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E S S A Y S
By The Number
Fig. 1.
e64
ON the AMBIGUITIES of LANGUAGE
A Probing Investigation into the Process of Questioning
Paradoxes
1. A logical statement that contradicts itself.
"I always lie" is a paradox because if it is true, it must be false.
2. A statement starting with something that is apparently true that leads to counterintuitive or unacceptable conclusions.
3. A person or thing having contradictory properties.
"I always lie" is a paradox because if it is true, it must be false.
2. A statement starting with something that is apparently true that leads to counterintuitive or unacceptable conclusions.
3. A person or thing having contradictory properties.
To briefly borrow from Wikipedia, "Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (ca. 490-430 B.C.) to support Parmenides's doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. Some of Zeno's nine surviving paradoxes are essentially equivalent to one another. Aristotle offered a refutation of some of them. Three of the strongest and most famous are that of Achilles and the tortoise, The Dichotomy argument, and that of An arrow in flight."
Because Aristotle recounted the Dichotomy paradox in particular, it is presented here in a highly paraphrased condition and serves as a basis only for what I have personally de-contructed and reassembled in my own words.
As defined, a paradox is a statement that seemingly contradicts itself and yet appears to be true. Most logical paradoxes are known to be invalid arguments, but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in certain definitions which were otherwise presumed as rigidly accurate, and have caused axioms of mathematics and logic to be re-evaluated. One example is Russell's paradox, which questions whether or not a "list of all lists that do not contain themselves" would include itself. The problem demonstrated that the numerical relationships described by set theory were flawed. Others, such as Curry's paradox or Aristotle's Dichotomy paradox, are apparently still unresolved.
Because Aristotle recounted the Dichotomy paradox in particular, it is presented here in a highly paraphrased condition and serves as a basis only for what I have personally de-contructed and reassembled in my own words.
As defined, a paradox is a statement that seemingly contradicts itself and yet appears to be true. Most logical paradoxes are known to be invalid arguments, but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in certain definitions which were otherwise presumed as rigidly accurate, and have caused axioms of mathematics and logic to be re-evaluated. One example is Russell's paradox, which questions whether or not a "list of all lists that do not contain themselves" would include itself. The problem demonstrated that the numerical relationships described by set theory were flawed. Others, such as Curry's paradox or Aristotle's Dichotomy paradox, are apparently still unresolved.
The Dichotomy Paradox
Part One
As originally recited by the Greek mathematician and philosopher, Aristotle, Zeno's Dichotomy paradox states: That which is in locomotion must arrive at the half-way stage before it arrives at the goal. The paradox is among those which have (apparently) never been satisfactorily explained. Until now? Yes, I understand how presumptuous it is for me to pretend that I've achieved new insights into some very old problems, but with a zany moniker like "Zeno's paradoxes", how could any writer (or thinker) resist? Especially when such stuff is right up my own wacky alley, so to speak.
Many variations of this particular puzzle are in circulation; the proposition is such that any number of different versions might be used to illustrate the deceptively simple nature of the problem involved. One of the most common examples imagines the fictional character of Homer as he races to catch a stopped bus before it leaves. In order to reach the vehicle, he must first travel half the distance between the bus and himself. Thereafter, he must traverse half the remaining distance, then cross half of the next leftover span, then half of the next, over and over again. The actual paradox is phrased in a manner that forces Homer to work backwards, meaning he never makes it past the first half of his journey, which is broken into sequentially smaller increments. As long as any distance remains between Homer and the bus, however, he must travel half of what there is, as he moves forward. Pretty straightforward, right? Only problem is that, in theory, poor Homer never reaches the bus; no matter how you slice it, or how close he gets, there's always just enough empty space to be cut by half. Right? That's precisely what we're here to find out.
This essay is probably my all-time favorite. The main reason, of course, is because yours truly believes (in his own warped way) he has solved it, yet without credible creds to my credit, it's likely that no one will take the time to read this. Maybe you'll take a moment or two and see what you think. Secondly, however, this one essay encapsulates a wonderful analysis of how language works, how it's at one and the same time the best tool we have, and yet the least efficient when it comes to communicating with one another. This is one of those dirty little secrets, in a manner of speaking, if it ever came to any kind of meaningful dialogue with an alien, extraterrestrial race. Meaning we can barely understand each other, let alone the thoughts and feelings of the ultimate foreigners.
