Go Figure / Uncategorized

GO FIGURE: Too Little Margin for Proof

by Susan Sechrist

“Go Figure” is a regular feature at Bloom that highlights and celebrates the interdependence and integration of math and literature, and that will “chip away at the cult of youth that surrounds mathematical and scientific thinking.”  Read the inaugural feature here.

1.

One of the first disagreements I had with my husband involved books. It wasn’t about favorite authors, or the deficiencies of certain genres, or the nature of truth in creative non-fiction. It was about page margins.

I am an unapologetic marginaliac, a glossing fool.

I write in my books. I circle words and underline sentences; I put stars around paragraphs of particular meaning; I write my own words about other people’s words, usually in pencil (I’m not a barbarian).

My husband finds this behavior sacrilegious, almost as bad as how I manhandle a book and crack its spine like so many knuckles. To me, a pristine book is an unread book, as undigested as an untouched meal: cold, congealed, wasted. My favorite books are not only written in, their pages are dog-eared and ornamented with sticky tags and post-it notes that hang over the edge and make them impossible to shelve. (My electronic books are also annotated, though I find the digital process less satisfying.)

Those margins are a sacred, liminal space; an invitation to converse with the author, the characters, and the concept of the book. Marginalia excavate the secret passageways that exist in every text, and are not just about adding words, they’re also about adding space—about using your thought as a lever under the author’s thought, to turn it over, change its position. Like how you stoke the last life of a fire by grabbing the poker and lifting that last smoldering log, adding just enough oxygen to ignite the charcoaled surface with new flame.

 

2.

Tom Stoppard’s play Arcadia features one of the most famous marginal ideas in history: mathematician Pierre de Fermat’s Last Theorem. Fermat’s idea is based on the Pythagorean Theorem, a familiar equation to most of us which describes the relationship between the three sides of a right triangle: a2+ b2= c2.

Fermat never published a proof of his theorem, but rather left a tantalizing note in Latin on the edge of a page of Greek mathematician DiophantusArithmetica: “I have discovered a truly marvelous proof… which this margin is too narrow to contain.”

In Arcadia’s first scene, Septimus, like any good tutor, uses Fermat’s Last Theorem to distract his student, the young Thomasina Coverly, from her interest in sex. When Thomasina asks him to explain the rumored “carnal embrace” between the visiting Mrs. Chater and a man not her husband, Septimus obliges, but challenges Thomasina to think instead about Fermat:

SEPTIMUS: Carnal embrace is sexual congress, which is the insertion of the male genital organ into the female genital organ for purposes of procreation and pleasure. Fermat’s last theorem, by contrast, asserts that when x, y, and z are whole numbers each raised to power of n, the sum of the first two can never equal the third when n is greater than 2.

THOMASINA: It is disgusting and incomprehensible. Now when I am grown to practice it myself I shall never do so without thinking of you.

My marginal note next to Thomasina’s reply says: “Sex or Fermat?”

Stoppard’s magnificent play features many of these kinds of clever balancing acts: between classicism and romanticism, order and disorder, science and poetry, but they are never with commensurate opposites: his pairings create relationships that are slightly off-kilter and out of alignment. It’s this dis-equilibrium that Thomasina senses as she contemplates why she cannot unstir the jam from her pudding; that and a notably precocious suspicion that neither God’s mysterious ways nor Newton’s determinism is an accurate description of the universe:

THOMASINA: …God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers?

SEPTIMUS: We do.

THOMASINA: Then why do your equations only describe the shapes of manufacture?

SEPTIMUS: I do not know.

THOMASINA: Armed thus, God could only make a cabinet.

SEPTIMUS: He has mastery of equations which lead into infinities where we cannot follow.

THOMASINA: What a faint-heart! We must work outward from the middle of the maze. We will start with something simple… I will plot this leaf and deduce its equation….

Thomasina describes her own mathematical breakthrough, like Fermat, in the margin of Septimus’ primer:

THOMASINA: ‘I, Thomasina Coverly, have found a truly wonderful method whereby all the forms of nature must give up their numerical secrets and draw themselves through number alone. This margin being too mean for my purpose, the reader must look elsewhere for the New Geometry of Irregular Forms discovered by Thomasina Coverly.’

Her discovery is even more brilliant than she could envision. Valentine, the play’s present-day mathematician, recognizes that Thomasina has created an iterated algorithm—a mathematical engine where the output of the algorithm is fed back in as input, creating a complex feedback loop that not only draws Thomasina’s apple leaf, but ultimately explains some of the behavior of many naturally unpredictable systems, like weather or economies or biological populations:

VALENTINE: …The unpredictable and the predetermined unfold together to make everything the way it is. It’s how nature creates itself, on every scale, the snowflake and the snowstorm. It makes me so happy. To be at the beginning again, knowing almost nothing … The ordinary-sized stuff, which is our lives, the things people write poetry about – clouds, daffodils, waterfalls, and what happens in a cup of coffee when the cream goes in – these things are full of mystery….

Valentine is dumbfounded that Thomasina was able to see such complexity without the benefit of a computer to run the numbers. She has found the heart of 21stcentury chaos theory, the mathematics of how natural order can arise from disorder.

Like Thomasina’s algorithm, Stoppard’s play works like an iterative engine: output is fed back into the system as input and so on, creating a kind of literary feedback loop. The present-day characters consume the clues left behind by those in the past, while the characters in the past provide a prescient, but not completely predictable, view into the future. The closer I looked at the play’s intertwined characters and plot, the more replication and recursion I saw: the play itself unfolds and expands into more detail with every zoom in, like a coastline that gets longer or a flat surface that reveals unseen crests and troughs with magnification.

 

3.

Manil Suri, a mathematician at the University of Maryland Baltimore County (UMBC), award-winning novelist, and a bloomer (he published his debut, The Death Of Vishnu, when he was 42), produced a video lecture on the mathematics of Arcadia. He describes one particular chaos theory concept, the Sierpinski triangle, that mirrors mathematically what Stoppard is doing with his narrative engine:

  1. Start with an equilateral triangle.
  2. Subdivide it into four smaller equilateral triangles and remove the central triangle.
  3. Repeat step 2 with each of the remaining smaller triangles forever.

The resulting form looks like a lovely, symmetrical lattice composed of differently-sized white and black triangles. Of course, if you truly do repeat step 2 forever, you get an object that is made up of mostly empty space. I think this is similar to how Stoppard makes his play seem frictionless and perpetually productive—it is more margin than text.

 

4.

My copy of Arcadia is riddled with fractured notes and marginal ravings, even a few sketches and doodles and equations. In those glosses, I see bits of Stoppard’s characters—obnoxious Bernard and earnest Valentine and genius Thomasina.  It’s like a weird kind of fan fiction. There is so much to explore, I feel a bit like the heartbroken Septimus, left behind with Thomasina’s brilliant idea and an insufficient forest of paper and pencils. There is too little margin of time, of space, of focus, of energy to understand and encapsulate and connect all of it.

Like Thomasina’s algorithm, the play “eats its own” in a glorious feast of inputs and unexpected outputs powered by infinitesimal tweaks–a misunderstanding or missed opportunity between characters; a prop that persists inexplicably between scenes across time; the elliptical plot centered simultaneously on proof and lack of proof, or on importance and triviality; and the final, impossibly four-dimensional waltz made visceral. Physically writing in the margins of this book was my effort to frame the organic, infinite, unprovable purpose of Stoppard’s ideas. When I turned to my computer to write this piece—to connect my glosses to the core text and confer some kind of emergent property upon the complexity of the whole system—I felt like Valentine trying to write one equation to fit a deluge of data. I revisited the writing iteratively, creating several possible drafts, all with different populations of ideas and words, all with different fates. Some died, some thrived. What has made this final analysis is just a crude drawing of Arcadia’s entire coastline.

 

5.

I had many ideas that didn’t make it in to this piece, some intriguing, some silly; the sheer volume of them and the fractals they produced were overwhelming, thanks to Stoppard’s wit and ingenuity and my blossoming love of the play. In the spirit of glossing and in honor of both Pierre de Fermat and Thomasina Coverly, I leave a few of them here, and claim too little margin for proof, and implore the reader to look elsewhere:

  • Heat death of the universe may be followed by the Poincare Recurrence, the possible but improbable return to order of a disordered system after a very long, but finite period of time.
  • If a simple bluebell can be described with an iterated algorithm, then why not a more complex rose; and if math can describe a rose, maybe math rather than rhetoric becomes the source of metaphor’s power–if indeed, love is a rose?
  • Equilateral edges of proof, primer, and palimpsest.
  • “Mean”– scarce, cruel, or average? Or mediator between concepts?
  • Reality is turtles all the way down (or, one turtle–Plautus).

Bloom Post End

Susan Sechrist is a freelance technical writer who is striving to better integrate her creative and mathematical sides. She published her first short story, the mathematically-themed “A Desirable Middle,” both in Bloom and the Journal for Humanistic Mathematics.

Photo credits

Frost Patterns – By Schnobby [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)%5D, from Wikimedia Commons
Right triangle  – CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=617373
Sierpinski triangle – https://en.wikipedia.org/wiki/Sierpinski_triangle

2 thoughts on “GO FIGURE: Too Little Margin for Proof

  1. Thank you for this intellectually-stimulating piece. Not being mathematically minded, I will have to read it again, perhaps twice, to absorb the wisdom. I feel confident commenting on the 1st segment. I also converse with the author by making margin notes in pencil as I read. When I reread the book later, sometimes I am amazed at my initial comments and revise them. The problem comes every few years when I have to cull my collection because there is no room in the house to move. Before donating the culled books to a fundraiser or the local used-book store I use up the entire kneaded eraser rubbing out my comments because I don’t want to impose them on the next reader.
    Cheers from Vancouver, Canada.
    Elizabeth McLean

    • Elizabeth, thank you for your thoughtful comment! I can relate to your margin experiences as well – after I read China Mieville’s The City & the City my husband expressed an interest in reading it, too; I bought him a new copy because mine was full of notes and spoilers! Your approach gives new meaning to the palimpsest!

      Thank you again,
      Susan

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