Formal Logic and Scientific Method

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

does not as you go to infinity scale,does not dimension do the same,in other words as something gets smaller, does not the whole scale of space do the same,right along with it? therefore minimum distance become relative?

if i'm way off please fell free to let me know if i'm way off. perhaps i do not fully understand the problem here.
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I think you are thinking of minimum length in terms of the smallest distance a measuring tool can resolve. The minimum length a meter stick can resolve for example is on the order of millimeters. We were speaking in terms of the hypothetical minimum length possible for matter and space to exist, whether this length can be measured or not by current technology. Since we have been taught in basic math that you can divide a number by as large a number as you can think of, it seems intuitive that this would also be the case with physical matter and space. I was trying to show that this was not the case- that matter and space can only be divided into minimum
parts- while jr and others tried, ineffectively, to show the opposite.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">I am not following your argument why there can only be 5 particles in the universe given a minimum length exist. Do you remember playing with ball and stick models in chemistry? Let's say I were to give you a very large set of balls and sticks but all of equal sizing, and I gave you a couple of simple rules for constructing 3 dimensional structures using those balls and sticks. 1. Before any two balls can be connected, there has to be a stick between them. 2. You are allowed to glue the sticks (or rather, the ends of the sticks stick to each other) to form longer connections between balls. The smaller the structures you construct, the less possible arrangements are possible. But when you start making larger structures, all kinds of shapes and angles become possible. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

The ball-and-stick analogy isn't really valid here. The problem is that, when you're working with a set of balls and sticks, you only have to worry about the distance between <i>adjacent</i> balls (those that are connected by a stick).

However, when we're talking about possible structures on the scale of a hypothetical smallest possible length, you not only have to worry about the distance between adjacent balls, but the distance between <i>all</i> balls. Consider the following structure:

o
|
o---o

If you remember my previous diagrams, you will recall that I stated that the hyphens were there only for placement purposes and were to be ignored. In this case, however, I'm using the hyphens (-) and the vertical lines (|) to indicate the presence of sticks connecting the balls. Hopefully nobody will be confused by this change. In the figure above, please assume that both sticks are the same length.

Anyway, in this structure, the top ball is <i>A</i>, the bottom left ball is <i>B</i>, and the bottom right one is <i>C</i>.

If your set of balls and sticks allows for 90-degree angles, then clearly this structure can be built. Furthermore, you can measure the length between ball <i>A</i> and ball <i>B</i> to be one stick length; the same is true of the distance between ball <i>B</i> and ball <i>C</i>.

However, think about the distance between balls <i>A</i> and <i>C</i>. There is no stick there, but clearly such a distance must exist regardless. This distance is in fact 1.414 stick lengths. As this is not an integer multiple of the length of a stick, this structure is impossible if we assume a minimum possible distance. Since this structure contradicts our starting assumption, it cannot exist in a universe in which this assumption is true.

Permit me to modify your ball-and-stick analogy so that it is better suited to the problem at hand. As in your analogy, we may assume that we have an indefinite number of sticks and balls, and that we are allowed to glue sticks together, but not cut them. The difference between this new analogy and yours is that every ball must be connected to every other ball by sticks. Why? Because the distance from every ball to every other ball is just as important as the distance from one ball to its adjacent balls; if any of those distances is not an integer multiple of a stick length, the structure cannot exist, as we would be breaking the rules.

So, using this more accurate analogy, it becomes apparent that only a very few unique structures can be built, and only a very few balls can be used in any structure. In the same way, a hypothetical universe in which we assume a minimum possible length can have only a very few unique structures, and only a very few particles. Either there will be no more than five particles in the universe, or there is an indefinite number of particles arranged in a straight line. Neither fits our observations of the universe.

The only way to resolve this difficulty is to reject the assumption that there is a minimum possible length for our universe.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Why don't you provide at least the minimal evidence that the smallest particles we know of are spherical?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

We don't know the shape of the smallest known particles; however, the smallest structures that we have been able to "visually" observe (atoms) are spherical. There is no evidence to suggest that smaller particles would be any different.

However, the problem is that no particle is possible in a universe in which there is a minimum possible length--not even a spherical one. A cube can have sides that are integer multiples of that length, but the distance from corner to corner will not be integer mutiples of the minimum possible length; the same would be true of prism-shaped particles as well. A tetrahedron can have sides that are integer multiples of the minimum possible length, but the height of the tetrahedrom will not be an integer multiple of that length; the same goes for pyramid-shaped particles. Finally, a sphere could have a radius that is an integer multiple of the minimum possible length, but its circumference cannot be an integer multiple of that length. So none of these particle shapes can exist in a universe in which there is a minimum possible length, because their dimensions must contradict that assumption.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">If your line of mathematical induction was any accurate representation of properties of our universe, you should have no problem finding more than the mere 100 or so elements on our periodic table , since there should exist particles and energy levels, according to infinite divisibility, at every possible scale.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

I disagree. I have heard many proofs for infinite divisiblity of <i>space</i>, but none for infinite divisibility of energy levels.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">By your logic, there cannot be any space left that is not filled with matter since such a discovery would immediately contradict the result obtained by the principle of infinite divisibility that there be no distances smaller than the smallest matter. This is trivially disproven by my typing of this post.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

I don't see any problem with the deduction that all space is filled with matter, nor do I see any observation that disproves it.

The smallest particles we observe would be surrounded by a flux that would appear uniform to us. However, under further magnification, that flux would be resolved into individual particle components. These particles themselves would be surrounded by a flux. As before, given enough magnification, it would be possible to resolve this flux into particles surrounded by a flux. And this would continue to infinity in both increasing and decreasing directions.

astro,
<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">The smallest particles we observe would be surrounded by a flux that would appear uniform to us. However, under further magnification, that flux would be resolved into individual particle components. These particles themselves would be surrounded by a flux. As before, given enough magnification, it would be possible to resolve this flux into particles surrounded by a flux. And this would continue to infinity in both increasing and decreasing directions.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

How succinct and clear.

