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With this column, the New Federalist is inaugurating a regular series of pedagogical demonstrations, designed to show that, within today's academic professions, most of what is taught as scientific fact, is an out-and-out fraud. As the method of the 18th-century Enlightenment has increasingly won out against the scientific method of the Italian Renaissance, true scientific method has disappeared.
As Lyndon LaRouche has often put it, if you have an academic degree, you should sue your university for consumer fraud.
But we don't want you just to take our word for it. Beginning with this column, we will be presenting a series of pedagogical paradoxes, designed to show you that formalism in any field leads to the denial of fundamental truths--starting with the nature of man as being a creative individual made in the image of God. In one issue, we'll present the paradox; in the next, the method by which a solution can be obtained. The point is not to get the right answer, per se, but to uproot habits of linear thinking, to actually provide access to the kind of creative thinking upon which all scientific breakthroughs depend.
We begin with what amounts to a prologue to the prime number paradox. It
should get the juices flowing.
--Nancy Spannaus
Most people today, be they scientists, economists, corporate planners, TV weathermen, or ordinary citizens, make life-and-death decisions based on a fraudulent conception of mathematics, even though, for the most part, they are totally unaware of it. In modern times, this state of affairs can be traced to the fraud perpetrated during the mid-1700s by the Swiss mathematician Leonhard Euler, who at the time was in the service of the pro-British oligarch Frederick the Great of Prussia. By sheer political thuggery, Euler attacked the work of Gottfried Leibniz, whose discoveries in science, philosophy, law, and economics, were the inspiration of the Founding Fathers of the United States, and the basis for all progress in science and technology to this day. Against Leibniz, Euler insisted on reducing all human knowledge to mathematical formalisms, a set of rules to be obeyed, no matter what. In Euler's world, if the mathematical formalism said the world should work a certain way, the mathematics was right, even if the evidence contradicted it. With the popularization of the modern digital computer, this slavish adherence to mathematical formalism, has taken on the proportions of a dangerous mass psychosis. How many times have you heard someone say, ``According to the computer...''? All creative thinkers, including the greatest mathematical scientists, were not as stupid as Euler and his followers. A famous anecdote from the childhood of the greatest mathematical scientist of the 19th century, Carl Friedrich Gauss, illustrates the point. One day, when Gauss was about 10, his teacher asked the students to find the sum of all the integers from 1 to 100. (In those days, before the Internet, teachers had to find other ways to dull the minds of students, and keep them busy at the same time.) After being given the assignment, the boys immediately set to work, doing their calculations on little slates. The usual procedure was for the first student who finished, to place his slate on a table in the middle of the room. The next to finish placed his slate on top of the first, and so on, until all were done. Scarcely had the teacher stated the assignment, than Gauss proudly walked to the front of the room, placed his slate on the table, and sat back down at his desk. The other students labored for more than an hour, while the teacher looked skeptically at Gauss the whole time. When the time came to check the answers, Gauss's slate had only one number on it, which was the correct answer. Can you figure out what Gauss did? Next week, we'll show you Gauss's method, and introduce you further into the world of Mind over Mathematics.
CAN YOU SOLVE THIS PARADOX?
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Last week in this column, we recounted a problem encountered by the great scientist Carl Friedrich Gauss when he was 10 years old. His arithmetic teacher had assigned the students, the mind-busying task of adding up the whole numbers, from 1 to 100. The students were to do their calculations at their desks, on small slates, and, when finished, place their slates on a table in the middle of the room, so the slate of the first one finished, would be at the bottom of the pile. The teacher had barely given out the problem, when the young Gauss, wrote one number on his slate, walked to the front of the room, placed his slate on the table, and returned to his desk. For more than an hour, the other students labored over the calculations, while the young Gauss sat quietly, enduring the skeptical stares of the teacher and his fellow students. When it came time to check the answers, Gauss's slate had written on it the number 5,050, which was correct. How did he do it? Gauss was able to arrive at the solution so quickly, because he didn't allow his mind to be bound by the formal rules of mathematics, implicit in the instructions of the teacher. When the teacher asked the students to add up the numbers from 1 to 100, all the students, and most of you, ``instinctively'' thought: Begin with 1, then add 2, then add 3, and so on and on. This type of thinking is not really an ``instinct,'' but merely an example of how the mind is governed by underlying assumptions, of which most people are unaware. In this case, the underlying assumption is that the so-called ``natural'' order of numbers, 1|+|1|=|2, +1|=|3, + 1|=|4, ... is somehow fixed by nature--when in fact, this ordering of numbers, and the numbers themselves, are creations of the mind. If your thinking is governed