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E S S A Y S
By The Number
Fig. 1. The smallest size in math is a Planck Length.
ON the AMBIGUITIES of LANGUAGE
A Probing Investigation Into the Process of Questioning
Continued
The Dichotomy Paradox
Part Two
Before going further, this is an appropriate time to mention what is called the "Planck Length" established by the German physicist, Max Planck at the turn of the century. Considered one of the founding fathers of Quantum physics, the Planck "unit" attempts to describe the smallest possible size, or span-of-distance between or among bits of matter. Sounds slightly apropos to the discussion at hand, doesn't it? Anything smaller, it is proposed, would necessarily occupy the same space as something else. Therefore in order to remain separate, but with no space between or among them, "points" can be no smaller than a Planck unit. See how simple this stuff is?
To repeat, the “Planck length” is the an increment of size or distance expressed in terms of Planck units. Such a "thing" is very tiny. It's not even possible to describe in words, how small one of these units really is. Try imagining the most miniscule, minutest thing possible, and a Planck unit is huge by comparison. For example, a hydrogen atom is about ten-trillion-trillion Planck lengths across. If that sounds as meaningless as it actually is, you're on the right track. If you see Homer, tell him we said, "Hello."
Physicists primarily use the Planck length to imagine (play with) stuff that is ridiculously itty-bitty. Specifically too teeny to matter (no pun intended). By the time you approach anywhere close to a Planck length, things stop making much sense with respect to discussing -- in any reasonable fashion -- the difference between any two points in proximity to one another. In my application of what are (in essence) Planck units, to the Dichotomy paradox, I may have found just such a sensibleness. Maybe not, but I had fun trying.
Basically, because of the uncertainty principle, which refers to measuring both the location of something and how fast it's moving, there’s no practically relevant difference between the physical positions of things when they are separated by such infinitesimally small gaps (distances). Once again, my essay here attempts to "dally" with just such a relevance. Since nothing fundamentally changes at the Planck scale, there seems little point in trying to deal with things that small. Except for weirdos like me, few people bother with finding meaning, either mathematical or philosophical, with this particular subject matter. Which is why I got so enthused when I stumbled upon it. Interestingly I had never even heard of a Planck unit when I made all my original calculations some two decades ago.
By the way, the smallest particle, the electron, is about 1,020 times larger than a Planck length (that’s the difference between a single hair and a large galaxy). Rather than something tangible or manipulatively manageable, the Planck scale is as much philosophical as it is a series of numbers. And it is precisely for this reason that a non-mathematician is as equally qualified (almost) as the finest astrophysicist at finding meaning in meaninglessness.
To repeat, the “Planck length” is the an increment of size or distance expressed in terms of Planck units. Such a "thing" is very tiny. It's not even possible to describe in words, how small one of these units really is. Try imagining the most miniscule, minutest thing possible, and a Planck unit is huge by comparison. For example, a hydrogen atom is about ten-trillion-trillion Planck lengths across. If that sounds as meaningless as it actually is, you're on the right track. If you see Homer, tell him we said, "Hello."
Physicists primarily use the Planck length to imagine (play with) stuff that is ridiculously itty-bitty. Specifically too teeny to matter (no pun intended). By the time you approach anywhere close to a Planck length, things stop making much sense with respect to discussing -- in any reasonable fashion -- the difference between any two points in proximity to one another. In my application of what are (in essence) Planck units, to the Dichotomy paradox, I may have found just such a sensibleness. Maybe not, but I had fun trying.
Basically, because of the uncertainty principle, which refers to measuring both the location of something and how fast it's moving, there’s no practically relevant difference between the physical positions of things when they are separated by such infinitesimally small gaps (distances). Once again, my essay here attempts to "dally" with just such a relevance. Since nothing fundamentally changes at the Planck scale, there seems little point in trying to deal with things that small. Except for weirdos like me, few people bother with finding meaning, either mathematical or philosophical, with this particular subject matter. Which is why I got so enthused when I stumbled upon it. Interestingly I had never even heard of a Planck unit when I made all my original calculations some two decades ago.
By the way, the smallest particle, the electron, is about 1,020 times larger than a Planck length (that’s the difference between a single hair and a large galaxy). Rather than something tangible or manipulatively manageable, the Planck scale is as much philosophical as it is a series of numbers. And it is precisely for this reason that a non-mathematician is as equally qualified (almost) as the finest astrophysicist at finding meaning in meaninglessness.
