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 1234567890</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 jrich</i>
You are correct that one cannot apply operations which contradict the assumption as that would be a form of logical fallacy. However, this does not mean that one cannot show that the assumption leads to logical or empirical contradictions. I provided such a proof in the previous topic on this subject last week. That was why I asked the question, to see if this ground had already been covered.

If assuming both finite or infinite divisibility in turn leads to contradictions, then it is a matter of personal preference since there is no way to scientifically distinguish them. However, when one assumption leads to contradictions and the other does not, then no matter our personal prejudices we must be accept the correctness of the logically consistent one. If one does not accept this then one is not practicing scientific, critical judgement and will not be taken seriously by most of the participants in this MB.


JR
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I assume you are referring to your argument regarding to touching spheres where the intermediate space directly above and below the
point of contact would be smaller than the diameter of each individual sphere? One obvious answer would be that at that distance, there are no spheres.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
Then give me another example of a particle geometry that does not violate the assumptions of smallest object and smallest distance?
<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Again, all the arguments leading to infinite divisibility appear to arise from a particular geometry , which carries with it a set of assumptions. If you start out with different assumptions, you get different results. And you can't say one is self contradictory unless you interpreted it using a different set of assumptions. It's like you can't say the shortest distance between two points is not a straight line leads to contradiction unless you used Euclidean geometry. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
One must be very careful not to introduce hidden assumptions. However, assumptions impose their own requirements and necessarily define constraints on the problem. These requirements and limits are not assumptions in themselves, but are part of the original assumptions. It makes no sense to talk of smallest anything except in the context of some goemetry. Every geometry will have the equivalent of "smallest", so this can be generalized to any geometry.


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>

Then give me another example of a particle geometry that does not violate the assumptions of smallest object and smallest distance?
JR
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Well, Astro has been trying to figure this out so read his examples.
But take a look at this photo


There are obviously smaller distances in between the "spheres" in the photo than the diameter of the spheres themselves, but does this fact change the size of the pixels on your monitor, which represents the smallest area resolvable by your monitor? No, it just means that the spheres are larger than the smallest pixels possible. Does the fact that the spheres in the photo exist lead to the "logical deduction" that there are an infinite number of pixels on your monitor or do their existence make impossible the existence of minimum sized pixels on your monitor?

I don't think so or you wouldn't be seeing the picture on your monitor. Minimum physical quantity is a property of existence,
the fact that we exist and take up a certain amount of space proves this. It is not a mathematical deduction that can be shown to be contradictory using some set of axioms.

And I'm sure it is much more a function of the properties of space and matter and energy of motion than it is a function of the supposed existence of continuous functions in calculus.

123,
<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Does wave motion lead to infinite divsibility then? Only if you first defined a wave as an indivisibile "particle" then tried to show that by another definition of a wave, that waves are the collective motion of many particles, waves are not indivisible. I think all you have shown using this method is that both definitions have mutually exclusive meanings. It is not a logical deduction.
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It is clear we first must come to an agreement what constitutes a wave. For example, do we assume that a wave is the collective motion of objects? Secondly, do we interpret this collective of objects as a medium?

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by 1234567890</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 jrich</i>

Then give me another example of a particle geometry that does not violate the assumptions of smallest object and smallest distance?
JR
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There are obviously smaller distances in between the "spheres" in the photo than the diameter of the spheres themselves, but does this fact change the size of the pixels on your monitor, which represents the smallest area resolvable by your monitor? No, it just means that the spheres are larger than the smallest pixels possible. Does the fact that the spheres in the photo exist lead to the "logical deduction" that there are an infinite number of pixels on your monitor or do their existence make impossible the existence of minimum sized pixels on your monitor?

I don't think so or you wouldn't be seeing the picture on your monitor. Minimum physical quantity is a property of existence,
the fact that we exist and take up a certain amount of space proves this. It is not a mathematical deduction that can be shown to be contradictory using some set of axioms.

