Infinity and Infinitesimals
"One is the Onliest Number"
I am going to devote considerable discussion to the topic of this chapter because the understanding of the true nature of quantity, or numbers, in terms of Reality is essential to changing the basis of one's personal paradigm from separation to Oneness.
In conventional mathematics, an infinitesimal is defined as a number that is greater than zero, but whose value is too small to be measured. Such a concept seems as refractory to understanding as the concept that Infinity is a value so large that it cannot be measured. Both Infinity and infinitesimals have the quality of being unmeasurable, so for purposes of space/time Experience, the infinitesimal and Infinity have been disregarded in favor of the fixed, tangible measurement.
However, measurements in general are of very limited value without a reference measurement. For example, to know what a "meter" is, you must have a reference meter that you can use in order to verify that, in fact, you have measured a true meter. Otherwise, a meter might be different for everyone who measures it. The surprise is that, even with a reference meter, no two meters are measured to precisely the same length because accuracy is dependent both upon the limit of resolution of the measuring device and upon the one doing the measuring.
So, is there a reference that we can use no matter what the measurement in question is? Truth is that the only reference we have that never changes is Infinity, but Infinity cannot be measured. