by Rolf Witzsche

There is a difference between being told about the truth and knowing the truth on the basis of one's understanding of the underlying principles that it represents. Allow me to illustrate this difference. Let us consider the case of one of our modern students being taught in school a truth that Pythagoras had discovered a long time ago. The student would be told that Pythagoras discovered that the surface area of the squares over the short sides of any right angled triangle, added together, are always equal to the surface area of the square over its large side, regardless of the ration between its shorter sides A and B. Today's students are taught the equation of this discover as an element of their curriculum in algebra. The students take this knowledge in and write it down. They may even apply the formula to a few examples, but do they really KNOW anything as the result of this kind of teaching? Sure, they can apply the formula to all kinds of uses, but do they really know its truth and understand the underlying universal principles? No, they don't! What the students experience in such a course of teaching is similar to a what a traveler experiences, who steps into an Airplane in Miami and steps out of it in Buenos Aires, having skipped over everything in between. In educational terms, this is a tragedy. The tragedy that is involved in the above type of 'teaching and learning' that skips over everything in between, is immense. It skips over the exploration of the universal principles that lays the foundation for all true knowledge. The fact is, we live in divine universe that operates exclusively on a platform of divine Principle and its countless reflection in that universe. The fact is that nothing is understood in any fundamental way of understanding, if this understanding is not grounded in a knowledge of the universal principles that are reflected in everything that is true. This recognition is important, because there exists not a multitude of different principles for a given aspect, like economics for instance, that one might choose from at will. There exists only one principle of economics, and the welfare of humanity depends on its understanding of this singular, universally applicable, principle. The economic mess that we find in the world right now testifies to the fact that a deep seated awareness of universal principle no longer a part of our culture. Indeed, similar experiences can be encountered in almost any department of life in today's world. The fact is, society has drifted far away from thinking in terms of universal principles. This rather sad fact was recently illustrated by an experiment a few young political activists performed in Detroit. They set up a table and a white-board and challenged anyone who would listen to step up and illustrate the process for doubling a square. A new square was to be created from a given square, with double the surface area of the original square. In order to get people interested, they offered $100 to the first person who could do it. Lots of people lined up for a chance to earn the money, but no one could present the solution. Not even a mathematics professor could. The professor could write down the formula, but he couldn't do it physically to prove it. The money was never paid out. Ironically, the money would have been paid out 2400 years earlier to a slave boy who had accomplished exactly this particular task, and did so without any formal education to his credit. The bottom line is that we have lost the ability to understand what an uneducated slave boy had been perfectly able to understand more than 24 centuries ago. Is it any wonder then, that our world is in such a mess? Actually, we have not really lost that ability. We just think we can't do it. We give up before we even try. The slave boy didn't have that luxury. The story of the slave boy that I am talking about is the one that appears in Plato's Meno dialog. Menon was a friend of Socrates and a wealthy young nobleman. One day the two men's conversation centered on the exploration of 'virtue.' Menon requested Socrates that he would teach him. Socrates replied that there is no such thing as teaching. There is no need for it. In order to prove his point he asked Menon to call up anyone of his servants for an experiment, and then observe if that person would be learning anything from him. A slave boy was chosen. Socrates drew a diagram like the one below, and asked the boy a number of questions about it that would prompt the boy to explore in his own way the principles reflected in structures that are made up of squares. For instance, Socrates asked the boy to consider if this is a four cornered space, and if all lines associated with this space are equal. The boy's answer was that this is so. Then Socrates asked him to consider if such a space might be made larger and smaller, and so forth. The boy recognized that the space is "four feet" big, according to size Socrates defined for it, with each side being two feet long as shown in the diagram.
So, the boy was asked to consider what would happen if one side of the structure were to be made only half as long. The boy discovered that the surface area would also be half in size, and the resulting structure wouldn't be a square anymore.
He was further asked to consider what would happen if one of the sides of the original structure were to be doubled in length. The boy discovered that the resulting surface area would be double in size, eight square feet, and again, the resulting structure wouldn't be a square anymore. At this point Socrates asked the boy if a space could be crated that is double in size of the original area, which would be eight square feet big as the boy had already discovered, but which retains the shape of the square. Socrates also asked, "how long would its sides have to be?" The boy considered what happens if both sides of the original square are doubled in length. He discovers that the area of the resulting structure is then four times as large, sixteen square feet, much too large, although it retains its shape as a square.
