Zeno’s Paradoxes Resolved: A single particle does not move smoothly across space-time, it  changes its position by λ-hopping. Achilles physical arrow (in the Achiles’ arrow paradox) is more complex than a single particle in that it is an assembly of particles which all jump in a complex process involving the mass distribution of the arrow. But even so the theoretical obstacle to its movement is removed and it also hops over time and space.  Click on the picture above for a very good timeline of Greek philosophers and to see a map of where they lived.

Achilles shoots an arrow. Zeno says the arrow cannot move continuously in space-time. Zeno’s argument holds, calculus has no foothold at all. See Section 9.

b. In the race between Achilles and the tortoise Paradox: If Achilles and the tortoise can overcome the logic that impedes their motion (from “a” above) then  the race can begin.

DWT says Achilles can mark his progress as he approaches the tortoise by noting that he halves the distance and then halves the distance again, but eventually he is within a λ-hop of the tortoise. Here he can calculate the distance to the tortoise as being less than his wavelength, but he cannot move that small a distance because he is forced (via quantum mechanics) to make a full λ-hop. And on his next hop he passes the tortoise with no problems. He is not anchored to space-time by the continuum of points.  It should be noted that Achilles can only approximately half his distance to the tortoise because he is limited to whole wavelength hops.

Calculus says that the infinite sum of times involved with Achilles approach to the tortoise, sum to a finite value. And since Achilles gets to the tortoise in a finite time there is no paradox. This          argument has some merit if you forget about the infinite number of steps it took to get to a finite result. Also note that Achilles still has not passed the tortoise and has only caught up to it. The calculus is interesting but it is also “ugly” and forced in terms of how we see the real world.

The race with the tortoise provides a little wiggle room for calculus, but it is all predicated on both Achilles and the tortoise being able to move, and from Zeno’s logic in the “Achiles Arrow Paradox” this is impossible.

Zeno vs. Newton
Zeno and Newton both took a mathematical look at how motion occurs. Zeno started with an object that was not in motion and showed that motion would be an impossibility for this object because there is no distance to the closest point and no motion is need to get to it.

Newton took a different tac and started with an object in motion. He defined that motion as a velocity v=Δx/Δt. He then made the calculus which logically showed that velocity existed for the object where the object existed. He did this by making the Δx’s and Δt’s smaller and smaller. This result of this calculus was a new velocity for the object v=dx/dt, an instantaneous velocity for the object at the position of the object. And general agreement was soon to follow, an object can have a velocity as a property. Just about everyone agrees Newton defeated Zeno and objects can mathematically have a precise velocity at a point in space-time. There was just a little problem.

Physical measurements showed we could not get a velocity for a particle at a point. Was the elegant math of calculus wrong? No, because Heisenberg saved it by showing that it is impossible to “know” position and velocity simultaneously (see section 3 to see how this is done). With this uncertainty principle a particle has a velocity at a point (calculus is intact), it is just that we cannot know it. Zeno blows a very loud raspberry here.

Digital Wave Theory questions the validity of taking v=Δx/Δt to v=dx/dt.
This derivative operation is elegant mathematics and as mathematics it is completely valid. However, as physics it is screwy. Velocity as v=Δx/Δt is valid as long as the Δx and Δt are not smaller than the deBroglie wavelength and period of the particle being considered.

Newton was incorrect in his deduction that a particle can have a velocity at a point. Particles when they manifest are static (with respect to the observer), they only get motion by “λ-hopping” across their wavelength. And yes, it is a big pill to swallow. Below the level of the particle wavelength we loose calculus as a tool for physics (but we may benefit by not needing renormalization in Feynman’s QED).