This installment concludes the series on Zeno's paradoxes (see Part 1 and Part 2).
The Arrow Paradox
This paradox investigates the motion of an arrow, and prior to the previous two paradoxes, focuses on the divisibility of time as an argument against motion. Again, Aristotle analyzed the validity of Zeno's reasoning in
Physics, VI.9.
The Story
Say an arrow is in flight in midair. As is characteristic from human observation, divide time into a rapid succession of static moments (think of a flipbook, e.g.). Now zoom in on a given moment, representing the present. In this state, the arrow is motionless, traveling no distance. Assuming every point in the past and future passes through this 'present' state, we're left with a series of static arrows, preventing it from
ever being in motion.
The Implications
Motion itself is an illusion, being physically impossible. But of course we have the sensation of traversing distance, leading to yet another paradox.
Proposed Solution
This paradox, like the others, relies on linear assumptions about the time-space continuum, as well as its infinite divisibility. This seems
very Newtonian. Luckily, twentieth century physics seems to have provided us with some tools to address these apparent inconsistencies. Einstein's
general relativity introduced us to the notion of wormholes, where one doesn't necessarily have to travel through every continuous point in space and time. Likewise,
quantum mechanics is able to place a limit on the divisibility of space and time. The so-called
Planck length and
Planck time are defined as 1.62×10
-35 m and 5.39×10
-44 s, respectively. Providing a bound on physical values leads to another limit: the ratio of the Planck-length to Planck-time is the definition of the speed of light in a vacuum, the upper bound for motion.