When I first heard of this paradox, many years ago, it was introduced to me as a scenario whereby a person stood holding a rock in their hand, then released their grip and dropped the stone to the ground. Naturally the rock falls and hits the dirt -- except it shouldn't. And can't. Not logically. So instead of Homer rushing to catch a bus, let's make things really simple and catch up with him while he's still at the bus stop, standing and waiting. The train tracks aren't far away and the railcar we used earlier (essay #62 Kaboom in the Caboose) is still available; everybody took off when they heard gunshots.
Once we've got Homer inside the caboose, his back against the forward wall, we hand him a gun and ask him to shoot a bullet into the rear, opposite wall of the caboose. Which he does, no problem. It doesn't get much more simple than that, right? No, wrong. If you've been paying attention, you now know that the bullet never reaches the wall, doesn't even nick it, because it couldn't have. This fella, Zeno, told us it wasn't possible and he must have been pretty smart. Let alone Aristotle, the Man, who gave the problem his own shot, so to speak. So what gives? There's a fresh bullet hole in the rear wall of the car, still smoking, yet logically, even mathematically, there shouldn't be. This particular thought experiment and logic problem drove me nuts for a long time. Since no calculus, trigonometry, or other higher math seemed to be the sole solution (I can barely do arithmetic), I wanted to solve the damn thing once and for all.
The so-called "Planck Length" represents the present model for dealing with the smallest of things, which is at the very core of this essay. But that's getting way ahead of ourselves and I'll mention a little more (no pun intended) about Planck units later on. Meanwhile, I had the distinct feeling that reasoning alone could solve the riddle, at least superficially, and we'd let the mathematicians do the equations some other time. Thus far, it seemed that folks a lot smarter than I had tackled the paradox and failed to resolve it, so who was I to come along and think I could achieve what others hadn't? Or couldn't. Well, I gave it a good run for its money, and after a year or so of grappling with both numbers and logic, here are my notes and answers from twenty years ago. If your uncle or aunt is a mathematician, please pass this along and have them verify my results. Thanks.
Many variations of this particular puzzle are in circulation; the proposition is such that any number of different versions might be used to illustrate the deceptively simple nature of the problem involved. One of the most common examples imagines the fictional character of Homer as he races to catch a stopped bus before it leaves. In order to reach the vehicle, he must first travel half the distance between the bus and himself. Thereafter, he must traverse half the remaining distance, then cross half of the next leftover span, then half of the next, over and over again. The actual paradox is phrased in a manner that forces Homer to work backwards, meaning he never makes it past the first half of his journey, which is broken into sequentially smaller increments. As long as any distance remains between Homer and the bus, however, he must travel half of what there is, as he moves forward. Pretty straightforward, right? Only problem is that, in theory, poor Homer never reaches the bus; no matter how you slice it, or how close he gets, there's always just enough empty space to be cut by half. Right? That's precisely what we're here to find out.
This essay is probably my all-time favorite. The main reason, of course, is because yours truly believes (in his own warped way) he has solved it, yet without credible creds to my credit, it's likely that no one will take the time to read this. Maybe you'll take a moment or two and see what you think. Secondly, however, this one essay encapsulates a wonderful analysis of how language works, how it's at one and the same time the best tool we have, and yet the least efficient when it comes to communicating with one another. This is one of those dirty little secrets, in a manner of speaking, if it ever came to any kind of meaningful dialogue with an alien, extraterrestrial race. Meaning we can barely understand each other, let alone the thoughts and feelings of the ultimate foreigners.
When I first heard of this paradox, many years ago, it was introduced to me as a scenario whereby a person stood holding a rock in their hand, then released their grip and dropped the stone to the ground. Naturally the rock falls and hits the dirt -- except it shouldn't. And can't. Not logically. So instead of Homer rushing to catch a bus, let's make things really simple and catch up with him while he's still at the bus stop, standing and waiting. The train tracks aren't far away and the railcar we used earlier (essay #62 Kaboom in the Caboose) is still available; everybody took off when they heard gunshots.