And I'm sure it is much more a function of the properties of space and matter and energy of motion than it is a function of the supposed existence of continuous functions in calculus.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
The picture of the spheres is composed of pixels. Therefore, the spheres are not the smallest particle. A smallest particle could not be decomposed into smaller "pixels", therefore your representation of the spheres as the smallest particle violates the assumption. The pixels in your example are the smallest particle. Now suppose that the spheres are exactly one pixel in size so that they may not be further decomposed on the monitor. The smallest distance is then the distance between the pixels. In your pixel universe no pixels may touch, but if the pixels may not touch, how will the spheres touch? Are you saying that in your assumed finitely divisible universe contact is impossible? Does your pixel universe represent the real universe or is it only a close approximation?

I am not trying to "logically deduce" that the universe is infinitely divisible. That would be impossible because deduction doesn't work that way. What I am attempting to do is assume finite divisibility and using logical deduction show that a contradiction will result. I'm even letting you define the properties of the particles, such as their geometry and whether contact is possible lest you object to the ones I might choose.


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 /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by 1234567890</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 jrich</i>

Then give me another example of a particle geometry that does not violate the assumptions of smallest object and smallest distance?
JR
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

There are obviously smaller distances in between the "spheres" in the photo than the diameter of the spheres themselves, but does this fact change the size of the pixels on your monitor, which represents the smallest area resolvable by your monitor? No, it just means that the spheres are larger than the smallest pixels possible. Does the fact that the spheres in the photo exist lead to the "logical deduction" that there are an infinite number of pixels on your monitor or do their existence make impossible the existence of minimum sized pixels on your monitor?

I don't think so or you wouldn't be seeing the picture on your monitor. Minimum physical quantity is a property of existence,
the fact that we exist and take up a certain amount of space proves this. It is not a mathematical deduction that can be shown to be contradictory using some set of axioms.

And I'm sure it is much more a function of the properties of space and matter and energy of motion than it is a function of the supposed existence of continuous functions in calculus.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
The picture of the spheres is composed of pixels. Therefore, the spheres are not the smallest particle. A smallest particle could not be decomposed into smaller "pixels", therefore your representation of the spheres as the smallest particle violates the assumption. The pixels in your example are the smallest particle. Now suppose that the spheres are exactly one pixel in size so that they may not be further decomposed on the monitor. The smallest distance is then the distance between the pixels. In your pixel universe no pixels may touch, but if the pixels may not touch, how will the spheres touch? Are you saying that in your assumed finitely divisible universe contact is impossible? Does your pixel universe represent the real universe or is it only a close approximation?

I am not trying to "logically deduce" that the universe is infinitely divisible. That would be impossible because deduction doesn't work that way. What I am attempting to do is assume finite divisibility and using logical deduction show that a contradiction will result. I'm even letting you define the properties of the particles, such as their geometry and whether contact is possible lest you object to the ones I might choose.


JR
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Can you see more pixels than the number of holes in the shadow mask of your monitor? How does the idea of these holes representing the smallest size possible on your monitor lead to contradiction then?
Can you divide these holes into smaller parts without changing the composite parts of your monitor, i.e., without changing the "universe" of the monitor?.The universe that comprises of the images on a specific monitor then is completely self consistent with the idea of a minimum distance and is incompatible with the idea of infinite divisibility. Any image appearing on your monitor can only be divided into integal parts of the smallest sized holes.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by 1234567890</i>
Can you see more pixels than the number of holes in the shadow mask of your monitor? How does the idea of these holes representing the smallest size possible on your monitor lead to contradiction then?
Can you divide these holes into smaller parts without changing the composite parts of your monitor, i.e., without changing the "universe" of the monitor?.The universe that comprises of the images on a specific monitor then is completely self consistent with the idea of a minimum distance and is incompatible with the idea of infinite divisibility. Any image appearing on your monitor can only be divided into integal parts of the smallest sized holes.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
Obviously, a finitely divisible monitor screen cannot be infinitely divided without voiding the warranty[] But what is your point? I'm not arguing that a universe that is assumed from the start to be only finitely divisible can somehow be made to be infinitely divisible. Why do you keep putting up this straw horse? I am arguing that a finitely divisible universe must necessarily have properties which are incongruous with the properties of our universe. The question at hand isn't whether logically consistent finite universes may be devised, the question is whether a finite universe with the properties of our universe is logically consistent. Do you not see the distinction? Do you have no understanding of logical proofs?


JR