Let us consider the case of modern education where students are being taught a truth that Pythagoras had discovered nearly 3000 years ago. In this case student would be told for example that Pythagoras had discovered that the surface area of the squares over the short sides of any right angled triangle, added together, are always equal in surface area to that of the square over its large side, and that this truth remains absolute regardless of the ratio between the shorter sides A and B. Today's students would be taught an algebraic equation. Thus the discovered truth would be expressed as an element of a curriculum in algebra. The students would take this knowledge and write it down. They might even apply the formula to a few examples to prove the truth of the algebraic equation. But do they really KNOW anything of value that improves the quality of the human culture as the result of this kind of teaching? Do the know the Truth? Sure, the resulting lesson would suffice to enable them to apply the formula to all kinds of uses, but would they really know the principle of the truth and thereby begin to understand the nature of universal principles and absolute Truth, and with that understand their own quality in the Sublime to know absolute Truth? This, they would not gleam from such a lesson! What the students would experience in such a course of teaching would be 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. In cultural terms it is actually something worse than a tragedy as it diminishes the human being and formalizes culture that has the potential to be a 'living' impetus. The undeniable fact is that we live in a Sublime universe that operates exclusively on a platform of harmonizing forms of universal Principle and their countless expression in the universe. This recognition is important, because there exists not a multitude of different principles for a given aspect, like numerous principles of economics for instance, that one might choose from at will. There exists only one principle of economics, and the welfare of humanity depends on society's 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 is no longer a front-center part of our culture. Truth no longer means anything. The political watchword has become: "In Lies We Trust!" Indeed, similar experiences can be encountered in almost any department of life in today's world. Society is drifting away from thinking in terms of universal principles. This rather sad fact was recently illustrated by an experiment a few young political activists. The experiment was performed in Detroit. They set up a table and a white-board and challenged anyone who would listen to step up to the board and illustrate the geometric process for doubling a square. The task was that s new square was to be created from a square of a given size, and in such a manner that the new square would be double in surface area. In order to get people interested in accepting the challenge, the science activists offered $100 to the first person who would complete the assignment. Guess what happened. Lots of people lined up for a chance to earn the $100 reward, but no one could present the solution. Not even a mathematics professor could do it. The professor could write down the formula, but he couldn't physically to prove it. At the end of the day the money was still in the kitty. Ironically, the money would likely have been paid out fairly early in the day in the event had been taken place 2400 years earlier. Plato described such an event in which a slave boy had accomplished exactly that particular task, and had done 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 before out time. Is it any wonder then, that our world is in such a mess? The story of the slave boy appears in Plato's Meno dialog. Menon was a wealthy young nobleman and friend of Socrates. One day the two men's conversation centered on the exploration of 'virtue.' Menon had requested Socrates that he would teach him. Socrates had 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 or would express an already existing quality for discovering truth that every human being has. A slave boy was chosen. Socrates drew a diagram into the sand 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 the size Socrates had 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 have to be eight square feet big as the boy had already discovered, but which would retain the shape of the square. Socrates then asked the boy, "how long would its sides have to be?" The boy considered what would happen if both sides of the original square were doubled in length. He discovered that the area of the resulting structure would be four times as large, sixteen square feet, much too large, although it would retain 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 certainly be smaller. However, he quickly discovers that the resulting structure would be nine square feet big, not eight, recognizing thereby that this kind of an approach to solving the problem doesn't work. Socrates asked him in his predicament to try a new approach that incorporates some of the principles he had already discovered. He asked the boy to go back to the original diagram of the space that is four square feet big. Then he asked the boy to ignore 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 according to the principle that he he had discovered.
After the boy had done so, Socrates suggested to the boy that he draw a new dividing line (diagonally) across each of the four squares, from corner to corner, which he called the diagonal. The boy did so, and while dividing every square in half he discovered that he could do it in such a manner that the dividing lines altogether formed a square. The boy recognized that the new square is half the size of the whole, which was four times of the original, and that this process did indeed create a square that is double in size of the original area. He recognized that the truth of this statement was self-evident since the original area contained two triangles (one green and one blue in the illustration), while the new square contained four (all green). Consequently he could say with absolute certainty, as an absolute truth, that the new square is double in size. The boy recognized that this process provided the answer that Socrates was looking for, that he had succeeded in doubling the square and understood the principles involved. Socrates then commented to his friend Menon that the boy understood that process without any teaching having been 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. How would one do this? Could one build on the principles already established in Meno dialog?
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 had 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 equal in shape to the qwerky square, and attach it to the lager form, which I did as shown above. You can see in this illustration that 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 therefore the more powerful solution. That's 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 had made these amazing fundamental discoveries in app. 500 B.C.. In fact some proof has been found that the same theorem has been discovered a thousand years earlier in Indus Valley of ancient India. 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 and continue the process and drive it forward. With the ongoing mental disassociation from universal principles in today's world, we've become disassociated from the universe in which we live, which is a Sublime universe operating on a platform that is expressed in the sublime nature of our humanity whereby the truth is knowable. In this universe of principles and absolute truth, nothing is arbitrary. All is Sublime. All expresses the lawful operation of universally operating natural principles. These universal principles, of course, are principles that cannot be overruled with the arrogance of human will, or by legislation. No legislator can degree that the principles, such as the Pythagoras had discovered shall 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 expressed as the principle of the general welfare or the principle of economics. The tragic loss that we suffer today as these powerful principles are no longer sufficiently understood shapes our civilization and the prospects in our life. But this can be altered. The tragic trend to hell can be reversed. Herein we find the foundation for creating a new Renaissance in our time. As the Meno dialog illustrates, none of the profound principles need to be taught in order for them to be discovered. Indeed, they could not be taught in the standard sense of teaching in the context of 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 knowing. And the discoveries built on this knowing 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 to be the Truth. The same method can also be applied to discovering and knowing the fast free energy resources that the Universe has in store for us in the form electric energy that powers our Sun and the Universe itself, and also surrounds the Earth as a resource that we can learn to utilize. |