by this underlying assumption, you will believe, falsely, that the only way to solve the problem, is to add the numbers one by one. But human beings need not be slaves to their underlying assumptions. Instead, human beings, unlike other creatures, are capable of making themselves conscious of the underlying assumptions which govern their thinking, and changing them. This is what the young Gauss did. Instead of being chained to the ``natural'' order of numbers, Gauss thought of all the numbers between 1 and 100 all at once, as a whole. Then he reordered the numbers, in his mind, into pairs. The first number, 1, and the last number, 100, when added together, total 101; the same for the second number, 2, and second to the last number, 99; It is immediately clear that there are 50 pairs of numbers which equal 101, making a total of 50|@ms|101, which equals 5,050. Once Gauss ``saw'' this ordering in his mind, the problem was solved. He didn't have to add up each number, or even each pair. It was the ordering of the numbers on which Gauss focussed his mind, not the numbers themselves. This is real mathematics. Here the numbers work for the mind, not the mind for the numbers. A valuable lesson can be learned from this simple example. When encountering any problem, the first question to ask is: ``How am I thinking about this problem? What assumptions are implicit in the way I'm thinking about the problem? How can I reorder the problem so that I am no longer constrained by those underlying assumptions?'' This, of course, is not the way we're accustomed to thinking, but it is the way all creative minds think. Most people assume that creativity is not something they can master. Not true. If we make the effort to replicate, in our own minds, those creative discoveries made by others, we'll soon learn to be creative ourselves. Next week, we'll investigate another underlying assumption about numbers, in examining the question, ``From what are whole numbers made?''
Odd and Even Numbers
--
Thank God for the Odd One
We last left our prisoner confronting the divergence of, on the one hand, the endless succession of days, one after the other, and, on the other hand, the actual ordering of those days according to the physical events that occur in them. This clash, between two concepts of number, sparks our prisoner to embark on a journey to discover the nature of number, beyond the realm of sense-certainty. Reflecting back on his childhood education, he realizes that his thinking about number is confined to a rigid set of rules and operations, mere manipulations of numbers as external objects, memorized, not discovered, to be re-called on command. Suddenly the liberating words of Nicholas of Cusa from {On Conjectures} come to his mind: "The essence of number is therefore the prime exemplar of the mind.... In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the Divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world." He begins to recall some happier memories of his childhood quest for knowledge, reminiscing how he once playfully discovered hidden relationships among numbers, while secretly exploring their nature with his mind only. Little things, oddities he kept to himself. Once, he had ventured to tell his teacher about one such discovery, only to be discouraged by the response, "Don't be an oddball. That has no practical application. You won't need that in later life." Now such canons and dogmas memorized in youth are of no use, if they ever were. He finds himself free to inquire anew, beginning first with those elementary principles, which, never simple (except to the simple-minded), unfold a rich bounty of profound ideas, if the underlying, seemingly subtle, paradoxes are sought out. He takes out a paper and pencil, and unfolds a series a numbers with the following construction: Begin with a unit * and add a unit ** and add another unit *** and another unit **** and so forth.... ***** ****** ******* ******** ********* ********** *********** ************ It seems apparent enough, from the method of construction, that each number is unique, differing from all others by its relationship to the process of adding one, just as each day follows another. But, seeking to shed the shackles from his mind, our prisoner tries to discover what is behind the numbers, by looking into the numbers on a different level, besides the succession of adding one. He tries the following experiment: With each number, he alternately marks one unit from each end, beginning with the first and last unit, then proceeding to the second and second to the last unit, until he can go no further. (The reader is required to make his or her own drawings by hand, rather than rely on computer generated images. Hand drawings, even crude ones, contain within them the cognitive process, whereas the computer images suppresses same.) What emerges, from this process, is that numbers are distinguished from one another by more than just adding one. Some numbers, (every other one) has a unit left unmarked in the middle, while in the others, no unit remains in the middle. Again the words of Nicholas of Cusa come to mind: "It is established that every number is constituted out of unity and otherness, the unity advancing to otherness and otherness regressing to unity, so that it is limited in this reciprocal progression and subsists in actuality as it is. It can also not be that the unity of one number is completely equal to the unity of another, since a precise equality is impossible in everything finite. Unity and otherness are therefore varied in every number. The odd number appears to have more of unity than the even number, because the former cannot be divided into equal parts and the latter can be. Therefore, since every number is one out of unity and otherness, so there will be