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 integrationAnd 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).