Once we've got Homer inside the caboose, his back against the forward wall, we hand him a gun and ask him to shoot a bullet into the rear, opposite wall of the caboose. Which he does, no problem. It doesn't get much more simple than that, right? No, wrong. If you've been paying attention, you now know that the bullet never reaches the wall, doesn't even nick it, because it couldn't have. This fella, Zeno, told us it wasn't possible and he must have been pretty smart. Let alone Aristotle, the Man, who gave the problem his own shot, so to speak. So what gives? There's a fresh bullet hole in the rear wall of the car, still smoking, yet logically, even mathematically, there shouldn't be. This particular thought experiment and logic problem drove me nuts for a long time. Since no calculus, trigonometry, or other higher math seemed to be the sole solution (I can barely do arithmetic), I wanted to solve the damn thing once and for all.
The so-called "Planck Length" represents the present model for dealing with the smallest of things, which is at the very core of this essay. But that's getting way ahead of ourselves and I'll mention a little more (no pun intended) about Planck units later on. Meanwhile, I had the distinct feeling that reasoning alone could solve the riddle, at least superficially, and we'd let the mathematicians do the equations some other time. Thus far, it seemed that folks a lot smarter than I had tackled the paradox and failed to resolve it, so who was I to come along and think I could achieve what others hadn't? Or couldn't. Well, I gave it a good run for its money, and after a year or so of grappling with both numbers and logic, here are my notes and answers from twenty years ago. If your uncle or aunt is a mathematician, please pass this along and have them verify my results. Thanks.
The Basic Proposition:
In order that something travel from a position designated as point A and arrive at a further designated point B, between both of which a space or gap exists, it must first move one-half the distance between the two points, A and B. This sounds stupidly obvious, I know, but give it a chance; this does get interesting. As I was saying, once having reached the half-way location, the something in motion (in this case a bullet) must again traverse one-half the remaining distance between itself and point B (in this case the rear wall of the caboose). Upon reaching this half-way point and still on its way to arriving at point B, the bullet must yet again move through one-half the remaining distance -- which is comprised of whatever space still exists between itself and point B. And so on and so forth.
Assuming that until such time as point B is physically contacted (struck), a gap -- regardless of how small -- exists between point B and the bullet moving towards it. Further, that if this remaining space can always be divided by one-half, then a paradox would occur wherein point B can never be reached. A bullet or anything else cannot travel from point A and arrive at point B because a seemingly infinite series of diminishing gaps would exist as an object approached ever closer to its destination point of B. Any remaining space between the approaching bullet and point B could, in theory, always be divided by one-half. By the way, it's understood that other increments such as quarters, eighths, sixteenths and so forth are involved, but to keep things as simple as possible, we'll take the paradox literally and deal only in halves.
Therefore a primary example of this enigma which suggests that nothing can ever get anywhere is "henceforth" (that makes it sound official) demonstrated by the discharging of a bullet from a pistol (a rifle is also okay). The Basic Proposition attempts to illustrate logically that a bullet thus fired, should not -- indeed cannot-- hit the rear wall of the caboose because it must first travel half-way there, then half the distance which remains, then half of that, ad infinitum.
Assuming that until such time as point B is physically contacted (struck), a gap -- regardless of how small -- exists between point B and the bullet moving towards it. Further, that if this remaining space can always be divided by one-half, then a paradox would occur wherein point B can never be reached. A bullet or anything else cannot travel from point A and arrive at point B because a seemingly infinite series of diminishing gaps would exist as an object approached ever closer to its destination point of B. Any remaining space between the approaching bullet and point B could, in theory, always be divided by one-half. By the way, it's understood that other increments such as quarters, eighths, sixteenths and so forth are involved, but to keep things as simple as possible, we'll take the paradox literally and deal only in halves.
Therefore a primary example of this enigma which suggests that nothing can ever get anywhere is "henceforth" (that makes it sound official) demonstrated by the discharging of a bullet from a pistol (a rifle is also okay). The Basic Proposition attempts to illustrate logically that a bullet thus fired, should not -- indeed cannot-- hit the rear wall of the caboose because it must first travel half-way there, then half the distance which remains, then half of that, ad infinitum.
A Question:
Obviously the bullet strikes the wall, and all manner of other things go from here to there all the time. How is it, though, that in light of the foregoing Proposition, things are able to go from points A to points B? How can objects move from here to there, from there to here, and how is the paradox -- as presented within the context of the Proposition -- resolved?
One Answer:
Before explaining the exact circumstances which solve the puzzle (to my satisfaction), it is first necessary to explore the subtleties, limitations, and deceptive qualities of language itself. The paradox in question is first and foremost a product of language -- both the spoken and written forms of thoughts and words. An understanding of all or most of the elements included within a stated position or question is crucial to comprehending that such things as contradictions, hypocrisies, or paradoxes even exist.
