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I have made some implicit assumptions. I have assumed, for example, that only conscious human beings will read the words I write and that such beings are rational. As a member of that class of being, I assume that I am representative of them, and that all thinking beings learn and think in more or less the same way that I do. Despite all the differences, everyone has that intuitive sense to look beyond what is experienced and represented to what actually is. They understand the common meaning of terms and concepts. There is much more common knowledge than many people care to admit. That's human nature, a part of which is the ability to even deny that such a nature exists.
What could be done with a pure line anyway? Being pure, it would be as unrestricted as possible; it would have no points, no parts; it would extend beyond the conception of it to infinity, with no beginning and no end. It could represent a pure relationship without bothering to consider the things related. The relationship itself could be purified by purging it of the notions of distance, of long or short, even of length. It could represent a continuum. By self reference, it could represent a oneness with self and the infinite.
Such a concept is pointless. Yes that statement is intended to have a double meaning. It has no points, and there appears to be no point, no purpose, in using it. In the sense of the humor that uses "double entendre", or in the more serious sense of analogy, it could have different meanings in different contexts at the same time. A conscious being could vacillate between those contexts, or frames of reference, and could see both the ridiculous and the sublime separated, or joined, by a line. It would not even require vacillation if it were an interface of frames of reference. Opposites, even contradictions, could take their meaning from each other as opposite ends of a continuum. Truth could define lies, as lies could define truth. Happiness could define sadness, as sadness defines happiness. Love could define hate; hate, love. Life could define death; death, life. Limits could define limitlessness; infinity, limits.
Have I succeeded in my second attempt at proving that we could have the same ideas? Did I get your intuitive sense to contemplate itself and its operations by trying to contemplate a pure concept? I know that we do have many of the same ideas. Perhaps you too knew it all along; you just never thought about it, not in just this way. I know that you believe it, too, even if you still will not admit it. Our ability to communicate implies it.
I have one more trick up my sleeve to make one more attempt. To use that trick, however, I must make a confession. I have already confessed that I am resorting to a trick. I must also confess that I play with my thoughts much like a child plays with toys. I had originally intended to try to present the results of my work without playing games. If I could do that, perhaps instead of being considered a foolish child, I might be considered a sophisticated intellectual. But what's in a name anyway?
Starting with the conception of a pure line, I shall pretend that I am a point on that line. Better, I shall pretend that I can send conscious points to represent me and to report back to me. Sending a point to that line might taint its purity, but the first report I got was that the point was embedded in the line and it appeared to cause no distortion of the line: the line was completely compatible with the point and could contain as many points as I might send. Viewing that line from the point's position, which was arbitrarily selected, I got the report that my point did not know where it was on that line, but that the line had length that extended beyond what could be seen.
Asked about how his conclusion went beyond what he could observe, he replied: "Since I could not tell where I was in the line, I marked the position of arrival, then went to the farthest position I could see, and marked that too. I cut a piece of string that I generated to record that distance to make sure I didn't get lost, to make sure I could get back to the position I was sent, so that I could be found and picked up. I did this many times, marking each new position with the same recorded length of string. The view from each position was always the same. I just knew that I could go on forever with the same results. I just knew there would always be something beyond."
I sent other observers to duplicate the expedition. They confirmed the results. They all came to the same conclusion. They noted that they passed other markings, but weren't sure what they meant, except that someone else had been exploring this frame of reference. The pieces of string they returned were not equal to that returned by the first observer, or to those of the others.
How could these observers, using different positions of reference, and different records of length, all come to the same conclusion? I sent more observers with the same results until I realized that I could see beyond what I could do, too. All of us were enriched by the conception of the beyond, the infinity we all just knew existed. No matter how many observers, the results would be the same. Despite the differences in direction, size, or number of steps from which the idea was derived, everyone had the very same idea.
I knew that all along. So did you. If you still wish to deny that our concepts could be the same, there is nothing more I can think of to do. Sooner or later you will recognize and appreciate many things that you already know in your own way, in your own time. That's the truth of the matter. Could the truth of this matter or any other really be that simple? So simple that it is available to everyone in their own way, in their own time? So simple that it could be recognized and understood by agreement, even though it is not even dependent on being observed at all? What a mind boggling implication! Truth can be known by agreement! What a startling result!
So far, I have not gotten too far. I set out in search of the meaning of the "relative measure of lines". In my thought experiment I discovered some interesting things, but I have obtained little more than a confirmation of the intuitive notions of linearity and of measure. Perhaps the embryo of a notion of "relative" was also obtained. All I have to work with are the "strings" returned by my observer points, which are not all equal to each other. They differ in length, yet all of them are almost perfect models of the pure line they were intended to represent.
My conceptual representation of length is then the limits imposed on the infinite linearity of the pure line that resulted from my attempt to understand linearity in anthropomorphic terms; i.e. "as far as the eye can see". The differences in the lengths of the strings was the result of the anthropomorphic differences of the observers who set different limits relative to their ability to observe; i.e., some could see farther than others. Thus the measures of length they used are relative in that sense.
Going one step further, the different lengths representing the relative measures of different observers can be related to each other; i.e., they are relative to each other. In fact, in observing that they are unequal to each other, I used a comparison of the strings, not the observers. I merely explained the cause of the differences in length by referring to the observers. The relative measure of any two lengths could therefore be the intuitive comparative relationship between them, which after all is part of the intuitive notion of measurement.