Fig. 2.
As we return to Homer who's been waiting patiently in the caboose, an example of how the original proposition insults the intelligence of persons who wish to take-it-on, lies in the manner with which the bullet is conveniently allowed to travel past the half-way point between A and B. Seemingly with no problem whatsoever. We will refer to this as point C .
A >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> C >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> B
barrel & bullet half-way to rear wall rear wall of caboose
Working our way backwards from point C towards A, we come upon another (previously traversed) half-way point between the two, which can now be labeled as point E.
A >>>>>>>>>>>>>>>>>>>>>E>>>>>>>>>>>>>>>>>>>>> C >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> B
barrel & bullet half-way to C
The bullet appears to have had no difficulty in crossing the gap between A and B, passing through C with ease, and yet, according to the proposition, the same rules should apply. The moving round should never reach the half-way point of C, between A and B, because it first had to travel half the distance between A and C, thus passing through point E. Subsequently, the bullet then encounters the same sequence of diminishing spaces between E and C, as is predicted should occur between C and B.
Although this sounds confusing, I assure you that a careful, methodical reading will not only make total sense, but be worth whatever time it takes to do so.
Now then, if we continue with this same retrograde sectioning, it begins to appear obvious, the importance of understanding and defining exactly what is meant by a term such as point. Having come this far, it is possible, of course, to indulge ourselves further and locate point F, which is half-way between A and E.
A >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> F >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> E
half-way to C
Point G which is half-way between A and F . . .
A >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> F
And so on and so forth . . .
A >>>>>>>> K >>>>>>>>> J >>>>>>>>>> I >>>>>>>>>> H >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G
half-way to F
So what happens as we continue to reverse course and locate half-way points indefinitely, constantly edging our way nearer and nearer to point A? Previously it was asserted and shown that an indefinite -- and not infinite -- number of points exist between two or more other points. Locations which appear to be separated from one another by some arbitrary, unspecified distance or space. However, as more and more points are demarcated along our journey back towards the muzzle of the gun itself (point A) an observer, at some point, must recognize that a single, last remaining gap is all that separates the bullet from making physicial contact with the metal barrel of the pistol (point A). Since any movement closer to the muzzle would involve the simultaneous occupation of the same space, there is theoretically no remaining space between point A (the end of the barrel) and the rear end of the bullet itself.
To repeat, it is very important to realize that the back of the bullet is not separated from point A by any space or gap -- other than that occupied by a final point between them. A single (and indivisible) Planck unit, if you will. Nor is a half-point or quarter-point "slice" possible; it is the point itself that entirely fills the space that remains. Furthermore, crossing this point (language fails at this level) is synonymous with arriving at point A -- thus overlapping the same space as that occupied by the barrel of the gun. In this case, with the bullet either entering the barrel of the gun or penetrating the rear wall of the caboose. Meanwhile the Basic Proposition pretends that some kind of mysterious void exists between A and B, that cannot be filled by anything other than a bullet or some other object.
As you no doubt noticed, instead of following the bullet on its journey to the rear wall of the caboose, we chose to dissect the problem in much the same way as others have tackled the Dichotomy paradox -- namely in retracing Homer's steps as he runs to catch the bus. I simply changed things around a bit, but we could just as easily have accompanied the bullet to the rear wall and then demonstrated how it makes contact with the wall of the caboose in the same manner as it leaves the barrel of the pistol in the first place.
Obviously a limited and indefinite number of Planck units separate everything from everything else. Once bridged, the outer energy "shells" that compose the seemingly material surfaces of matter merely bang into one another. A final notation is that any and all points are utterly arbitrary -- in both size and number -- and extend indefinitely in all directions. More importantly, no empty space or gap exists either between two points, or an infinite number (chain) of them. The Homeric proposition simply (and incorrectly) locates two separated points along an indefinitely extended span of connected points, and proposes that some kind of mystical nothingness exists between the points.
How easily our language systems can deceive us into thinking about (and struggling with) mysterious, nonsensical propositions which cannot withstand the most rudimentary scrutiny, and confound us with intense contemplations based on illogical fluff.
