A poem, a contradiction . . .

     One strategy for proving a mathematical theorem is a "proof by contradiction."  In such a proof one begins by supposing the opposite of what is to be proved -- and then reasons logically to obtain a statement that contradicts a known truth. This contradiction verifies that our opposite-assumption was wrong and that our original statement-to-be-proved is indeed correct.   (An easily-read introduction to "proof-by-contradiction" is given here.)
       Peggy Shumaker is an Alaskan poet whom I had the pleasure of meeting at a reading at Bloomsburg University where I was a math professor a few years ago.  Her poem, "What to Count On," below, has a beautiful surprise after a sequence of negations -- and reminds me of the structure of a proof-by-contradiction.

What to Count On     by Peggy Shumaker

Not one star, not even the half moon       
       on the night you were born
Not the flash of salmon
       nor ridges on blue snow 
Not the flicker of raven’s
       never-still eye 

Artificial Intelligence in the Library . . .

     Libraries are wonderful places and library book sales are temptations impossible to resist -- and so, during a recent trip to Boston and exploration of the historic public library buildings on Boylston Street, I purchased a copy of Living Proof  (Florida International University Press, 1985) by Edmund Skellings (1932-2012).  Born in Boston and a poet laureate of Florida, Skellings was a pioneer in the application of computers to the arts and humanities.  The word "proof" in his title was enough to make me pick up the book and I have relished the opportunity to turn up memories of a long ago graduate course in AI while reading this poem:

Artificial Intelligence     by Edmund Skellings

Euclid rolled over in his bones
When Newell & Simon instructed
Their machine to look for new proof
For bisecting the ordinary triangle.   

Math fun with song lyrics

Song-writer Bill Calhoun is a faculty member in the Department of Mathematics, Computer Science and Statistics at Pennsylvania's Bloomsburg University (where I also hung out for many years). He belongs, along with colleagues Erik Wynters and Kevin Ferland, to a band called "The Derivatives."  And Bill has granted permission for me to include several of his math lyrics (parodies) here. (In this previous post, we consider the connection between song parodies and mathematical isomorphism.)  My first Calhoun selection deals with difficult mathematical questions concerning classification of infinite sets and decidability.  Following that, later lyrics consider proving theorems and finding derivatives.

Questions You Can’t Ever Decide*      by Bill Calhoun

(These lyrics match the tune of  "Lucy in the Sky with Diamonds" by Lennon and McCartney.)

Picture yourself in  a world filled with numbers,
But the numbers are really just words in disguise.
Gödel says “How can you prove you’re consistent,
If you can’t tell that this is a lie?”    

Poet as mathematician

     Lillian Morrison (1917-2014) was a NYC poet and librarian whose work I first met in the poetry-with-math anthology, Against Infinity.  Here is one of her poems from that collection.

       Poet as Mathematician    by Lillian Morrison

       Having perceived the connexions, he seeks
       the proof, the clean revelation in its

       simplest form, never doubting that somewhere
       waiting in the chaos, is the unique

       elegance, the precise, airy structure,
       defined, swift-lined, and indestructible.

Morrison's insightful poem disappoints me in one important way:  her mathematician-poet is "he."  Another Morrison poem, "The Locus of a Point," may be found in my posting for 15 September 2014.

Prove It

After observing that

               1  =  1
and         1 + 3  =  4
and         1 + 3 + 5  =  9
and         1 + 3 + 5 + 7  =  16
and         1 + 3 + 5 + 7 + 9  =  25

it seems easy to conclude that, for any positive integer n, the sum of the first n odd integers is n².

Excitement of Proving a Theorem

Wow!  From first sighting, I have loved this description:

       I prove a theorem and the house expands:
       the windows jerk free to hover near the ceiling,
       the ceiling floats away with a sigh.

These lines from "Geometry" by Rita Dove express -- as well as any string of twenty-four words I can think of -- the excitement experienced from proving a theorem.

Split This Rock 2014

     Plan now to attend the 4th national biennial Split This Rock Poetry Festival: Poems of Provocation & Witness in Washington, DC, March 27-30, 2014.  The sixteen poets to be featured at the 2014 festival are:  Sheila Black, Franny Choi, Eduardo C. Corral, Gayle Danley, Natalie Diaz, Joy Harjo, Maria Melendez Kelson, Yusef Komunyakaa, Dunya Mikhail, Shailja Patel, Wang Ping, Claudia Rankine, Tim Seibles, Myra Sklarew, Danez Smith, and Anne Waldman.   The website SplitThisRock.org offers photographs and more information about the festival.  It will be awesome!  