numbers in which the unity prevails over the otherness, and others in which the otherness appears to absorb the unity." A smile comes across the prisoner's face as he now sees the once familiar concept of even and odd numbers (thrust at him as an almost trivial distinction in his youth), in a new light. His joy is mixed with a tinge of anger, as he realizes this new light is not new at all, but, in fact, an ancient discovery, he should have relived as a youth. Unlike what he was taught in school, the concept of even and odd, is not a mere description about a particular number, but a concept associated with the {nature} of number itself. The doctrine he was taught in school seemed to work, but because of it, his mind didn't. His anger abates as he turns back to his inquiry. He leaves it to others to uncover how these ancient discoveries were written out of the curriculum. The infinity of all numbers, has now been divided by two, according to the nature of the individual numbers, when each of them is divided by two. The discovered principle of even and odd, divides the infinity of numbers into two {types} -- those numbers in which "otherness prevails over unity," and those numbers in which, "unity prevails over otherness." From the original construction of all numbers by adding one, no number is equal to any other number. But now, he discovers some numbers are alike but not equal to others. To bring this discovery into a One, the prisoner is taken to Gauss' concept of congruence. All numbers of the same {type} are congruent to each other, and those of a different {type} are non-congruent. There are two {types}. So under Gauss' concept, all even numbers are congruent to each other relative to modulus two. Likewise, all odd numbers are congruent to each other relative to modulus two. And, all even numbers are non-congruent to all odd numbers relative to modulus 2. Seeing this, the prisoner desires to continue the exploration, dividing the numbers again. This time, he starts with the even numbers only, taking the parts created from the first division, and marking off the units from each end until he can go no further. (The reader is required to complete this step for himself.) Now the even numbers have been divided into two {types}; those whose parts when divided in this way, leave no unity -- called even-even, and those whose parts, when divided this way, still have a unity in the middle of the part -- called even-odd. The odd numbers, in turn, are divided into two {types}. Those which have even numbers on each side of the unity left in the middle -- called odd-even and those which have odd numbers on each side of unity that was left in the middle -- called odd-odd. The infinite has now been divided four times! Again, numbers of the same type are not equal, so we go to Gauss' concept of congruence, to bring this new discovery into a One. Each number of any of these four types -- even, odd, even-odd, even-even, odd-even, odd-odd, is congruent to all other numbers of that type, relative to modulus 4. Yet there is nothing self-evident, from the construction of numbers by the addition of one, from which the now-discovered distinction between even, odd, even-even, even-odd, odd-even, and odd-odd, logically follows. To be able to envisage, from this small distinction among numbers, a different domain, other than the linear domain of adding one, the prisoner must free himself from the constraints of his formal thinking. That domain is characterized, not by linearity, but by curvature, of which the principles of even and odd are but a reflection. The nature of that curvature will be further discovered, by new investigations to which the prisoner looks forward. As the prisoner contemplates his next experiments, he's interrupted. Oddly enough, it's time for lunch. [bmd]
An Exporation of the Relationship -
Among Number, Space, and Mind -
I can conceive in the mind of six objects, whose relationship to one another I wish to investigate. Their character as real objects does not interest me, but only that quality which makes them distinct, thinkable. They are, thus, objects in thought. I will label them with the number designations 1 to 6, though I might equally denote them by letters, or any other symbols which allowed me to keep them distinct in my mind. I am interested in discovering the number of different ways these six distinct objects can be formed into pairs. Their representation by numbers, allows a convenient means of investigating this. I first list all the pairs of 1 with the other 5, then all the pairs of 2, and so forth. The result is summarized in the table: 12 13 23 14 24 34 15 25 35 45 16 26 36 46 56 == == == == == 5 4 3 2 1 Counting the number of pairs in each column and summing them, produces 5 + 4 + 3 + 2 + 1 = 15 pairs. In another form of representation, I can imagine the six objects as points on a circle, and portray their pairing as the straight lines connecting any two. Drawing them produces a hexagon, and all the straight lines that may be drawn between its points. Counting all the connecting lines, we find 15, the same as the number of pairs above! The mind rejoices in the discovery of the equivalence of the two representations. Closer examination of the second form of representation, now reveals also a difference with the first. In the first, nothing distinguished one pair from the next, except the symbols used to designate them. In the second, we discover three distinct species of relationships among pairs, each characterized by a different length of connecting line. We have (i) the six lines forming the sides of the hexagon; (ii) the six somewhat longer lines connecting every other vertex (i.e., 13, 24, etc.); (iii) the three longest lines connecting diametrically opposite vertices (14, 25, 36). Where, before, the