We are surrounded by and inundated with, ideas, philosophies, religions and ideologies that, in many cases, possess no more of a factual basis than the foregoing, paradoxical proposition. Yet we sacrifice our entire lives sometimes on their behalf and in their defense. It is for this reason that questions are always superior to their answers, and why our ability to question our questions, and not just our answers, should be counted among the greatest of human attributes.
. . .
A >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> C >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> B
barrel & bullet half-way to rear wall rear wall of caboose
Working our way backwards from point C towards A, we come upon another (previously traversed) half-way point between the two, which can now be labeled as point E.
A >>>>>>>>>>>>>>>>>>>>>E>>>>>>>>>>>>>>>>>>>>> C >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> B
barrel & bullet half-way to C
The bullet appears to have had no difficulty in crossing the gap between A and B, passing through C with ease, and yet, according to the proposition, the same rules should apply. The moving round should never reach the half-way point of C, between A and B, because it first had to travel half the distance between A and C, thus passing through point E. Subsequently, the bullet then encounters the same sequence of diminishing spaces between E and C, as is predicted should occur between C and B.
Although this sounds confusing, I assure you that a careful, methodical reading will not only make total sense, but be worth whatever time it takes to do so.
Now then, if we continue with this same retrograde sectioning, it begins to appear obvious, the importance of understanding and defining exactly what is meant by a term such as point. Having come this far, it is possible, of course, to indulge ourselves further and locate point F, which is half-way between A and E.
A >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> F >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> E
half-way to C
Point G which is half-way between A and F . . .
A >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> F
And so on and so forth . . .
A >>>>>>>> K >>>>>>>>> J >>>>>>>>>> I >>>>>>>>>> H >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G
half-way to F
So what happens as we continue to reverse course and locate half-way points indefinitely, constantly edging our way nearer and nearer to point A? Previously it was asserted and shown that an indefinite -- and not infinite -- number of points exist between two or more other points. Locations which appear to be separated from one another by some arbitrary, unspecified distance or space. However, as more and more points are demarcated along our journey back towards the muzzle of the gun itself (point A) an observer, at some point, must recognize that a single, last remaining gap is all that separates the bullet from making physicial contact with the metal barrel of the pistol (point A). Since any movement closer to the muzzle would involve the simultaneous occupation of the same space, there is theoretically no remaining space between point A (the end of the barrel) and the rear end of the bullet itself.
To repeat, it is very important to realize that the back of the bullet is not separated from point A by any space or gap -- other than that occupied by a final point between them. A single (and indivisible) Planck unit, if you will. Nor is a half-point or quarter-point "slice" possible; it is the point itself that entirely fills the space that remains. Furthermore, crossing this point (language fails at this level) is synonymous with arriving at point A -- thus overlapping the same space as that occupied by the barrel of the gun. In this case, with the bullet either entering the barrel of the gun or penetrating the rear wall of the caboose. Meanwhile the Basic Proposition pretends that some kind of mysterious void exists between A and B, that cannot be filled by anything other than a bullet or some other object.
As you no doubt noticed, instead of following the bullet on its journey to the rear wall of the caboose, we chose to dissect the problem in much the same way as others have tackled the Dichotomy paradox -- namely in retracing Homer's steps as he runs to catch the bus. I simply changed things around a bit, but we could just as easily have accompanied the bullet to the rear wall and then demonstrated how it makes contact with the wall of the caboose in the same manner as it leaves the barrel of the pistol in the first place.
Obviously a limited and indefinite number of Planck units separate everything from everything else. Once bridged, the outer energy "shells" that compose the seemingly material surfaces of matter merely bang into one another. A final notation is that any and all points are utterly arbitrary -- in both size and number -- and extend indefinitely in all directions. More importantly, no empty space or gap exists either between two points, or an infinite number (chain) of them. The Homeric proposition simply (and incorrectly) locates two separated points along an indefinitely extended span of connected points, and proposes that some kind of mystical nothingness exists between the points.
How easily our language systems can deceive us into thinking about (and struggling with) mysterious, nonsensical propositions which cannot withstand the most rudimentary scrutiny, and confound us with intense contemplations based on illogical fluff.
We are surrounded by and inundated with, ideas, philosophies, religions and ideologies that, in many cases, possess no more of a factual basis than the foregoing, paradoxical proposition. Yet we sacrifice our entire lives sometimes on their behalf and in their defense. It is for this reason that questions are always superior to their answers, and why our ability to question our questions, and not just our answers, should be counted among the greatest of human attributes.
. . .
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