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 have not fully considered the additional assumption of requiring distances to be integer multiples of the minimum length , but if that was the case then your arrangement of particles would be impossible since it would violate that assumption. I also don't believe that the presence of lengths not in integer multiples of the minimum distance logically contradicts the arguments I made. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

They would have to be integer multiples of the minimum length. Let's again refer to the minimum length as <i>d</i>. What if we have a distance of 1.5<i>d</i>? When we say 1.5<i>d</i>, what we are really expressing is <i>d</i>+0.5<i>d</i>. One <i>d</i> plus half of <i>d</i>. And if our assumption is that <i>d</i> is the minumum length, then there cannot be such a thing as half of <i>d</i> in our model, because that would mean that <i>d</i> is divisible, contradicting the core assumption. The only possible lengths would be 0, <i>d</i>, <i>d</i>+<i>d</i>=2<i>d</i>, <i>d</i>+<i>d</i>+<i>d</i><i>d</i>, and so on--all integer multiples of <i>d</i>.

So any arrangement of particles or points that includes a distance that is not an integer multiple of <i>d</i> would contradict the assumption that d is the smallest possible distance. This is extremely restrictive.

In fact, there are only two possible arrangements of particles that I can think of that wouldn't violate such an assumption. The first is three particles arranged in an equilateral triangle:

-o
o o

However, this would require that there could only be three particles in existance in the entire universe. Obviously, this is not the case.

The second is an indefinite number of particles arranged in a straight line:

o o o o o o o o ...

This arrangement would allow for any number of particles to exist in the universe. But it requires that all particles in the universe be in a straight line. As before, this is not the case.

If you can think of any other possible arrangements, I would like to hear about them. But I cannot picture any possible arrangements other than the two above, and neither matches our observations of the structure of the universe.

The logical consequences of the assumption of a minumum possible length do not agree with reality. Hence the assumption must be incorrect.

<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 /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">I have not fully considered the additional assumption of requiring distances to be integer multiples of the minimum length , but if that was the case then your arrangement of particles would be impossible since it would violate that assumption. I also don't believe that the presence of lengths not in integer multiples of the minimum distance logically contradicts the arguments I made. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

They would have to be integer multiples of the minimum length. Let's again refer to the minimum length as <i>d</i>. What if we have a distance of 1.5<i>d</i>? When we say 1.5<i>d</i>, what we are really expressing is <i>d</i>+0.5<i>d</i>. One <i>d</i> plus half of <i>d</i>. And if our assumption is that <i>d</i> is the minumum length, then there cannot be such a thing as half of <i>d</i> in our model, because that would mean that <i>d</i> is divisible, contradicting the core assumption. The only possible lengths would be 0, <i>d</i>, <i>d</i>+<i>d</i>=2<i>d</i>, <i>d</i>+<i>d</i>+<i>d</i><i>d</i>, and so on--all integer multiples of <i>d</i>.

So any arrangement of particles or points that includes a distance that is not an integer multiple of <i>d</i> would contradict the assumption that d is the smallest possible distance. This is extremely restrictive.

In fact, there are only two possible arrangements of particles that I can think of that wouldn't violate such an assumption. The first is three particles arranged in an equilateral triangle:

-o
o o

However, this would require that there could only be three particles in existance in the entire universe. Obviously, this is not the case.

The second is an indefinite number of particles arranged in a straight line:

o o o o o o o o ...

This arrangement would allow for any number of particles to exist in the universe. But it requires that all particles in the universe be in a straight line. As before, this is not the case.

If you can think of any other possible arrangements, I would like to hear about them. But I cannot picture any possible arrangements other than the two above, and neither matches our observations of the structure of the universe.

The logical consequences of the assumption of a minumum possible length do not agree with reality. Hence the assumption must be incorrect.
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

What about any right triangle whose integer squared sums yield
a hypotenuse with another integer squared sum? Like a 3, 4, 5
right triangle for example? If you are given an infinite number
of rulers even if they only came in 1 size, I think one can still make much more than an equilateral triangle shape.

The only restriction is that new shapes and sizes have dimensions that are multiples of the integer 1. 1 representing the smallest
"length" of a dimension.

Note how easy one can define distances when the method of using natural numbers is employed. The distance between 5 and 1 is just 4 units of 1. The distance between 1 and 2 is one unit of 1. Thus it makes sense to say that the distance between 5 and 1 is 4 times greater than the distance between 1 and 2.

When infinite divisibility is employed however, finding distances between two "points" is not as self-evident. How do you even locate two points in space if there is no smallest dimension?

[123...] "The only restriction is that new shapes and sizes have dimensions that are multiples of the integer 1."

Now try something a little harder. Like a square.




[123...] "How do you even locate two points in space if there is no smallest dimension?"

Measurement systems. Units. Same difference either way. (There are some books ... )


The Really Short Version (in your "smallest possible length" universe) -

1) Pick something to be the standard of distance. (This could be the diameter of a particle that happens to be the smallest possible lenght. Or some integer multiple of that smallest possible length.)

2) Give a name to this standard distance. Call it a "min-dee".

3) Use it to measure things.


Really Short Version (in an "infinitely divisible" universe) -

1) Pick something to be the standard of distance. (This could be the diameter of the same particle we used above. Or some multiple of that diameter.)

2) Give a name to this standard distance. Call it a "measure".

3) Use it to measure things.

The bottom line is there is no way to tell for sure which sort of universe we live in. Logic points clearly to the infinitely divisible universe. But there are no guarantees.


===

You behave as if studying is a painful thing for you. (You obviously avoided it in the past, and you seem to be avoiding it now.) Perhaps you should look for an interest area that doesn't require as much studying as physics?

Or ... get busy. Crack some books. Make your parents proud.

Good luck to you,
LB

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">What about any right triangle whose integer squared sums yield a hypotenuse with another integer squared sum? Like a 3, 4, 5
right triangle for example?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Ok, that one would work. (And some time after I made that post, I realized that four particles arranged in a tetrahedron would still work.) However, a minimum possible length would still be extremely restrictive, even including special right triangle arrangements. It only allows us to add these five possible arrangements (I'm not counting congruent figures as unique arrangments) to our short list:

#1

o


o---o

#2

o---o


o---o

#3

----o


o---o---o

#4

o


o---o


o

#5

----o


o---o---o


----o

Even with these additional arrangements, a universe in which there is a minumum possible length can contain no more than five particles, in only eight possible configurations. As you know, the universe contains many more particles than that, and in many more arrangements. So there is still a severe discrepancy between the logical consequences of a minimum possible length and our observations of reality.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">How do you even locate two points in space if there is no smallest dimension?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

We arbitrarily select a length, usually based on some physical quantity or process (for example, the meter is defined in terms of the speed of light), to be used as a stardard unit, and then note how many of those standard units must be combined to connect one point to another.

1234567890,

Does your finite divisible universe also require that matter have some smallest size? Is there a smallest particle?


JR

There should be no smallest particle. Matter should be divisble infinitly. Our observation is limited to our technology. For instance, attempting to view electrons with electrons give uncertainty. A graviton based system (if possible at all)used to view electrons may show them as macroscopic objects, and so on and so on. This is analogous to our more easily visible universe where structures repeat from the atom to the galaxies. Our senses have a range of scales that we can observe but there could be more. MV