Wierstrass and Zeno (the Eleatic) Come to Haunt Us

[BaalChatzaf]

1 Zeno, the Eleatic, posed conundrums or paradoxes concerning motion. It was Zeno's contention that motion as given to us by our senses and the associated conceptualization and intuition is paradoxical. He left eight paradoxes of one I will cite one, the Paradox of the Arrow. It goes like this: At every instant of time an arrow is in a definite place. Therefore the arrow is at rest at every instant and it is impossible that it can move. How can something that is at rest at all times move?

2. The concept of motion in classical physics is that a point object is at a position which is a function of a time and time is a real variable which varies over a compact set (no holes and containing all of its limit points). The "instantaneous" velocity of a point object is determined by seeing it position at different times. Let p(t) be the position of the object. To determine its velocity at time t we construct a sequence of time value t1, t2, .... where the sequence tn concerges to t as n ->oo and tn < t for all n. Construct the ratios

(p(t) - p(tn))/(t - tn) and take the limit of the ratios as n -> oo. This is a sequence of velocities of the point object over the intervals (tn, t). The limit of the velocities so defined is the instantaneous velocity of the point object at t. When this definition was proposed it was assumed that every continuous function had derivatives almost everywhere (it could lack derivatives and jump on at most a discrete set of values).

3. In 1872, Wierstrass construction a function that was continous over an interval but had a derivative nowhere in the interval. There is nothing in the laws of physics that prevents us from representing the motion of a point object by such a function. See

http://planetmath.org/encyclopedia/WeierstrassFunction.html

for the definition of such a function. A point with such a motion has velocity nowhere, yet it is supposed to move. It the point is a mass point it is in motion but has no kinetic energy (lacking a velocity anywhere). This is a counterexample to the idea that we can associate an instantaneous velocity to a body almost everywhere on a time interval.

3. What is going on? What is going on is that our visual apparatus has fooled us into believing that infinitesimal time intervals are possible and that the motion of an object is describable by a continuous function. In point of fact our visual apparatus will produce the impression of continuous motion even where there is no such motion. Example: motion pictures which project a sequence of still images at intervals less than 1/30 th of a second. We will see the sequence as continuous motion. Like with t.v. pictures which are traced out by a moving lit up pixel sweeping the screen once every 1/30th of a second. And the pixel jumps. If our senses were square with us we would not see a t.v. image of a moving object. What we would see is what is there, to wit, a jittery jumpy lit up dot jumping back and forth from the top of the screen to the bottom. In short, motion is an optical illusion in this case.

How about "real" motion. If a body jittered at a sufficiently high rate of jumpiness we would see it as a continuous motion.

4. Our visual field is also an optical illusion. We have a "blind spot" near the fovea where the optic nerve penetrates the retina. Why don't see the blank area (blind spot). Because our eyes continuously move back and force, up and down and the "smeared image" is averaged out by the visual cortex. So our apparent visual field is compact and without holes or gaps. Likewise motion was we see it, is created by a combination of averaging and persistence of vision (which is why movies move). Not only that but our retinas are not continuous. They are pixilated with rod and cone receptors at discrete positions. Yet we perceive or visual field as continuous. This is analogous to printed photos which are dense configuration of dots of various colors giving us the illusion of continuous extension.

So it would appear Zeno is right. Motion as we perceive it and conceive it (i.e. a continuous function from point instants of time to specific positions in space) is a construct, not the real thing. Also the idea of an instant of time (0 length time interval) is a construct rather than a reality. Over a zero length instant of time, no energy could go from on object to our retina and be mapped into the visual cortex. But we do receive such energy. Hence time can only exist as intervals and not as instants. We identify the continuous motion of a harmonic oscillator with time. That is a habit we develop from out early youth.

Even the orientation of the image is an illusion. The lenses of the human eye are simple lenses which invert the image of whatever is out there sending light photons to our retinas. Nevertheless we see everything right side up. Experiments have shown that if a subject wears prismatic lenses that invert the input to his eyes, within a matter of days he learns to undo the inversion and see things right side up.

