Physics

He and I approached each other along a thoroughfare.

I reasoned thus: "If I continue along my present trajectory, we needs must collide. But if I swerve (to right or left, it matters not which), he'll swerve too (to left or right). He's human like me, our fate is symmetry. Calculations of the rational animal... A gesture of genius is required, the genius of simplicity. To show for once and for all who I am and who he is not... Let me stride boldly forward!"

He reasoned thus: "If I continue along my present trajectory, we needs must collide. But if I swerve (to left or right, it matters not which), he'll swerve too (to right or left). He's human like me, our fate is symmetry. Calculations of the rational animal... A gesture of genius is required, the genius of simplicity. To show for once and for all who I am and who he is not... Let me stride boldly forward!"

And so we walked right into each other. We walked right through each other. When we emerged, everything has changed. I was now he, he was now I...

Quite simple, really.