Still, there is something missing! What is the reference of comparison? Each string is the measure of the others in terms of longer, or shorter. That's all. That's not good enough for exact measurement. What seems to be required is the relative measure of the relative difference of comparison of two relative measures.
The embryos of the concepts of counting and number are present in this context. I have not permitted their birth because I did not want them to be misconceived nor did I want them to mature in an environment wherein they might be deformed by it. It is not absolutely necessary to introduce them at this time, but it might be helpful. Instead of speaking about iterations of equals without number, I could then define the process of one iteration and count the number of times that iteration must be repeated to achieve a desired result.
For example, each of the strings returned by the observer points is just as good as any other one is. That was really not appreciated until after the trial. A trial became necessary because there were accusations that some points misrepresented their observations.
It was assumed before the experiment that all points were equal. It had also been suspected that the greatest fear was the fear of the unknown. That fear seemed to be borne out by the experiment as an explanation for each observer returning a string. They instinctively generated the string out of the apparent universal fear of getting lost; the string established a means of recording their movements.
To insure a fair trial, each observer was instructed to signal when an approaching point came into view: then his string was brought out to compare this result with the one he had recorded as the measure of "as far as he could see". This procedure could at the same time account for the inequalities of the strings as well as it could determine those points that fudged their results.
During the trial, a number of other unexpected differences were observed. Some points could not see an approaching point from one direction, but had no difficulty when approached from the opposite one. Thus some were considered positive points and others negative ones; in either case there was no effect on the comparison of their strings so that the "true", properly recorded strings could be separated from the defective, fudged ones.
Some of the points, having noticed other markings, or having heard rumors that all the strings were not equal, claimed that they fudged the result out of fear of returning a "wrong" string. They did not want to be considered different or defective or inferior. Some of the points claimed that they simply forgot to stop generating the string as they went from one position to another. Since these were accepted as frailties of point nature, they were not banished from the domain, or punished in any way.
Some of the points, however, faced with the evidence, admitted that they fudged in order to be considered superior to the other points. This was not as readily accepted as a frailty of point nature, and some punishment, short of banishment, seemed appropriate. The attorney for the defense then presented the following argument:
Since the fudging did not account for all of the inequalities of the strings, the motives for fudging were not relevant. They could not be guilty , as charged, of misrepresenting the pure line, even if that were their intention. They could merely exaggerate the differences found in such representations: each of the lines was just as good as any of the others; the only difference was that they were represented by different limits. They could not be blamed for the inability of the court to explain the remaining differences. They could not be blamed for the ignorance of the court. Each line represented the linearity and infinity of a pure line; lengthening the string by repetition they could represent an infinity of largeness in either direction. Even by shortening the string they could do so without end so long as there was a finite length - after which they could represent an infinity of smallness - which any point already represented by its nature of non-dimensionality.
He was right of course. All the accused points had to be acquitted.
The prosecutor was not happy. He brought a new charge against one of the points, the one that had returned the longest string. The charge was pretense of superiority. Everyone was not in agreement that such a trial should take place, but the prosecutor argued that the point that returned the longest string had to be the most pretentious and should be used as an example. Such pretense had to be dishonest, and it should be made clear that such behavior was frowned upon. The trial was allowed.
The accused appeared confused by the proceedings. He did not respond to an approaching point until it was almost or actually touching him. Assuming he was a negative point rather than a positive one, the test was repeated from the opposite direction with the same result. Could he be a neutral point? He was the first to be able to respond to a point from either direction without turning around, but the approaching point had to be so close it was unbelievable that he could have generated the longest string. He did not appear pretentious at all; he seemed to be trying his best.
A motion to dismiss the charge on the basis of his incompetence to stand trial was denied, and the accused was very upset by it. He asked for the opportunity to prove himself by repeating his exploration of a pure line, which he did twice, in opposite directions, producing strings that exactly matched the original one. The court was perplexed.
It was finally determined that he was blind. How that was determined is an interesting story in itself because it was a totally unknown condition up to that time. The blind point had interpreted the command to move as far as he could see in the only way he could understand it, by moving as far as he could until he got tired. His system for keeping track of his movement turned out to be another important discovery that later became known as counting. It came about in the following way:
When he was born it was thought that he was a stillborn - a point that could, or would, never move - a fixed point. At first he was afraid to move for fear of getting lost. Then he discovered by alternately representing himself as larger and smaller, by stretching the point that he was, he could move in the manner of an inchworm. He kept track of how far he moved by creating names for each step: e.g., 1, 2, 3, ... et cetera. He realized that he did not have to generate a string at all; the last name or number he used was as exact a measure of the string, as the string was itself. He continued to do so because that's what points did, and it was useful to leave a trail that others could see even if he could not. Not only could such a number line represent the linearity of a pure line, it could also represent its infinity of largeness, since new names could always be conceived. The infinity of smallness is also retained in the ever diminishing representation of dimension until the point disappears into a motionless, dimensionless representation of itself.
Of course, all charges against that point were dropped. Instead of receiving the expected punishment, he was honored as the father of number systems. It is somewhat of a shame that he is no longer remembered by all those who have used, and continue to use, his discovery. Such is the fleeting nature of fame. Perhaps, had he not been forgotten, some of the misapplications of number to geometry would not have taken place, and there would be no disagreement with my third proposition. In the light of these discoveries, I shall now review the mathematician's conclusion in terms of simple geometric analyses that yield some interesting results.
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