"If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.”
—Aristotle, Physics VI:9, 239b5
In any "instant" nothing is moving, it's already there, inbetwen actions. After the instant it is moving, before the instant it has moved, so in any given instant, it's not a flying arrow from that perspective, it's just a sitting arrow that has moved and will move.
Who says if something occupies an equal space makes something at rest anyway? I suppose that's besides the point.
From another perspective which defines motion as something in the process changing distance, and in-process be defined as needing no additional force in or after that instant to continue changing distance while also being provable through observation or other means that it's moving you can determine that by observing things that are apparently being affected by its movement, such as broken glass around an arrow in mid air. This would be a good way to determine movement, not considering the funny thing two paragraphs below.
In any "instant" nothing is moving according to the popular definition of movement. You can think of an instant as an infinitely small measurement of time, so small in fact that time itself is sitting still. Either that or another popularly assumed, thus viable definition of an instant would be where only one thing in all of existence has change position, even if it is the smallest existing thing that has moved. Time after all, is defined by movement, isn't it?
I think something funny may be that how in the world could you know that any one thing you're observing is not the single most thing at rest in the universe and that all things seeming to move it are merely coincidence and all things seeming to move with it are merely moving in a method that provides an illusion that it is being moved with it. Take a spring grounded on the Earth for example that seems to have pushed a pin from it, but really, the spring has pushed itself and the Earth from the pin. Perhaps the arrow was never fired, though the archer thought it was fired, but coincidentally kept the Earth spinning by pushing it off of the arrow. Perhaps that's impossible :D
Definitions are things deserved of respect, though they belong to language, they are also a math problem, math after all, is a language that should seek its elders... etymology.