Opposites, Balance

     Recently, and perhaps always, opposites have interested me.  For example, the complementary and sometimes  conflicting nuggets of advice contained in "Pinch a penny, waste a pound" and "It is best to prepare for the days of necessity."   And in  "Kindness effects more than severity" and "Spare the rod, spoil the child."   Maybe what I like best is the challenge of synthesizing opposite truths.
     Mathematics contains many pairs of entities that are, each in some different sense, opposites:

2 and -2      2 and 1/2

horizontal and vertical   differentiation and integration

And there are some arbitrary subdivisions that often are treated as if they are disconnected opposites:

pure vs. applied (creating mathematics vs. solving problems)

teaching and learning, creating vs. teaching, arts and sciences

In an ideal world, opposites exist with "Balance" -- which is the title of the following lovely and contemplative poem by Adam Zagajewski :

       Balance    by Adam Zagajewski     translated by Clare Cavanagh

       I watched the arctic landscape from above
       and thought of nothing, lovely nothing.
       I observed white canopies of clouds, vast
       expanses where no wolf tracks could be found.

       I thought about you and about the emptiness
       that can promise one thing only: plenitude—
       and that a certain sort of snowy wasteland
       bursts from a surfeit of happiness.

       As we drew closer to our landing,
       the vulnerable earth emerged among the clouds,
       comic gardens forgotten by their owners,
       pale grass plagued by winter and the wind.

       I put my book down and for an instant felt
       a perfect balance between waking and dreams.
       But when the plane touched concrete, then
       assiduously circled the airport's labryinth,

       I once again knew nothing. The darkness
       of daily wanderings resumed, the day's sweet darkness,
       the darkness of the voice that counts and measures,
       remembers and forgets.

Zagajewski was born in Poland but has lived in many cities and has taught at several US universities.  "Balance" may be found at poets.org and is from his collection Eternal Enemies (Farrar, Straus & Giroux, 2008). More Zagewski poems are online here.
      The theme of opposites herein also connects to earlier considerations of zero as something or nothing (see, for example, my postings for  24 December 2014 and 3 January 2015).