To return to Zeno‘s paradoxes, the solution to all of the mentioned paradoxes then, that there isn‘t an instant in time underlying the body‘s motion (if there were, it couldn't be in motion), and as its position is constantly changing no matter how small the time interval, and as such, is at no time determined, it simply doesn't have a determined position.In the case of the Arrow paradox, there isn‘t an instant in time underlying the arrows motion at which it‘s volume would occupy just —one block of space“, and as its position is constantly changing in respect to time as a result, the arrow is never static and motionless.
The paradoxes of Achilles and the Tortoise and the Dichotomy are also resolved through this realisation: when the apparently moving body‘s associated position and time values are fractionally dissected in the paradoxes, an infinite regression can then be mathematically induced, and resultantly, the idea of motion and physical continuity shown to yield contradiction, as such values are not representative of times at which a body is in that specific precise position, but rather, at which it is passing through them. The body‘s relative position is constantly changing in respect to time, so it is never in that position at any time. Indeed, and again, it is the very fact that there isn‘t a static instant in time underlying the motion of a body, and that is doesn‘t have a determined position at any time while in motion, that allows it to be in motion in the first instance.
Moreover, the associated temporal (t) and spatial (d) values (and thus, velocity and the derivation of the rest of physics) are just an imposed static (and in a sense, arbitrary) backdrop, of which in the case of motion, a body passes by or through while in motion, but has no inherent and intrinsic relation. For example, a time value of either 1 s or 0.001 s (which indicate the time intervals of 1 and 1.99999….s, and 0.001 and 0.00199999…. s, respectively), is never indicative of a time at which a body‘s position might be determined while in motion, but rather, if measured accurately, is a representation of the interval in time during which the body passesthrough a certain distance interval, say either 1 m or 0.001 m (which indicate the distance intervals of 1 and 1.99999….m, and 0.001 and 0.0019999….m, respectively).
Therefore, the more simple proposed solution mentioned earlier to Achilles and the Tortoise and the Dichotomy by applying velocity to the particular body in motion, also fails as it presupposes that a specific body has precisely determined position at a given time, thus guaranteeing absolute preciseness in theoretical calculations before the fact i.e. ∆d/∆t=v. That is, a body in motion simply doesn‘t have a determined position at any time, as at no time is its position not changing, so it also doesn‘t have a determined velocity at any time.
Lastly, and to complete the mentioned paradoxes, William James‘ variation on the Dichotomy is resolved through the same reasoning and the realisation of the absence of a instant in time at which such an indivisible mathematical time value would theoretically be determined and static at that instant, and not constantly changing. That is, interval as represented by a clock or a watch (as distinct from an absent actual physical progression or flow of time) is constantly increasing, whether or not the time value as indicated by the particular time keeping instrument remains the same for a certain extended period i.e. at no time is a time value anything other than an interval in time and it is never a precise static instant in time as it assumed to be in the paradoxes.
5. Closing Comment
To close, the correct solution to Zeno‘s motion and infinity paradoxes, excluding the Stadium, have been set forward, just less than 2500 years after Zeno originally conceived them. In doing so we have gained insights into the nature of time and physical continuity, classical and quantum mechanics, physical indeterminacy, and turned an assumption which has historically been taken to be a given in physics, determined physical magnitude, including relative position, on its head. From this one might infer that we‘ve been a bit slow on the uptake, considering it has taken us so long to reach these conclusions. I don‘t think this is the case however. Rather that, in respect to an instant in time, it is hardly surprising considering the extreme difficulty of seeing through something that one actually sees and thinks with. Moreover, that with his deceivingly profound and perplexing paradoxes, the Greek philosopher Zeno of Elea was a true visionary, and in a sense, over 2500 years ahead of his time.
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