouble (still) is that the operations seem actual to me–the sun really does multiply a field to get flowers!

Kaz said as a comment to Bob’s Blog:
There is a bit of a disconnect here. All mathematics is based in metaphor not just mathematical poetry. The problem Gregory had was that he was trying to delineate mathematical poetry from pure mathematics by claiming that mathematical poetry works by analogy and Pure mathematics doesn’t.
If you read George Lakoff’s book “Where mathematics comes from” then you will come to realize that all mathematics is based in metaphor. Not just mathematical poetry.

Sarah Glaz's Definition

I asked the mathematician and co-editor of "Strange Attractors Poems of Love and Mathematics", Sarah Glaz for her definition of mathematical poetry and here is what she had to say:

Mathematical poetry is an umbrella term for poetry with a strong link to mathematics in either imagery, content, or structure. The mathematics involved in mathematical poetry does not have to be mathematically significant. Some poems I would call mathematical involve just arithmetic, or counting. How significant are those in the scheme the entire body of mathematical knowledge? Certain mathematical components do not make a poem mathematical, and this is expressed through the words "strong link to mathematics." For example, all formal poetry has a built in mathematical structure, but we would not call every sonnet, for example, a mathematical poem just because it has 14 lines. If the link to mathematics is in the poem's structure, there has to be something non standard, or unusual, about the use of mathematics in the poem's structure to make the poem a mathematical poem. I left, on purpose, the term "poetry" undefined because I want to include in this definition poems that have only mathematical symbols. Although my preference is for poetry that includes words, I would like the term mathematical poetry to embrace all poetic mathematical forms, even those that come to us from the depth of mathematical silence in symbol form.

What is Mathematical Poetry?

Lately, there has been a bit of passionate yet conflicting talk debating the definition of Mathematical Poetry among those who care. I will present six definitions. You pick what you like best or come up with your own.

Here is Bob Grumman’s:
A mathematical poem is a poem some or all of whose verbal elements undergo a mathematical operation centrally important to the poem that is simultaneously both significantly mathematical and significantly verbal–in the opinion of those capable of appreciating the poem.

Here is Karl Kempton’s:
A visual poem must contain a visual element consciously composed so that the poem must be seen to fully grasp meaning and experience, a mathematical poem must contain a mathematical operation, such as a addition, to fully grasp meaning and experience. a mathematical poem can or not be a visual poem.

Here is Gregory Vincent St. Thomasino’s ‘working’ definition:
The “mathematical poem,” if it is to be, or to contain, poetry, must have some poetic elements, as well as some formal symbols and operations of math.
I want to emphasize that by “operations of math” I do not mean that the poem will be “doing math.” What I mean is that the poem will be, in some way or in some sense — be that metaphorical, allegorical, but for the most part figurative — mimicking or imitating or finding a trope in that operation (whichever that operation may be). (I emphasize: I do not mean that the poem is “doing math.” Math does math. The poem is representational.)

Here is Kaz Maslanka’s: Mathematical Poetry is a umbrella term that covers any poetic expression involving Mathematics. Maslanka has broken mathematical poetry into five categories – they can be viewed here

Here is Sarah Glaz's: Mathematical poetry is an umbrella term for poetry with a strong link to mathematics in either imagery, content, or structure. -click here for more-

Here is JoAnne Growney's: Years ago when I first began to bring poetry into my mathematics classrooms, I used the term “mathematical poetry” to refer to poems in which some of the imagery involves mathematics; it was a sort of “applied mathematics.” Now, after lots of reading and exploring, the possibilities for math-related poetry seem nearly endless--including shaped poems, functional poems, permutation poems, various Oulipian structures, and then--on the Internet--a myriad of possibilities including animated poems, interactive poems (including linked hypertext), and so on. These days, I mostly avoid the term “mathematical poetry” (since I can’t formulate a definition that satisfies me). Instead, I think of the multiple possibilities as intersections of mathematics and poetry. (See, for example my blog: “Intersections -– Poetry with Mathematics.”)

Bob Grumman expresses what he calls True Mathematical Poetry

The following is a comment by Bob Grumman on my delineations for four types of mathematical poetry; with his vote that the only real kind of mathematical poetry is what I call Equational Poetry. I think his argument is pretty good so I am posting here so that everyone can see it.
I also agree with his assessment about number poetry … to me the beauty of number poetry IS the beauty of mathematics. Sure it has rhythm in it but so does virtually everything that the mind can remember due to memory's existence being dependent of repetition. And sure it can be visualized but does visualizing something make it art? Once again I will express that I think art is an expression of culture and ‘Pure Mathematics’ is cultureless.


VizPo-Central has left a new comment on your post "Four Types of Mathematical Poetry":

Number poetry gives one an appreciation of pure math but doesn't seem to me to be poetry. Appreciation of it takes place in one's mathematical awareness only, it seems to me.

The more I think about it, the less I know what to call it. It's not visimagery (i.e., visual art). I guess I would call it number art--it's numbers arranged in order to elicit mathematical pleasure. It's not a kind of mathematical poetry, but an equal art.

As for who "dominates" the term, "mathematical poetry," I say let there be competition; let all who want to define it have their say, and hope that reason prevails. What usually happens in picking terms for kinds of art is what has happened with the term, "postmodernism." A catchy worthless term is coined, probably by an ignorant academic, and someone even more ignorant but with a lot of readers makes it fashionable, and the morons run with it before people of intelligence have had a chance to analyze it and perhaps find a better term.