<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>

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">
The ball-and-stick analogy isn't really valid here. The problem is that, when you're working with a set of balls and sticks, you only have to worry about the distance between <i>adjacent</i> balls (those that are connected by a stick).

However, when we're talking about possible structures on the scale of a hypothetical smallest possible length, you not only have to worry about the distance between adjacent balls, but the distance between <i>all</i> balls. Consider the following structure:

o



|
o---o

If you remember my previous diagrams, you will recall that I stated that the hyphens were there only for placement purposes and were to be ignored. In this case, however, I'm using the hyphens (-) and the vertical lines (|) to indicate the presence of sticks connecting the balls. Hopefully nobody will be confused by this change. In the figure above, please assume that both sticks are the same length.

Anyway, in this structure, the top ball is <i>A</i>, the bottom left ball is <i>B</i>, and the bottom right one is <i>C</i>.

If your set of balls and sticks allows for 90-degree angles, then clearly this structure can be built. Furthermore, you can measure the length between ball <i>A</i> and ball <i>B</i> to be one stick length; the same is true of the distance between ball <i>B</i> and ball <i>C</i>.

However, think about the distance between balls <i>A</i> and <i>C</i>. There is no stick there, but clearly such a distance must exist regardless. This distance is in fact 1.414 stick lengths. As this is not an integer multiple of the length of a stick, this structure is impossible if we assume a minimum possible distance. Since this structure contradicts our starting assumption, it cannot exist in a universe in which this assumption is true.
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The distance doesn't exist since the particles only interact
with the sticks.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">
Permit me to modify your ball-and-stick analogy so that it is better suited to the problem at hand. As in your analogy, we may assume that we have an indefinite number of sticks and balls, and that we are allowed to glue sticks together, but not cut them. The difference between this new analogy and yours is that every ball must be connected to every other ball by sticks. Why? Because the distance from every ball to every other ball is just as important as the distance from one ball to its adjacent balls; if any of those distances is not an integer multiple of a stick length, the structure cannot exist, as we would be breaking the rules.

So, using this more accurate analogy, it becomes apparent that only a very few unique structures can be built, and only a very few balls can be used in any structure. In the same way, a hypothetical universe in which we assume a minimum possible length can have only a very few unique structures, and only a very few particles. Either there will be no more than five particles in the universe, or there is an indefinite number of particles arranged in a straight line. Neither fits our observations of the universe.

The only way to resolve this difficulty is to reject the assumption that there is a minimum possible length for our universe.

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Actually I was thinking of springs and rubberbands because they would stretch.
Also, if there was a force field like gravity or em to bend the sticks
then more arrangements can be made using those minimum length sticks,
no?

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">
<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Why don't you provide at least the minimal evidence that the smallest particles we know of are spherical?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

We don't know the shape of the smallest known particles; however, the smallest structures that we have been able to "visually" observe (atoms) are spherical. There is no evidence to suggest that smaller particles would be any different.
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The keyword here being "structures". Has it been shown that atoms and molecules are spherical in shape first of all? The spherical shape is only applicable to elements whose electrons reside in the 2s and 1s orbitals (Beryllium and below) I think and even there, since electrons in the atoms are described by a probability cloud in QM,
and it's not even clear whether they are particles or waves,
their geometric relation with respect to the nucleus are not
very well defined. In fact, the de broglie hypothesis (matter-waves)
makes the geometric shape of protons, electrons, and other "particles" pretty ambiguous.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">

However, the problem is that no particle is possible in a universe in which there is a minimum possible length--not even a spherical one. A cube can have sides that are integer multiples of that length, but the distance from corner to corner will not be integer mutiples of the minimum possible length; the same would be true of prism-shaped particles as well. A tetrahedron can have sides that are integer multiples of the minimum possible length, but the height of the tetrahedrom will not be an integer multiple of that length; the same goes for pyramid-shaped particles. Finally, a sphere could have a radius that is an integer multiple of the minimum possible length, but its circumference cannot be an integer multiple of that length. So none of these particle shapes can exist in a universe in which there is a minimum possible length, because their dimensions must contradict that assumption.
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I disagree. A smallest particle doesn't need to have 3 dimensions. More likely than not it only has one. So there are no longer diagonals, circumferences, non-integer bases, etc, to worry about.
Take the set of naturals- if we forbid the division of these numbers by a divisor that would yield a non-integer, we get a mathematical construct based on the idea of a minimum quantity, namely 1. If we can build it, such a universe be come s possible. I mean, this is a simple example of course but you are still using geometric constructs from a geometry that requires infinite divisibility to argue that it is impossible in one that is not infinitely divisible. And we haven't even considered the effects that gravity and other fields would have on these geometric constructs.

Nevertheless, the fact that the diagonal of a cube is not an integer length if its sides are does not mean it can be physically divided into smaller parts, though it may appear so from our daily activities like slicing a toast in half. On the smaller scales,
like in particle accelerators, "particles" regularly collide and become energy instead of breaking into smaller "particles".

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">
<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">If your line of mathematical induction was any accurate representation of properties of our univer

123,

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">But if everything around us is occupied by matter, how can we move at all?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Perhaps ask the question, "How can I swim at all when everything around me is occupied by water?" ([])

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jan</i>
<br />123,

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">But if everything around us is occupied by matter, how can we move at all?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Perhaps ask the question, "How can I swim at all when everything around me is occupied by water?" ([])
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If there is no space left, you wouldn't be able to swim since the
water couldn't get displaced.