So much for our "infallible" senses. Motion as we perceive and conceive it is illusory. The visual field -as perceived- is a construct. This is not to say that nothing moves. Not at all. If you assume motion is jittery and subject to visual jitter, things indeed do move, but we do not perceive them as they actually do move. Fortunately the quantum jitters are evened out in reality to the same degree that our visual cortex averages and smears the jumpy images that out eye presents to our brain (the eye constantly jumps up/down left/right). If our eyes stood still we would not see motion as we normally do. Lucky us. Our visual cortex smooths out the real motion of things sufficiently well for our survival. Even so, our eyes can be fooled by camouflage and sleight of hand. Our vision is good enough to enable our survival in the wild.

Conclusion: The usual calculus reply to Zeno's paradoxes really do not answer the problems raised. What the calculus approach does do for us it to give us good enough approximation to motion (jittery and grainy as it is) sufficient for shooting clay pigeons and driving our cars. But the mathematical model we know and love is at best a heuristic and not a true description of the motion. Time is not what we think, nor is space. Which is why we need quantum physics to deal with motion as it is and not as it appears. Zeno still haunts us. What the calculus reply to Zeno did was to sweep the basic problem under the rug.

There is a well written book which deals with the problems associated with the Zeno paradoxes: -The Motion Paradox- by Joseph Mazur. It has some math, but is not overwhelming.

Ba'al Chatzaf

[tjohnson]

It goes like this: At every instant of time an arrow is in a definite place. Therefore the arrow is at rest at every instant and it is impossible that it can move. How can something that is at rest at all times move?

Personally I don't think the conclusion follows from the premise. Why should one conclude that because the arrow is "somewhere" at a given time that it is also at rest?

Edited January 4, 2009 by general semanticist

[Brant Gaede]

I don't know the value of this discussion, but aren't we talking about the necessary illusion of the past, present and future? Aren't these the best concepts we have for coherent thinking involving us personally and time? As time is simply the measurement of motion points in time are of course a contradiction, but it is not really points in time but blocks of time which also or did or will contain motion. The stop motion is only in our heads, such as the 17th Century or the Age of Reason or WWII or what I am doing right now in the present.

--Brant

[tjohnson]

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it. All this illustrates is that mathematics only applies to the physical world in a limited fashion - to expect otherwise is unsane.

[Dragonfly]

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it.

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

[tjohnson]

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

I don't think so It makes no sense to speak about "an infinite number of steps" in the physical world. In mathematics we may speak about an infinite series but this exists only in our imagination.

[Michael Stuart Kelly]

This whole exercise rests on the premise that something can exist without time and our eyeballs can filter time out.

The reality is that time is a fundamental part of everything and our eyeballs evolved to perceive that part (time) just as much as the other parts.

This paradox comes off to me like trying to imagine sight without light waves, then claiming a paradox exists.

Michael

[Dragonfly]

I don't think so It makes no sense to speak about "an infinite number of steps" in the physical world.

Try doing physics without calculus...

In mathematics we may speak about an infinite series but this exists only in our imagination.

A circle or an ellipse also exist only in our imagination, but they are still quite useful to describe practical situations, as long as you don't confuse the mathematical construct with the physical reality.

[BaalChatzaf]

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it.

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

Zeno was being ironic. He knew that motion is possible (he moved about) and one does get from Here to There. He was trying to prove our ideas about motion were paradoxical and he was basically right. The calculus "solution" or resolution to Zeno's paradox does not really answer the underlying problem. It is a heuristic that happens to get the right answer. The real answer is that the world is not infinitely divisible and topologically compact. So the ontology of the calculus (or analytic approach) is wrong. In the physical world there are quantities that are not infinitely small (infinitesimal) or infinitely divisible (such as space and time). At the very bottom reality is grainy and gritty. Unfortunately, mathematics based on an atomic unit of time or length is intractable which is why we use topologically compact analytic models.

Ba'al Chatzaf

[BaalChatzaf]

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it.

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

That is right. He had no concept of convergence. However his arrow paradox cannot be dismissed to easily. How does any object that is motionless at every instant of time move. The answer is that there is really no such thing as an infinitesimal interval or point of time. That is how we model time, but that is because it leads to a tractable mode of analysis. Time is not really like that. The Planck interval is probably irreducible as is Planck length.

Ba'al Chatzaf

[tjohnson]

I don't think so It makes no sense to speak about "an infinite number of steps" in the physical world.

Try doing physics without calculus...

I don't understand you DF. All I am saying is that in reality it only takes a finite number of steps to reach your destination because of physical constraints you reach a point where you cannot 'halve' the distance anymore. It is not a question of doing physics without calculus.