mind celebrated the sameness, it now rejoices at the difference of the two forms of representation, and is impelled to look for its cause. We hypothesize that the difference must reside in a property of the spatial mode of representation. We may reflect that, from the manifold ways we might have chosen to arrange our six points in space, we chose to place them on the circumference of a circle, equally spaced. An arbitrary arrangement of six points in a plane would have produced another, less-ordered relationship among the pairs. Another arangement, a spiral perhaps, would have produced a richer ordering. Thus, from the positing of relationship among things in the mind, we moved to two modes of representation of that relationship, then to their sameness and difference, then to the causes of that difference. Having hypothesized that the latter is the result of the spatial form of representation, we are next led to explore the variety of such representations. Of the great variety of possibilities, we choose now to rise above the plane, in order to examine the relationship among six points in three-dimensional space, the familiar backdrop for our visual imagination. Just as the circle aided us in ordering the points in the plane, here its counterpart, the sphere, comes to our aid. Six points, spaced evenly around the surface of a sphere, form the vertices of the Platonic solid known as the octahedron. We can picture two of its six points at the north and south poles of a globe, and four more forming a square inscribed in the circle of the equator. Connecting each point to its nearest neighbor, we find the 12 lines which form the 8 equilateral triangles, which are the octahedron's faces. But we have not yet connected the six points in all the ways which space allows. Each point can yet be connected to its opposite, forming 3 more lines, which are diameters of the circumscribing sphere. Behold, again, the 15 paired relationships of six objects, now clothed in a new ordering, this time of two species! We may now compare the three modes of representation our mind has invented to investigate these pairings: 1) By number, which produced the series 1 + 2 + 3 + 4 + 5 = 15. 2) In planar space, using the circle, which produced the three species of lines connecting the points of the hexagon. 3) In space, using the sphere, which produced the two species of lines connecting the vertices of the octahedron. In turn, each of these modes of representation suggests new investigations. For example, with respect to the first (i.e., number), we may inquire into the pairwise combinations of other numbers of things, from which we soon discover that, in general, for "n" things, the number of pairs that can be formed is equal to n(n-1)/2, and we may next inquire, what is the expression for combinations three-wise, four-wise, or n-wise? With respect to the second (the distribution of points on a circle and their combinations), we discover that there exist species beyond the regular polygons, which are known as the star (or Poinsot) polygons. These cannot be generated out of any arbitrary number of points, but only when the number of points, and the order in which we take them, are relatively prime to each other (that is, have no common divisor). The first of the star, or Poinsot, polygons, appears when we take five points on a circle, and connect every second one until the figure closes (that is, 1 to 3, 3 to 5, 5 to 2, 2 to 4, and 4 to 1). The result is the star pentagon, or pentagram, which is conveniently described as 5/2. We can then discover the 7/2 and 7/3, the 8/3, the 9/2 and 9/4, and so forth. With respect to our third mode of representation of the pairwise combination of things (the distribution of points on a sphere), a new ordering principle arises: that a perfectly even distribution is only possible in the cases of 4, 6, 8, 12, and 20 points. When we investigate these, we find species of pairwise combinations called edges, diagonals, diameters, and some others, the greatest variet of species occurring in the 20-point figure. Now, let us reflect on the higher ordering principle: All of the representations we have given, even the spatial, are creations of mind, products of the arithmetic or visual imagination. Yet, so real do these creations of the mind seem to us, we may be tempted to marvel at them as if they had some existence outside of the mind. ("But Platonic solids are {real}. I can build them!" you say. Perhaps you never have. Anyone who has tried, soon discovers a, sometimes gooey, massiness where massless points are supposed to be, a very finite thickness to the infinitely thin lines of the edges, and a, sometimes wrinkly, bulk to the massless surfaces. Even three-dimensional space, the forgiving medium of all our constructions, which seems so certain, so real, is only the ingenious work of the mind, the visual imagination. All are products of the mind.) But when, in nature, the mind discovers forms just like these we have just created (thought), put there not by us, but by something like to us in mind, yet much vaster, then may we truly marvel, and reflect: What makes nature makes us. What we make in mind, think, is then nature -- and may be so in a higher form than what we perceive outside us. (The proof of this truth, well-known to readers of this publication, need not be repeated here.) So in the ordering, number, space, and mind, the mind stands at both ends of the series, as both creator of its own images, and perceiver of others; the one is called imagination, the other, reality. Yet they are both real, as we just showed, and even both imagined, in so far as the perceived external is {known} only through the images of mind. With such considerations, true science begins. [lmh]
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