Tuesday, September 7, 2010

Against Intuition

One of my favorite poets (mentioned previously for her poem, "Pi" in my September 6 posting) is the Polish Nobelist (1996) Wislawa Szymborska.  Her language is apt and spare, her thoughts are wise, and her gentle humor is frequent. 

One of Szymborska's poems that I like best is "In Praise of My Sister,"  a soft reminder of how sisters differ without being better or worse than one another.  A deeper look into Szymborska's work suggests that she enjoys grappling, as I do, with counter-intuitive ideas--paradoxes, apparent contradictions.  Her poem, "The Three Oddest Words," introduces us to that topic--and, following the poem, several mathematical paradoxes are listed.

     The Three Oddest Words     by Wislawa Szymborska

     When I pronounce the word Future,
     the first syllable already belongs to the past.

     When I pronounce the word Silence,
     I destroy it.

     When I pronounce the word Nothing,
     I make something no nonbeing can hold.

This translation (by Clare Cavanagh and Stanisław Barańczak) is one of five Szymborska translations available (with the original Polish versions nearby) at NobelPrize.org.  Also, many of Szymborska's poems are available at Info-PolandThe websites Wolfram Alpha and Wikipedia are both good sources of information about paradoxes.

Paradoxes inherent in a single sentence include these:
     The set S of all sets must contain S as a member.  (Cantor, 1899)
     This statement is false. (Eubulides, 350 BCE)
     One thing I know is that I know nothing  (Socrates, 430 BCE)
In statistics we find Simpson's Paradox which is illustrated by this example:  in each separate college of a university the rejection rate of men is be higher than that for women, while the overall university rejection rate of women is higher than that for men. 

From German mathemtician David Hilbert (1862-1943) comes the Grand Hotel paradox about infinite sets that is true but counter-intuitive. (In essence, the paradox is that the situations "every room is occupied" and "no more guests can be accommodated" are not equivalent when the hotel has an infinite supply of rooms.) Consider the Grand Hotel which has infinitely many rooms--#1, #2, #3, and so on--each filled with an overnight guest.  A visitor arrives at the hotel asking for a room and the hotel clerk rearranges to make room-for-one-more by moving the guest in #1 to #2 and the guest in #2 to #3, and so on.  And, this process can be repeated when the next guest arrives, and the next.  And still more is possible.

Returning to poetry, here next is a poem by Janet Loxley Lewis (1899-1988) that reminds us that sometimes, in life as in mathematics, apparent paradoxes -- such as flight through the air of objects much more dense than the air -- can, with a proper point of view, be resolved.

   The Hangar at Sunnyvale: 1937       by Janet Loxley Lewis (1899-1998)

   Above the marsh, a hollow monument,
   Ribbed with aluminum, enormous tent
   Sheeted with silver, set to face the gale
   Of the steady wind that filled the clipper sail,
   The hanger stands. With doors now buckled close
   Against the summer wind, the empty house
   Reserves a space shaped to the foundered dream.
   The Macon, lost, moves with the ocean stream.

   Level the marshes, far and low the hills.
   The useless structure, firm on the ample sills,
   Rises incredible to state again:
   Thus massive was the vessel, built in vain.
   For this one purpose the long sides were planned
   To lines like those of downward pouring sand,
   Time-sifting sand; but Time immobile, stayed,
   In substance bound, in these bright walls delayed.

   This housed the shape that plunged through stormy air.
   Empty cocoon! Yet was the vision fair
   That like a firm bright cloud moved from the arch,
   Leaving this roof to try a heavenly march;
   Impermanent, impractical, designed
   To frame a paradox and strongly bind
   The weight, the weightless in a living shape
   To cruise the sky and round the cloudy Cape.

   Less substance than a mathematic dream
   Locked in the hollow keel and webbéd beam!
   Of the ingenious mind the expensive pride,
   The highest hope, the last invention tried!
   And now the silver tent alone remains.
   Slowly the memory of disaster wanes.
   Still in the summer sun the bastions burn
   Until the inordinate dream again return.

From The Selected Poems of Janet Lewis edited by R.L. Barth, (Swallow Press, 2000).

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