Next the boy discovers that if one were to make the sides of the new square only one and a half times as big as the original, the resulting surface would be smaller, but would it be eight square feet big? He discovers that it would be nine square feet big, not eight, so that this kind of an approach to solving the problem doesn't work. Socrates asked him in predicament to try a new approach that incorporates some of the principles he has discovered. He asked the boy to get back to the original diagram of the space that is four square feet big. Then he asked the boy that he ignoring the original dividing lines, and that he then double the length of the sides of the original square to create a space that is four times as big; the principle for which as he had discovered.
Next he suggests to the boy that he draw a mew dividing line (diagonally) across each of the four squares, from corner to corner, dividing everyone the squares in half. He suggests that he do it in such a manner that the dividing lines themselves form a square. The boy recognized that the new square is half the size of the whole, and that this process does indeed create a square that is double in size of the original area. He recognized that the original area contains two triangles, while the new square contains four. The boy recognized that this process provides the answer that Socrates was looking for. The process creates a space that is four times as big as the original one, which, when divided into half, creates a space that is two times as big as the original one, which doubles the original square. The boy recognized that the original square contains two triangles (one green and one blue) and that the new square contains four triangles (all green). Consequently he could say with absolute certainty, as an absolute truth, that the new square is double in size. Thus, the boy recognized that he succeeded in doubling the square and understood the principle involved. He understood that process without any teaching being involved. He understood that the doubling results from the process of dividing along the diagonal, which he may have recognized to be yet another universal principle. Now lets get back to solving the puzzle that Pythagoras has but on the table. The puzzle involves developing the proof that in every case of a right angled triangle the sum of the A-square, and the B-square is equal in size to the area of the C-square. Can this be done? It took me the better part of an hour to figure this out. Yes, it can be done. It can be done by utilizing some of the principles illustrated in the Meno dialog. The boy developed the solution for doubling the square by working with four squares that made the original construction four times as big, which he then divided into triangles.
I reasoned that I could use the same principle and adapt it to working with four rectangles, that too, can be divided along the diagonal into triangles.
The boy had created a new square that was made up entirely of the diagonals. I realized that I could do the same with rectangles.
I also realized that the diagonal represents the side C of my triangle, and that the new square that is being created by the diagonals has the surface area of the C-square.
I realized also that the slave boy solved his puzzle, to double the square, by creating a space four times as big, and then dividing it in half. I wondered if I could utilize this principle. For instance, if I were to create a space twice as big as C-square, would the resulting shape be such that I could fit into this double space two complete A-squares and two B-squares. Well, I gave it a try to test the 'hypothesis.'
I realized that the blue triangles are of the same size as the dark triangles within the C-square. I also realized that the whole construction would therefore be double in size were it not for the qwerky square in the middle that isn't a part of the triangles.
It became obvious that if I wanted to create an area twice the size of C-square I would have to add an area of equal shape as the qwerky square, to the lager form, which I did shown above. As you can see, the blue area is now exactly equal in size to the C-square (pink and brown), so that the combined area of the whole thing is twice as big as the C-square. My proposition is, that if I can fit two A-squares and two B-squares into the larger area, with nothing left over in any way, I have delivered proof beyond the shadow of a doubt that the area of C-square, which is half as big as the whole thing is equal in size to a single A-square and a single B-square. So, let's test the proposition.
As you can see, it is certainly possible to fit a single B-square (with both sides as along as the side B of the triangle) into the larger area.
It is also possible to fit another B-square into it, except that this second B-square overlaps the qwerky area that is already covered by the first B-square. It so happens that this overlapping poses no problem, because we have an area set aside that is equal in size to the overlapped space.
Now that we have fitted the two B-squares into this area, there remains just enough space for two A-squares (shown pink) to be added.
