Zeno revisited

There is some misunderstanding here because "indefinitely small" (but still finite) is very different from "infinitely small" (infinitesimal). In my example that you cited here, I used the former, not the latter. -|Tom|-

Tom,

I agree with that. I actually did say above; "as small as you please," which means "indefinately small." But the difference is not essential to my argument. I'll think about this a little more to see if I want to try to reformulate it or not. These are, it seems to me, epistomological questions. Very dificult to resolve.

Neil

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by nderosa</i>
<br />{Larry) &lt; BTW, I agree that the universe is what it is, and that if there is a smallest possible size, well, then there is (a smallest possible size). But I can see no reason to deduce a smallest possible size at this time.

I agree. I guess my point is that without empirical facts, one deduction is as good as another<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
When dealing with paradoxes there is always an empirical fact since it's the apparent conflict with the deduced conclusions that creates the paradox!


JR

Neil,

I understand your complaint regarding the use of infinitely small points in a discussion of Zeno's paradox, but I don't think it makes any difference.

Let's try to duplicate Tom's reasoning with points of non-zero dimension. For the sake of argument we'll introduce a smallest possible distance and call it <i>d</i>.

Now let's arrange three particles <i>X</i>, <i>Y</i>, and <i>Z</i>, each of size <i>d</i>, as I attempted to diagram below:

o
o o

Although the limitations inherent in making pictures out of text characters prevented me from diagramming them so that they are actually touching, I hope that it is understood that they are supposed to be touching. Particle <i>X</i> is the top particle, <i>Y</i> is the leftmost of the bottom particles, and <i>Z</i> is the rightmost of the bottom particles.

Particles <i>X</i> and <i>Y</i> are touching, as are particles <i>Y</i> and <i>Z</i>. Also, their centers are separated by a distance of <i>d</i>. However, <i>X</i> and <i>Z</i> are not touching; using the Pythagorean Theorem we can determine that their centers are separated by a distance of <i>d</i> * SQR(2), or approximately 1.414<i>d</i>. The surfaces of <i>X</i> and <i>Z</i>, then, are separated by a distance of approximately 0.414<i>d</i>. Thus the surfaces of the particles in such an arrangement must be separated by a distance smaller than the minimum possible distance. Hence any such model in which a minimum possible distance is assumed--regardless of whether the dimension of the points is zero or nonzero--is internally inconsistent.

It gets worse. In such a model, the diameters of the particles are <i>d</i>. But what of the radius of the particles? Mathematics is quite clear: the radius would be 0.5<i>d</i>. So we would then have to add the stipulation that the minimum possible radius for a particle is <i>d</i>. The minimum possible diameter, then, would have to be 2<i>d</i>. (Notice, however, that the problem described above still remains.)

But all is STILL not well. What about the particle's equatorial circumference? If the minimum possible radius is <i>d</i>, then the minimum possible equatorial circumference is 2 * pi * <i>d</i>, or approximately 6.283<i>d</i>. So there is yet another internal contradiction inherent in the concept of a minimum possible distance: all lengths must be integer multiples of <i>d</i>, but the radius, diameter, and equatorial circumference of a sphere can NEVER simultaneously be integer multiples of <i>d</i>.

So it follows that there can be no minimum possible length. Length must be infinitely divisible.

I was preparing an objection to Astrodelugeologist's assertion that the 0.5<i>d</i> radius and 6.238<i>d</i> circumference also are proofs that <i>d</i> cannot be the smallest distance. But then I realized that my objection also invalidated my own proof as well as the others. We have forgotten that Zeno's paradox is not concerned with a minimum possible distance in space, but a minimum possible distance <i><b>of motion</b></i> in space. There may be arguments against a minimum distance of motion, but those presented in this topic up to now are not valid.


JR

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by jrich</i>
<br />We have forgotten that Zeno's paradox is not concerned with a minimum possible distance in space, but a minimum possible distance <i><b>of motion</b></i> in space.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">It is true that all of Zeno's eight paradoxes deal with motion in one form or another. But motion means a space interval traversed in a time interval. Zeno was not concerned with whether or not there was a minimum non-zero increment of motion because motion is not a dimension and because it is always relative. (Absolute motion is not a well-defined concept.)

So Zeno was trying to address whether the <i>dimensions</i> of space and time are infinitely divisible or have minimum increments. This remained true even in his "motion" paradox with one arrow trying to overtake another, but requiring an infinite number of half-the-remaining-distance steps to do so. He was still looking at space and time intervals, not at speed increments. Zeno's arguments do not make sense except as applied to dimensions. -|Tom|-

Zeno's Paradox is not a Paradox at all, if the turtle took steps of half the distance each time he would indeed never reach the second point in the journey. The fact that a turtle or a human is unable to take such half steps does not mean the half step does not exist. The fact is measurable half steps are infinite, this does not however require the component parts of any structure are infinate in scale. If you divide a quark 1/1,000,000,000,000,000,000, onto infinity it could still be a quark or a homogenious piece of one. This is the point to be argued, does the half step require that independent quanta exist infinatly, if so, why?