A paradox might exist for two reasons only: One, that a true contradiction exists and is based upon one or more tested, proven, and infallible premises. Or two, that an apparent paradox seems to exist, but is actually the result of a flawed proposition and an incorrect questioning process.
"Why did the teeth of the toothless dinosaur run slowly forwards, then rapidly back into the rocky clouds and moist dryness?"
A paradox might exist for two reasons only: One, that a true contradiction exists and is based upon one or more tested, proven, and infallible premises. Or two, that an apparent paradox seems to exist, but is actually the result of a flawed proposition and an incorrect questioning process.
"Why did the teeth of the toothless dinosaur run slowly forwards, then rapidly back into the rocky clouds and moist dryness?"
The only thing wrong with the above interrogatory sentence is that it simply doesn't make any sense. Although an extreme example, this serves as an indication that language is based upon something other than grammar and sentence structure. Rational thinking demands a coherent thought regimen and we are led astray by varying degrees of our inability to use language or express in either verbal or written form, exactly what we are thinking. The precise nature required of answers is exceeded only by that which should be considered absolutely critical, of questions.
"If a person's unlisted phone number is listed at the phone company, and contained in a list of unlisted numbers, is the person's unlisted number really a listed, or unlisted number?"
"If one single copy of everything that's ever been written was contained within a master catalog, should the catalog contain a copy of itself?"
"If a tree falls in the forest, and no one is there to hear it, does it make any sound?"
This last question about a tree in the forest is somewhat similar to the original proposition posed earlier. For this reason it can be briefly analyzed and serve as a preparatory examination of the original problem.
First, the question contains a contradiction of terms within its own framework. This should be noted prior to any effort in acquiring an answer. The contradiction lies in the use of the terms hear and sound. The word sound is an after-the-fact event which implies and assumes that something has created a noise which is audible. The question doesn't propose that the tree produced wave-vibrations which, if picked up by an ear (or recording device), could be interpreted or transduced as sound, but rather states clearly within in its own context, that the actual cause of the sound is understood to exist for some unstated reason. The real query is not whether the falling tree makes any sound, but instead whether anyone could hear it, even if they are not in the forest. Thus the question is worded in such a way as to make little or no sense, disguised by a lingual veneer of comprehensibility.
In defining the word sound scientifically, as well as what it means to hear something, an approximate answer is possible given that the question is re-worded properly: "If a tree falls in the forest, does it produce wave-vibrations of air molecules which could, if a person (or recording device) were nearby, be heard as sound?"
It is important to observe that the contents of a question -- its rational purity -- is paramount. Prior to any investigation which seeks answers, should be an exhaustive dissection of both the question and the questioning process itself. It is understood that certain subatomic particles behave as waves under one condition, and as particles under other condtions. One insight into this conundrum arose from a determination that the observing process itself influenced the results. It is as though when looking directly for a specific particle, the particle appears to exist as a wave, and when looking directly for wave responses, physicists observe particle characteristics. Using this analogy, we should be cautious in our own, more mundane methods of observation and equally if not more so, our questioning processes.
How a scientific experiment is arranged and organized affects the observed results. And how a question is structured and worded may well determine whether it favors a plausible answer, or an insoluble paradox. With this in mind, it is now be possible to review the proposition and discover whether the problem yet has a tenable answer or solution. It is not necessary to attempt rewording the proposed scenario as it is understood that bullets shot from guns do indeed strike their targets; it is simply my intention to illustrate the fallacious nature of the overall premise involved.
The combination of the proposition and the questions raised make about as much sense as, "How long is a piece of string?" The somewhat silly answer, of course, is that its length is "from one end to the other." The question, in one respect, asks, "How long is one-half of a piece of string?" To which we can say it is "one-half the distance from one end of the string to the other." The Basic Proposition, in presenting a flawed, vague, and ambiguous premise that assumes its own integrity and authority, produces questions which are proportionately even more vague and more ambiguous. Other than an exercise in superficial, linear reasoning, all of it is utter gobbledygook.
The proposition of Homer in the caboose fails to present a case which can be substantiated by meaningful and or provable assertions of logic. On its surface, however, it would appear to be quite reasonable and an inquiry that defies immediate explanation. This is the danger and entrapment that lies poised within the structure of language, and which can easily create powerfully confusing presentations, jarring our usual abilities of understanding.