The end result is that we have proven, that the area of the entire construction is exactly of the size required to fit two B-squares into it, and two A-squares. This proves Pythagoras' theorem which states that the area of C-square, which in our case is half as big as the entire area of the contraction above, is equal in size to the sum of a single A-square and a single B-square, exactly as Pythagoras has stated (below). I also realized that this proof applies to every kind of right triangle that one cares to create, regardless of the ratio of side A to side B, because the principle of that proof remains the same in every case. The proof can be seen visually in one's own mind. Just count the triangles and the center areas. I do realize of course, that without my prior understanding of the principles illustrated in the Meno dialog for doubling the square, and the process of building on these principles, the illustrated proof of Pythagoras' theorem would likely not have been created by me in the manner illustrated above. While there are more 'elegant' ways possible for doubling the square, the one that Plato presented with the Meno dialog opens the gateway for the next step ahead. It becomes the more powerful solution. That the ideal method in the process for discovering truth. Still, I find it amazing that all of this was already known 2500 years ago when Pythagoras and his society of advanced thinkers made these amazing fundamental discoveries in app. 500 B.C.. The tragedy that we face today by no longer thinking in terms of universal principles is illustrated by the above exploration, which involves the application of principles that are built on other principles. This building process simply cannot occur if we don't focus our efforts on discovering universal principles in the search for what is absolute truth in a way that is knowable. When the ongoing mental disassociation from universal principles, in today's world, we become disassociated from the universe in which we live, which is a divine universe operating on a platform of divine Principle manifest in countless different universal principles. In this universe of principles and absolute truth, nothing is arbitrary. All reflects lawfully the operation of these universally operating, natural principles. These universal principles, of course, are principles that cannot be overruled by the arrogance of human will, or by legislation. No legislator can degree that the principles that Pythagoras discovered will no longer apply, although a lot of legislators arrogantly do this kind of thing in respect to other universal principles, like the Principle of Universal Love. The tragic loss occurs when those powerful principles are not sufficiently understood. And so, in this manner, the shape of our civilization and the prospects in our life, are directly determined by our recognition of those principles, and our understanding and knowing absolute truth. That is how universal principles that shape our universe. Here is is where we find the foundation for creating a new Renaissance in our time. The Lyndon LaRouche organization has utilized this method for a long time already, towards creating a new Renaissance. The method is utilized by LaRouche in the form of scientific pedagogical exercises that bring into focus the great discoveries in humanity's past, and their underlying principles, as a means for deriving a better understanding of our presents world and how to uplift our civilization. The pedagogical bring into focus all the wondrous aspects of our world that are skipped over in the course of "modern education" so-called. In this sense, the LaRouche organization is also one of the world's foremost educational institutions as many university and college students have recognized to be true in their personal experience. It would be surprising if the great scientific and spiritual pioneer of the 19th Century, Mary Baker Eddy hadn't likewise recognized the value of this method for discovering and knowing truth. Indeed she was aware of it. Beginning in 1875, everyone of the major aspects of her discoveries and her work was brought into structural conformity with a complex of structures that have all been designed in the form of pedagogical challenges for the moral, spiritual, and scientific development of humanity. They were designed by her as a method to bring into focus universal principles that would normally not be recognized, but which she recognized to be essential for the protection and the advance of civilization. The principles so brought to light, led in turn, to the discovery of higher principles, which were built on those principles. Also, as the Meno dialog illustrates, none of these principles need to be taught, to be discovered. Indeed, they could not be taught in the standard sense of teaching as a process of conveying facts. As Socrates asked Menon: "What do you think? Was there one single opinion which the boy did not give as his own?" Menon had to agree that all the discoveries of principles that the boy had made in the course of exploring the challenge set before him, were ultimately discoveries based on his own perception. And the discoveries could be understood by the boy to such an extent that he knew them to be correct because he also knew the principles on which the discoveries were made to be correct. This rendered the proof for what he had discovered. Mary Baker Eddy evidently also understood that this kind of pedagogical platform is the only platform on which her discovery of Christian Science can ultimately be understood, which rests totally on the operation of divine Principle in its many forms of manifestation, which are not arbitrary principles. Since this pedagogical foundation in her work is not being recognized to even exist, much less is being utilized, the natural consequence would have to be that her discovery of Christian Science becomes irrelevant in the world and even becomes non-functional in comparison to its potential. This appears to be indeed the present state of it. That of course, can change at an instant once society begins to recognize the treasure it has in the great historic achievements of the past and the present, from Homer to Mary Baker Eddy, to LaRouche, and so on, that are its heritage and its open door to the future. |