What is meant by the falconer, falcon and gyre?
The “falconer,” (valkenier) representing humanity’s attempt to control its world, has lost its “falcon” (valk)
in the turning “gyre” (the gyre is an image Yeats uses to symbolize grand, sweeping historical movements
as a kind of spiral). These lines also suggest how the modern world has distanced people from nature (represented here by the falcon) and it’s clear that whatever connection between falcon and falconer has broken, and now the human world is spiralling into chaos.
Indeed, the poem suggests that though humanity might have looked like it was making progress
over the past “twenty centuries” through ever -increasing knowledge and scientific developments, the First World War proved people to be as capable of self-destruction as ever. The “best” people lack “conviction,” they're not bothering to do anything about this nightmarish reality, while the“worst” people seem excited and eager for destruction. The current state of the world, according to the speaker, proves that the "centre" — that is, the foundation of society — was never very strong.
In other words, humanity’s supposed arc of progress has been an illusion.The promises of modern society
—of safety, security, and human dignity — have proven empty. And in their place,a horrific creature has emerged (see second part).