Zeno revisited

Astro's post is as clear as any other on why my assertion is mistaken that there can be a minimum possible distance in the universe, so I'l refer to it here. There also seems to be a clear unanimity among the other posts that the universe is indeed infinitely divisible thus making TVF's assumption of "infinate scale" seem quite logical.

Though "infinate scales" does not seem "economical" to me, (any more than do "multiple universes"),I admit that that alone is not a good argument against that concept. Still, I want to try again to make a logical argument--hopefully without repeating myself.

Start with a simple picture and some imagination. Let's suppose that LeSage's gravitons are actually the smallest entities in the universe. It doesn't matter if we know this to be true--but just suppose it is. Now it is still true as Astro and others have said, that there remain several paradoxes which can be mathematically demonstrated. I can't argue with that either.

My point is very simple and it goes back to William James's Pragmatism. It's this; if this smallest graviton particle is not divisible in fact, then any speculation about the mathematical properties of such a division is meaningless speculation because that graviton can not be divided. That's what I mean by the term "arbitrary construct."

All I'm saying is that the universe may indeed be infinately divisible, but we can not arrive at that conclusion deductively. We can not make any first, or major premise in a syllogism without reference to what really exists in the actual world. That's why induction is co-equal with deduction. The universe is what it is and scientists have to go find out the hard way.

[Neil] "... the universe may indeed be infinately (sic) divisible, but we can not arrive at that conclusion deductively."

Sure we can. Dr. Van Flandern just did it. But this may be semantics again. What we can't do is prove that the universe actually is (or actually is not) physically infinitely divisible.

LB

PS - Repetition is often annoying, but it is also sometimes the only way to work something out. After four or five circuits through a particular loop with the same people, a lurker might decide to share a thought that actually resolves the issue.

Or not. But it could happen.

<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 />Start with a simple picture and some imagination. Let's suppose that LeSage's gravitons are actually the smallest entities in the universe. It doesn't matter if we know this to be true--but just suppose it is.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">We could also assume that the universe began with a Big Bang, or that Santa Claus exists. But if any assumption hangs from clouds instead of being based on reason, argumentation, observation, experiment, or at least citation, then it will not accomplish what we value most in a hypothesis: that it be useful in understanding and predicting nature.

Far from helping us understand or predict, your proposed assumption creates numerous logical paradoxes and unanswerable questions, such as those I itemized in chapter one of <i>Dark Matter…</i> [DM]. For example, consider this quote:
<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><font color="yellow"><b>[DM]: One might use this argument to conclude that there is a smallest possible unit of matter or substance. Imagine such a "unit particle." It must be utterly uncomposed. It therefore cannot be broken or divided, nor even deformed by spin or collision -- since these are properties of bodies composed of yet smaller particles. What then are we to assume will happen when two such unit particles collide? What density will the unit particle have? Indeed, will there be anything inside it at all? (It would seem that the substance in its interior could never contribute in any way to anything in the universe outside the particle, since it can never interact with it.) What would the unit particle's "surface" be like? Could it be hollow inside? With what thickness of shell? Would two colliding unit particles have to stick, since they can't rebound elastically? If they rebounded, with what resultant velocity? What about the slightest of grazing collisions? Would the unit particles be spherical in shape? Why would they have finite space dimensions, yet infinite dimension in time? Or do they come into and go out of existence constantly? Where and when would they appear and disappear?</b></font id="yellow"><hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote"><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[nderosa]: if this smallest graviton particle is not divisible in fact, then any speculation about the mathematical properties of such a division is meaningless speculation because that graviton can not be divided. That's what I mean by the term "arbitrary construct."<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The problem with the hypothesis is that it leads to paradoxes and unanswerable questions, and it fails to lead us to a better understanding or ability to predict nature.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">All I'm saying is that the universe may indeed be infinitely divisible, but we can not arrive at that conclusion deductively. We can not make any first, or major premise in a syllogism without reference to what really exists in the actual world.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Chapter one of <i>Dark Matter…</i> shows how to arrive at that conclusion deductively and without any non-gratuitous reference to what actually exists, except for arbitrary examples used as analogies (such as crossing the street). But we could have reached the same conclusion using pure abstractions such as the set of real numbers, without invoking any physical entities. So that chapter contains a counter-example to your assertion here.

I suspect the problem really is with the most difficult point in the logical syllogism: the fact that sets can be infinite even while all their members are finite, accompanied by the need to use one-to-one correspondences to understand the properties of these infinite sets. Especially troubling to many people is the property that a set containing an infinite number of elements (such as half-the-remaining-distance intervals when crossing a street) can nonetheless have a finite sum. In fact, the non-intuitive nature of that property is why Zeno's paradoxes exist and are famous even today.

To see why a set can have an infinity of elements with a finite sum, we need to use one-to-one correspondences. So the intervals in the street-crossing example can be put into a one-to-one correspondence with the infinite series [1/2 + 1/4 + 1/8 + 1/16 + ...]. And by mathematical induction, we can prove that this sum is exactly 1. This is because we can show that the sum will eventually exceed any real number less than 1, but cannot exceed 1 even after an infinite number of terms are included.

