torsdag 13 februari 2014

Zeno's Arrow Paradox Still Unresolved after 2500 Years

Russell's "resolution" of Zeno's Arrow Paradox says that an arrow with position $f(t)$ changing continuously with time $t$, is moving. This not a true resolution but only an empty tautology.

Zeno's Arrow Paradox has haunted physicists and philosophers of physics since its formulation by Zeno around 450 BC:
  • How can an arrow be moving in space from one point to another,  if at each instant of time the arrow is still and is not moving?
In Stanford Encyclopedia of Philosophy the paradox is described as follows:
  • The third is … that the flying arrow is at rest, which result follows from the assumption that time is composed of moments … . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (Aristotle Physics, 239b.30)
  • Zeno abolishes motion, saying “What is in motion moves neither in the place it is nor in one in which it is not”. (Diogenes Laertius Lives of Famous Philosophers, ix.72)
  • This argument against motion explicitly turns on a particular kind of assumption of plurality: that time is composed of moments (or ‘nows’) and nothing else. Consider an arrow, apparently in motion, at any instant. First, Zeno assumes that it travels no distance during that moment—‘it occupies an equal space’ for the whole instant. But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, Zeno concludes, the arrow cannot be moving.
Zeno was so clever in formulating his arrow paradox that it has resisted convincing resolution into our days. The commonly accepted "resolution" is the "at-at" theory put forward by Bertrand Russell, which simply says that the motion of an arrow appearing at different positions at different times can be described by a continuous function $f(t)$ where $f(t)$ is the position at time $t$. With a non-constant continuous function $f(t)$ of the real variable $t$, the arrow is thus changing position with increasing time and thus is moving. Russell writes in The Principles of Mathematics in the chapter on motion
  • Motion consists merely in the occupation of different places at different times, subject to continuity as explained in Part V. There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number which is the limit of a certain set of quotients. The rejection of velocity and acceleration as physical facts (i.e. as properties belonging at each instant to a moving point, and not merely real numbers expressing limits of certain ratios) involves, as we shall see, some difficulties in the statement of the laws of motion; but the reform introduced by Weierstrass in the infinitesimal calculus has rendered this rejection imperative. 
The recent survey Can Continuos Motion be an Illusion? by Shan Gao (2013) points to the hollowness of Russell's view:
  • According to the “at-at” theory, it is fallacious to conclude from the fact that the arrow does not travel any distance in an instant that it is at rest. Motion has nothing at all to do with what happens during instants; it has instead to do with what happens between instants. In short, motion is merely being in different locations at different times, and that is that. If an object has the same location at the instants immediately neighboring, then we say it is at rest; otherwise it is in motion. Therefore, since the arrow in flight has different positions at different instants, it is surely moving. 
  • The “at-at” theory is a static theory of motion. In Henri Bergson’s cynical words, “movement is composed of immobilities.” (Bergson 1911, p.308) Continuous motion is simply the occupation, by an object, of a continuous series of places at a continuous series of times. There are no states of motion at an instant, and no instantaneous properties indicate that an object is moving or not. 
We understand that the "at-at" theory is no theory, just a tautological play with words: To say that an arrow is moving because its position is changing with time is simply a truism or tautology and thus is empty of physical content. To disguise the emptiness, a reference is often made to Calculus and the definition of a continuous function through the concept of limit, but the physics is still lacking. Another form of hand-waving is to say the motion of an arrow is explained by the theory of relativity.

The net result is that Zeno's arrow paradox still after 2500 year lacks a real physical resolution. An attempt to such resolution was made in the previous post.

1 kommentar:

  1. I agree, the at-at theory is just a statement of what happens, not an explanation. Just like Newton's first law.

    To resolve it first you need to accept that distance is composed of tiny indivisible unit lengths and not continuous as mathematics and physics would have us believe.

    Then it can be resolved because the object is at one unit, disappears and appears at another unit without moving continuously at all. Like a line of light bulbs flashing in sequence can simulate motion, that is what motion is. There is no motion as we think of it.

    So a fast moving object will "jump" over more units than a slow moving object per time step which is also discrete.

    This involves calculation. Each object must carry its speed and direction with it and a calculation must be done each time step to find the next location unit.

    The most obvious conclusion and answer to this paradox is that we are living in a computer simulation which calculates object paths. Dumb objects cannot calculate their own straight line path and the continuous motion idea also fails to explain the passing of velocity from past to present and on to the future.