Symbolism, Imagery, Allegory
Get out your oxygen tanks, because people can't stop talking about breath in this play.
That's right: the focus on breath is a little apocalyptic in Timon of Athens. Timon kicks it off by declaring: "Breath infect breath, that their society, as their friendship, may be merely poison" (4.1.30-32). He wants everything to be infected—just one infected part of society isn't enough for him. The very breath people inhale should be destructive.
But he doesn't stop there: Timon goes on to tell the courtesans, "And he whose pious breath seeks to convert you—Be strong in whore, allure him, burn him up" (4.3.141-142). Let's break this down. Breath is supposed to be a sign of life; if you look at it one way, it's the very thing giving you life.
Timon wants Athens to be corrupted even at that level. He doesn't just want some people to suffer; he wants the very thing that gives life to be destroyed. That's intense.
You Breathe in Vain
Alcibiades picks up the significance of breath and includes it in his speech outside of Athens. He declares that they
Have wandered with our traversed arms and breathed
Our sufferance vainly. Now the time is flush,
When crouching marrow in the bearer strong
Cries of itself 'No more:' Now breathless wrong. (5.4.5-10)
The imagery Alcibiades creates with this speech makes us think of someone having difficulty breathing. Try as they might, they just can't get those breaths to fill their lungs with oxygen. Alcibiades is saying the same about the city.
It turns out Timon's wish for Athens has come true: everyone is choking on the infectious lifestyle that Athens has produced. This society is corrupt at the level of its breath—translation: it's totally corrupt, all the way down to the core.
But Alcibiades, unlike Timon, wants to restore the breath to the people by giving them pure oxygen—translation: he wants to give them justice. Back when Alcibiades was trying to plead his buddy's case to the Senate, they told him: "You breathe in vain" (3.5.60). At that time, he was helpless, just wasting his breath. Now, he wants the whole city to have a breath of fresh air.
Timon wants to destroy, but Alcibiades wants to foster a rebirth.