Consider Pascal

     Mathematician Blaise Pascal (France, 1623-1662) is known for his explorations with computing machines, for his ideas concerning probabilities, for trying to make rational a decision to believe in God and eternal life, for his explorations of the cycloid and the limacon (curves generated by rolling circles) and a host of other topics.
     I was introduced to Melbourne poet, novelist, and mathematician (he teaches at Victoria University of Technology), Tom Petsinis by South-African editor of Poetry-International, Liesl Jobson.  Here from Petsinis' collection, Naming the Number (Penguin, 1998) is "Pascal's Tooth," (a poem also available at the Poetry-International site).  In the grip of severe pain, Petsinis ponders the ideas of Pascal.  

What is not possible?

     It is impossible for a number to be greater than 2 if it is not greater than 1.  It is impossible to find a rational number whose square is 2.   Up to now it has not been possible to show that π is a normal number.  Mathematicians like the challenge of the impossible.  To challenge, to prove, to refute.
     In the poem below Chelsea Martin devises an entertaining web of circular reasoning to explore the impossibility of eating at MacDonald's.

McDonalds Is Impossible       by Chelsea Martin

Eating food from McDonald's is mathematically impossible.
Because before you can eat it, you have to order it.
And before you can order it, you have to decide what you want.
And before you can decide what you want, you have to read the menu.
And before you can read the menu, you have to be in front of the menu.
And before you can be in front of the menu, you have to wait in line.
And before you can wait in line, you have to drive to the restaurant.
And before you can drive to the restaurant, you have to get in your car.
And before you can get in your car, you have to put clothes on. 

Ode to Alan Turing

In this week of announcements in the US about evolving views concerning human sexual preferences, it seems fit to offer a second poem (see also May 9) honoring British code-breaker and computer scientist, Alan Turing (1912-1954).  Here is "Ode to Alan Turing"  by Saskatchewan poet, Mari-Lou Rowley. 

Universal and Particular

Poet Yves Bonnefoy (b 1923) is one of France's greatest living poets. And Bonnefoy's university studies included mathematics. I read recently of Bonnefoy in the Wall Street Journal Bookshelf posting for 11 February 2012 by Micah Mattix entitled "The Pursuit of Presence." This reminder sent me to my bookshelf to review the poet's work with mathematics in mind. I found a bit of attitude toward the subject in a prose poem entitled "Devotion" when he used the phrase "stern mathematics." And Section 1 of "Trial by Ordeal" (offered below) ends with the word "proof."
     Mattix opened his Bonnefoy article with a quote: If I had to sum up in a sentence the impression Shakespeare makes upon me," the poet Yves Bonnefoy wrote in an early essay, "I should say that, in his work, I see no opposition between the universal and the particular." This universal-particular pairing (evident in Bonnefoy, as in Shakespeare) led my thoughts to the mathematical pairing, global-local, which I explore briefly following Bonnefoy's poem.* 

A mathematical woman

As in an earlier posting (20 December 2011), today's feature includes verse by Lord Byron (1788-1824). This time the source is Byron's satiric poem Don Juan. In Canto I, the poet describes Don Juan's mother, Donna Inez, as learned and "mathematical." Here are several stanzas about her -- sagely seasoned with words like "theorem," "proof," and "calculation."

Mathematical Induction -- principle, perhaps poem

One of my teachers -- I think it was Mr Smith in "College Algebra" during my freshman year at Westminster -- gave me these words to remember:

     When confronted
     with a statement
     that seems true
     for all positive integers
     the wise student
     uses mathematical induction
     as her proof technique. 

Mathematicians at work

     About her collecton, The Scottish Café (Slapering Hol Press, 2002), Susan Case offers this note:
     This series of poems is loosely based upon the experiences of the mathematicians of the Scottish Café, who lived and worked in Lvov, Poland (now L'viv, Ukraine), a center of Eastern European intellectual life before World War II, close to the area from which my own ancestors emigrated to the United States.  A book, known as the Scottish Book, was kept in the Café and used to write down some of their problems and solutions.  Whoever offered a proof might be awarded a prize.
     Here is "Fixed Points," the opening poem from Case's collection:

Theorem-proof / Cut-up / poems

     For mathematicians, reading a well-crafted proof that turns toward its conclusion with elegance and perhaps surprise -- this mirrors an encounter with poetry.  But can one have that poetry-math experience without being fluent in the language of mathematics?  Below I offer a proof (a version of Euclid's proof of the infinitude of primes) and a "cut-up" produced from that proof-- and I invite readers (both mathematical and non-mathematical) to consider them as poems.