"If a person's unlisted phone number is listed at the phone company, and contained in a list of unlisted numbers, is the person's unlisted number really a listed, or unlisted number?"
"If one single copy of everything that's ever been written was contained within a master catalog, should the catalog contain a copy of itself?"
"If a tree falls in the forest, and no one is there to hear it, does it make any sound?"
This last question about a tree in the forest is somewhat similar to the original proposition posed earlier. For this reason it can be briefly analyzed and serve as a preparatory examination of the original problem.
First, the question contains a contradiction of terms within its own framework. This should be noted prior to any effort in acquiring an answer. The contradiction lies in the use of the terms hear and sound. The word sound is an after-the-fact event which implies and assumes that something has created a noise which is audible. The question doesn't propose that the tree produced wave-vibrations which, if picked up by an ear (or recording device), could be interpreted or transduced as sound, but rather states clearly within in its own context, that the actual cause of the sound is understood to exist for some unstated reason. The real query is not whether the falling tree makes any sound, but instead whether anyone could hear it, even if they are not in the forest. Thus the question is worded in such a way as to make little or no sense, disguised by a lingual veneer of comprehensibility.
In defining the word sound scientifically, as well as what it means to hear something, an approximate answer is possible given that the question is re-worded properly: "If a tree falls in the forest, does it produce wave-vibrations of air molecules which could, if a person (or recording device) were nearby, be heard as sound?"
It is important to observe that the contents of a question -- its rational purity -- is paramount. Prior to any investigation which seeks answers, should be an exhaustive dissection of both the question and the questioning process itself. It is understood that certain subatomic particles behave as waves under one condition, and as particles under other condtions. One insight into this conundrum arose from a determination that the observing process itself influenced the results. It is as though when looking directly for a specific particle, the particle appears to exist as a wave, and when looking directly for wave responses, physicists observe particle characteristics. Using this analogy, we should be cautious in our own, more mundane methods of observation and equally if not more so, our questioning processes.
How a scientific experiment is arranged and organized affects the observed results. And how a question is structured and worded may well determine whether it favors a plausible answer, or an insoluble paradox. With this in mind, it is now be possible to review the proposition and discover whether the problem yet has a tenable answer or solution. It is not necessary to attempt rewording the proposed scenario as it is understood that bullets shot from guns do indeed strike their targets; it is simply my intention to illustrate the fallacious nature of the overall premise involved.
The combination of the proposition and the questions raised make about as much sense as, "How long is a piece of string?" The somewhat silly answer, of course, is that its length is "from one end to the other." The question, in one respect, asks, "How long is one-half of a piece of string?" To which we can say it is "one-half the distance from one end of the string to the other." The Basic Proposition, in presenting a flawed, vague, and ambiguous premise that assumes its own integrity and authority, produces questions which are proportionately even more vague and more ambiguous. Other than an exercise in superficial, linear reasoning, all of it is utter gobbledygook.
The proposition of Homer in the caboose fails to present a case which can be substantiated by meaningful and or provable assertions of logic. On its surface, however, it would appear to be quite reasonable and an inquiry that defies immediate explanation. This is the danger and entrapment that lies poised within the structure of language, and which can easily create powerfully confusing presentations, jarring our usual abilities of understanding.
One Solution
Fig. 2. The illusory nature of "points".
The Basic Proposition (Homer in the caboose) begins and ends with casual references to points, as if points were something real. As soon as one questions what and where these points are located, the structure and integrity of the proposition rapidly collapses. How big or small is a point? If we assume for the sake of argument that points are allowed to represent a position in space, a point can be said to be indefinitely large or indefinitely miniscule. A point cannot, however, be infinitely huge or infinitely minute because the entire universe could then be thought of as a single point, and nothing could theoretically move from one point to another. Motion (movement) requires, as a minimum, the ability to travel one point in any direction. In a three dimensional world, the minimum number of points which allow maximum directional movement is twenty-seven.
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
front view top view side view
This is equal to three vertical tiers of nine equidistant points each, which from center, allow a one point movement in all directions. It is important to note that no gap or space exists between or among the points and that whether they are indefinitely large or small, the same minimum number of points applies equally. The proposition ignores this level of evaluation and neatly assumes it is not subject to question. Had the word, "zenomorph" been substituted for "point", the reader could not have been more diverted. As used in the first sentence of this paragraph, the term equidistant refers only to the uniformity of point alignments; also as mentioned, no empty space whatsoever exists between one point (dot) and another; they are all theoretically "touching".