I will admit that my definition of mathematical poetry fits the kind of math-related poems I compose. So what? What matters is not whether my self-interest is involved, but whether the definition is effective or not.

Aside from what I'm calling "number art," it seems to me there are three kinds of math-related poetry: poetry that is about math, poetry that is generated by some kind of mathematical formula (like make a poem out of every third word in a given dictionary, and poetry in which some mathematical operation is aesthetically central.

I don't think poetry about math should be considered poetry because, to make it simple: poetry about chemistry is not called "chemical poetry," poetry about Bach would not be called "musical poetry," poetry about Picasso's paintings would not be called "visual poetry," poetry about Maria Tallchief would not be called "choreographical poetry," and so forth.

Similarly, mathematically-generated poetry (like sonnets, which are generated in part by the rule that they be ten by fourteen unit rectangles, or that kind of poem each of whose lines has a number of words in it equal to the sum of the number of words in the preceding two lines, or whatever it is) are no more mathematical poems than a bridge of building is a mathematical bridge or mathematical building because generated in part by mathematics. The end product is not mathematical.

Sorry about the slip up regarding "mathematical visual poetry," and I do see the difference. I wasn't able to type my post and read your entry at the same time, and forgot your designation. Anyway, my opinion remains the same: a visual poem that has mathematical symbols in it that don't carry out any mathematical operations is simply a visual poem with mathematical content.

I agree that my long division poems are equational. But some of my other math-related poems are just terms, Like one that is just an ampersand with an exponent of three. "Andness" multiplied by itself twice. I suppose you could call it an equation, half of which is implied.

Yes, I'm sure our little controversies will disappear into some void or other--"exiled history" sounds okay. Better than "non-history."

all best, Bob

Five Types of Mathematical Poetry

These are some delineations of "types" of mathematical poems that I have constructed from my experiences through my survey of mathematical poetry and mathematical poets. While it is true that I am writing these delineations they are not necessarily based on my personal beliefs they are based on what I have gathered from others who claim to be mathematical poets. Personally I have problems with some of these ideas and I may or may not address my objections later. However, I think it is important to draw some lines in the sand for discussion. Obviously these lines may move through further discussion and I can imagine that this page will be edited in the future.

I might add that numerous mathematical poems that I have experienced have facets or elements that extend into more than one of these types. In other words, very few "Mathematical Poems" can be described by just one category.

They are:
1.)“Mathematics Poetry”
2.)“Mathematical Visual Poetry”
3.)“Equational Poetry”
4.)“Visual Mathematical Poetry”
5.)“Pure Maths Poetry” which encompasses ”Number Poetry”


1.)‘Mathematics Poems’ are lexical poems that are influenced by the field of mathematics - There are many examples of these on the internet. This type of poem is the most lexical yet the least like “Pure Mathematics” in the sense of performing mathematical operations on the elements in the visual field. JoAnne Growney seems to be the biggest supporter of these types of poems on found on the internet.
Here is her blog

2.)“Mathematical Visual poetry” uses words and images mixed with/and/or mathematical symbols into a visual field. The mathematical symbols may or may not follow the rules for the formal language of mathematics. This type is much more open and encompasses everything between visual poetry and equational poetry. Because of the wide range of intent it is difficult to place a work on a scale between lexical poetry and pure mathematics However, I believe that if it is more toward visual poetry then it is less like pure mathematics and if it is more like equational poetry then it functions closer to pure mathematics. Or should I say it follows the rules of pure mathematics. Examples of mathematical visual poetry would be in the body of work from Karl Kempton, Scott Helmes, Pi.O. and Bob Grumman

3.)“Equational Poetry” is more rigid than “Mathematical Visual Poetry” in its use of mathematical elements. The rules of mathematics are explicitly used within the structure of the mathematical poem. The explicit use of mathematical rules is what separates “Equational Poetry” from “Mathematical Visual Poetry”. Within the equations words serve as metaphors as well as nested metaphors (metaphors inside metaphors) An example of this type of work would be the mathematical poems at this link. Also Bob Grumman approaches his work with elements of equational poetry and I must also mention the work of Craig Damrauer, which also falls into this catagory. If there are no words in the equation then it is not equational poetry

4.)“Visual Mathematical Poetry” follows the rules of mathematics the same as ‘Equational Poetry’ however the terms for the mathematical poem are purely visual as opposed to textual. In other words the metaphors are visual as opposed to lexical, yet, in essence they function mechanically the same.

5.) “Pure Maths Poetry” is the viewpoint that pure mathematical statements are a poetic expression. What
separates Pure Maths Poetry from the other types is that there are no words/lexical statement. Relative to all types of mathematical poetry “Pure Math Poetry” is the least like “Lexical Poetry.”
Number Poem is a visual formation of numbers who have a verifiable mathematical relationship to each other. The main poetic element in number poems is rhythm or pattern and can be seen by repetitions of certain numbers or operations. Number poems function correctly only when the rules of mathematics are observed. Richard Kostelanetz work from the 1970’s serves as one example of these poems yet, Magic squares and Yang Hui’s Triangle would be examples that are hundreds of years old. Number poems Number Poetry would be a subset of pure math poetry.
Toni Prat also does number poems, however, where Koselanetz focused on mathematical beauty, Prat focuses on paradox which some say is the crux of mathematical metaphor.