If you don't "get" this point, then it will appear that the syllogism in DM is broken and the conclusion (infinite divisibility of matter) did not necessarily follow. If you confirm that is the stumbling block for you, then we should analyze this point in greater depth, ideally until we all get it or have pinpointed any fallacy in it. -|Tom|-

As an onlooker here I am learning why models are prefered to actual events. If you work with a model you have rules to play by and that decides issues like you guys are kicking around(never to be resolved issues). Small is fixed in QM. Reading the history of why QM was adopted and Einstein's views on QM make sense after seeing how real questions can get bogged down over issues like these paradoxes. Even the BB model has its good points and the real problem with that model is it is accepted as a real representation of the universe. The MM model has one idea not found in BB-recycling. Everything in MM modeling is recycled and in the BB nothing is recycled. Revised BB models do recycle somewhat but not in a logical process-the MM model has a better process for how recycling really occurs in the universe. All in all, if real issues get bogged down in paradox, then models are a better option.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Astrodelugeologist</i>
<br />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.
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

If length is infinitely divisible, limits do not exist in calculus
since the whole process involves the idea of dividing intervals smaller and smaller until we reach the limit of the function
at a specified x value. 0 would not exist, nor would any other point on any curve.

Therefore there exists a minimum length for existence (besides
0 that is). Since we have defined apriori a minimum distance,
beyond which is 0, all your operations on this minimum distance above are inapplicable. Motion wise, all we need to assume is that things must move at least this minimum distance every time they move, which requres energy of motion to have a minimum as well.
And the values would need to be discrete as well in increments of the minimum allowed value.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by tvanflandern</i>
<br /><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 />Start with a simple picture and some imagination. Let's suppose that LeSage's gravitons are actually the smallest entities in the universe. It doesn't matter if we know this to be true--but just suppose it is.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">We could also assume that the universe began with a Big Bang, or that Santa Claus exists. But if any assumption hangs from clouds instead of being based on reason, argumentation, observation, experiment, or at least citation, then it will not accomplish what we value most in a hypothesis: that it be useful in understanding and predicting nature.

Far from helping us understand or predict, your proposed assumption creates numerous logical paradoxes and unanswerable questions, such as those I itemized in chapter one of <i>Dark Matter…</i> [DM]. For example, consider this quote:
<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><font color="yellow"><b>[DM]: One might use this argument to conclude that there is a smallest possible unit of matter or substance. Imagine such a "unit particle." It must be utterly uncomposed. It therefore cannot be broken or divided, nor even deformed by spin or collision -- since these are properties of bodies composed of yet smaller particles. What then are we to assume will happen when two such unit particles collide? What density will the unit particle have? Indeed, will there be anything inside it at all? (It would seem that the substance in its interior could never contribute in any way to anything in the universe outside the particle, since it can never interact with it.) What would the unit particle's "surface" be like? Could it be hollow inside? With what thickness of shell? Would two colliding unit particles have to stick, since they can't rebound elastically? If they rebounded, with what resultant velocity? What about the slightest of grazing collisions? Would the unit particles be spherical in shape? Why would they have finite space dimensions, yet infinite dimension in time? Or do they come into and go out of existence constantly? Where and when would they appear and disappear?</b></font id="yellow"><hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote"><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[nderosa]: if this smallest graviton particle is not divisible in fact, then any speculation about the mathematical properties of such a division is meaningless speculation because that graviton can not be divided. That's what I mean by the term "arbitrary construct."<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The problem with the hypothesis is that it leads to paradoxes and unanswerable questions, and it fails to lead us to a better understanding or ability to predict nature.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">All I'm saying is that the universe may indeed be infinitely divisible, but we can not arrive at that conclusion deductively. We can not make any first, or major premise in a syllogism without reference to what really exists in the actual world.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Chapter one of <i>Dark Matter…</i> shows how to arrive at that conclusion deductively and without any non-gratuitous reference to what actually exists, except for arbitrary examples used as analogies (such as crossing the street). But we could have reached the same conclusion using pure abstractions such as the set of real numbers, without invoking any physical entities. So that chapter contains a counter-example to your assertion here.

I suspect the problem really is with the most difficult point in the logical syllogism: the fact that sets can be infinite even while all their members are finite, accompanied by the need to use one-to-one correspondences to understand the properties of these infinite sets. Especially troubling to many people is the property that a set containing an infinite number of elements (such as half-the-remaining-distance intervals when crossing a street) can nonetheless have a finite sum. In fact, the non-intuitive nature of that property is why Zeno's paradoxes exist and are famous even today.

To see why a set can have an infinity of elements with a finite sum, we need to use one-to-one correspondences. So the intervals in the street-crossing example can be put into a one-to-one correspondence with the infinite series [1/2 + 1/4 + 1/8 + 1/16 + ...]. And by mathematical induction, we can prove that this sum is exactly 1. This is because we can show that the sum will eventually exceed any real number less than 1, but cannot exceed 1 even after an infinite number of terms are included.

If you don't "get" this point, then it will appear that the syllogism in DM is broken and the conclusion (infinite divisibility of matter) did not necessarily follow. If you confirm that is the stumbling block for you, then we should analyze this point in greater depth, ideally until we all get it or have pinpointed any fallacy in it. -|Tom|-

<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

If a series never ends, there can be no sum either. That would be like trying to calculate the life expectancy of our universe. So,
no, we don't "get" this point.