Special note: For the purists in the crowd, it might be more accurate to describe the points as not actually touching, but rather still separated by a space so small that for all practical purposes, it is the same as nothing. Called a Planck Length, or Planck unit, it is the only terminology available -- via the limitations of language -- to describe how two or more things can be so close together, that to be any closer would mean occupying the same space. That's not only a mouthful, but a brainful also. Think of a Planck unit as being so tiny that it is no longer divisible -- not by half, not by a quarter or any other "slice". As used in this essay, points themselves are likely individual Planck units, but for clarity, I just think in terms of depleting a finite number of points that obviously separate (or connect) one thing from (to) another. So does that make everything a lot clearer? I figured it would. More on this Max Planck person later on.
Another reason that points cannot be indefinitely large or small is due in part to the conflict which arises from disparate infinities. One infinity cannot reasonably be larger or smaller than another infinity, as it would be a contradiction of terms. If an infinite number of points existed between A and B, equal to a certain (finite) distance between them -- and a further distance was then added away from A, which we will call D, the following diagram illustrates the case:
A >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> C >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> B
barrel / bullet half-way to rear wall rear wall of caboose
A >>>>>>>>>>>>>>>>>>> C >>>>>>>>>>>>>>>>>>>> B .>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> D
rear wall of caboose
How is it that the infinity between A and B is equal to the infinity of A and D? Or that the infinity of A and D is greater than the infinity of A and B? The answer, of course, is that by their very nature, infinities are usually always equal -- especially when dealing with finite distances and indefinitely small point characteristics. In some instances it can be demonstrated that certain infinity-series sets contain larger or smaller infinities, but these are mathematical abstractions which do not apply here. However, the relevance, meaning, and differences between infinite and indefinite are critically important to this discussion.
We thus have a situation whereby an indefinite but finite number of points exist between A and B, while a still larger number of indefinite points exist between A and D. Again it is crucial to realize that these points are connected together, with no gap or space between them. Any or all spaces are filled or occupied by the same indefinite number of points -- until such time that one point physically touches another, or if any closer, would occupy the same space as another.
The Dichotomy Paradox continues on page NOU21.
Special note: For the purists in the crowd, it might be more accurate to describe the points as not actually touching, but rather still separated by a space so small that for all practical purposes, it is the same as nothing. Called a Planck Length, or Planck unit, it is the only terminology available -- via the limitations of language -- to describe how two or more things can be so close together, that to be any closer would mean occupying the same space. That's not only a mouthful, but a brainful also. Think of a Planck unit as being so tiny that it is no longer divisible -- not by half, not by a quarter or any other "slice". As used in this essay, points themselves are likely individual Planck units, but for clarity, I just think in terms of depleting a finite number of points that obviously separate (or connect) one thing from (to) another. So does that make everything a lot clearer? I figured it would. More on this Max Planck person later on.
Another reason that points cannot be indefinitely large or small is due in part to the conflict which arises from disparate infinities. One infinity cannot reasonably be larger or smaller than another infinity, as it would be a contradiction of terms. If an infinite number of points existed between A and B, equal to a certain (finite) distance between them -- and a further distance was then added away from A, which we will call D, the following diagram illustrates the case:
A >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> C >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> B
barrel / bullet half-way to rear wall rear wall of caboose
A >>>>>>>>>>>>>>>>>>> C >>>>>>>>>>>>>>>>>>>> B .>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> D
rear wall of caboose
How is it that the infinity between A and B is equal to the infinity of A and D? Or that the infinity of A and D is greater than the infinity of A and B? The answer, of course, is that by their very nature, infinities are usually always equal -- especially when dealing with finite distances and indefinitely small point characteristics. In some instances it can be demonstrated that certain infinity-series sets contain larger or smaller infinities, but these are mathematical abstractions which do not apply here. However, the relevance, meaning, and differences between infinite and indefinite are critically important to this discussion.
We thus have a situation whereby an indefinite but finite number of points exist between A and B, while a still larger number of indefinite points exist between A and D. Again it is crucial to realize that these points are connected together, with no gap or space between them. Any or all spaces are filled or occupied by the same indefinite number of points -- until such time that one point physically touches another, or if any closer, would occupy the same space as another.
The Dichotomy Paradox continues on page NOU21.
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