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Jan 25, 1994 02:18 PM

by eldon

There is considerable philosophy in even simple mathematics, and with a little thought we can find great truths in it, throught the law of analogy. Mathematics is one of the keys to the Mysteries, and as a key, it does not necessarily give specific numbers to cycles as much as it reveals deep philosophical truths. Consider the natural numbers, 1, 2, 3, ..., on to infinity. They represent a quantity of some type of thing, of some object. Natural numbers give us a count of group membership, but only deal with quantity when they deal with something that can only be measured in whole units. Quantity implies more than one of a certain type of being, or object. We have, for instance, three chairs in the room. To have more than one of anything, the objects need to share in a particular quality, to have a quality in common. In this case, the three wooden chairs all share in the nature of "chairness", they are prototypes of the archetype of a chair. Group membership, or influence, is represented, then, by natural numbers. Coming to the number zero, we don't have a quantity, but a lack of existence. We can have zero of any possible or impossible thing. To take on an attribute of zero imples going into non-existence. This non-existence need only go as far as the realm of thought. When we say, for instance, that we have zero tables, we need the thought of the table's absence, and the thought of what a table is, in order to say that we have zero of them. Our thought gives the absence of the table a limited sort of reality. It is non-physical, but patterned after what we consider a table to look like. That thought may be lacking in some detail, but there is something to it. This is a different form of zero than when there is no one to think of the table's absence. In that case, the table exists neither physically nor in thought, but the archetype of what a table is still is, although completely unmanmifest. There are certain properties of numbers that hold true, regardless of the type of object that they refer to. If we say that 13 > 10, and 10 > 3, we can say that 13 > 3. When we talk about quantity and measure, we are really talking about attributes of the manifest world. The Sanskrit term "maya" means to measure, and refers to the illusory nature of the world. We see things, and perceive them wrongly. A branch in the roadway is at first mistook for a snake. The branch is real, our idea that it was a snake was wrong, a maya. The attributes of manifest existence are all maya in the sense that they deal with a misperception of the true nature of things. Consider a triangle. A triangle does not exist, but is real in its own sense. An object can be patterned after it, and have a triangular attribute. Many objects can participate in this influence, and take on a triangular nature. Those objects are not part of a "triangle group soul", all conscious embodiments of a triangle being, but are separate objects, each with a self-chosen, in a sense, situation of being subject to that influence. Adding negative numbers, we now go from natural numbers to integers. Negative numbers do not exist. We never *have* -1 of anything. When we say -3, for instance, we are talking about the absence of three of an object, not of anything that exists. Negative numbers imply *change* or *comparison.* We could have nine apples, and take three away, leaving us with six. Or we could consider our nine apples as three less than the twelve our friend has. With change, we are taking three apples into zero, from something else. Our quantity of nine apples has seen three of them disappear. Or we could consider the relationship of our nine apples to the friend's twelve. We have the same group of apples as our friend, and three taken away, 12 - 3 apples. The integers deal with whole quantities, and although we are comparing quantities, we are just dealing with addition and subtraction. When we compare sizes, and deal with multiplication and division, we come to the next type of numbers, rational numbers. A rational number deals with the relationship between two integers, between two whole quantities. Examples are 3/1 or 4/7. It tells us how much bigger or smaller one is than the other. There are two parts to a relationship. One component is what is in common between the two--a form of identity. The other is what is different between the two--the real part of the relationship. When we compare 91 to 21, for instance, we get 91/21 = (13/3) * 7. The identity between the two numbers is seven, and their ratio or relationship is 13/3. With beings, there is a part of their life that is in common, and must be factored out before considering their true relationship. This is arriving at the least-common denominator. After rational numbers, we come to the irrational. These numbers are magical in a sense. They are not based upon the ratio of any two integers, but can only be derrived by mathematical means, and never exactly. The numbers can only be approximated, but never fully expressed except in symbolic form. Irrational, numbers like *pi* show transcendental truths or realities. They show relationships to things that are bigger than life, that relate to the universal and not to specific individuals or beings. There is a lot that could be said about them. When we combine the rational numbers with the irrational, we get real numbers, which precisely measure anything that exists in the world. But there is still one more step to take in measuring things, one more step beyond measures of that which can directly exist, and that is the imaginary numbers. Each measurement, to be complete, could be considered to be a complex number, containing a real and an imaginary number. The imaginary numbers are shown in mathematics to have to exist, and they are needed to solve equations that deal with things that happen in the world, but what thing, that exists, could exist in an imaginary quantity? Imaginary numbers have an implied or derived existence. Mathematics requires their existence, as numbers, but objects of imaginary quantities cannot exist. The imaginary quantity of an object is like a Zen koan that must be solved, but has no answer. It is there, but cannot possible be! Consider the square root of -1. When we compute a square root, we are asking the question, what number, when multiplied by itself, gives the number we are considering. For 4, we would say that its square root is 2, for 2 times itself is 4. But we know that two negative numbers, when multiplied together, give us a positive number (like -2 * -2 is +4) and two positive numbers, when multiplied together, give us a positive number (+2 * +2 is +4). So how do we come up with the square root of a negative number like -4? We would treat it as a complex number, with a real and imaginary part, and say what other complex number, multiplied by itself, gives us this complex number. We'd say that 2i * 2i = -4. In other words, we have a non-manifest, karmic potential of two, times itself, giving us the manifest, active negation of four. The meaning of this is something to puzzle over, and we'll have to come back to it. We need to consider what a negative number means, what an imaginary number refers to, and what is implied by a multiplication. Let's consider what it means to multiply two numbers. We are not considering them as two separate quantities, but rather as coming together to make a greater whole. After the multiplication, we get a higher dimension, with a new unit of measure. The relationship of the two defines a higher space than either, when taken by itself. Say we have two lengths, three inches, and four inches. When we multiply them, we have something new: twelve *square inches*. Something new has been created, between these two lines, that occupies a higher space, a two dimensional measure. By coming together, they contain something bigger than themselves, and of a different nature. We experience this in a sense, when we enter into our buddhic consciousness. When we rise to the higher part of our natures, and dwell in our relatedness to all of life, we experience a different dimension to life than we do while seated in Manas, in the sense of fixed, personal self. Next is the meaning of division. On aspect of division involves a shifting of quantity from one part of the whole to another. We have 1 group of 16 apples. We shift the quantity from apples to groups. We now have 2 groups of 8 apples. The total of apples in a group is the same: 16 x 1 is 8 x 2, but the apples component of the whole has been divided by two, at the same time that the number of groups has been multiplied by two. If we were considering things from the standpoint of a single group, we would say that we have divided the number of apples by two. There are half as many as before. But there is also an implied multiplication of the other part of the whole, of the number of groups. What we have is a state change in the wholeness. There was a shift from one part of it to the other, but there the overall system stayed the same. In a relationship, we find both parties are changed, and there is a form of conservation of energy. We also use division to infer one part from the whole. We know the overall, AB, and one part, A, and from the two infer the other part, B. We have, for instance, 12/3 = 4, or the whole, 12, taking away part A, 3, gives us part B, 4. When talking about something, we can take its measure, and have its number. But we need to carry the attributes along with the number. The *what* of the number is as important as the *quantity* of the number. Say we have an acceleartion of 1.2 ft/sec^2. It could be composed of 2 ft/sec * 0.6 1/sec. When we divide the 1.2 by 0.6, we are not just separating the 1.2 into two numbers, we are also separating the attributes from ft/sec^2 into ft/sec and 1/sec. We have two quantities with associated attributes, and each is of a different nature than the whole. And the same is true with multiplication. We have two numbers, each with associated attributes, and create something new, with combined numbers and combined units or attributes. Two inches times three inches gives us six square inches, something different than before! With multiplication and division, we are dealing with the nature of composite beings and the unity of life, albeit in an abstract manner, and there is much that can be learned from the symbolism. It is only the beginning to draw some elementary analogies. We can find abstract philosophy, and the Teachings themselves, in the simplest of mathematics, because both are structured on how life works. A hard, undistracted concentration is required, a steady contemplation, a fixed mind is needed, and we can break through to profound insights. There is a mathematical key to the Mysteries, and we have the power of turning it, and unlocking many aspects of the deepest philosophy! -- Eldon Tucker (eldon@netcom.com)

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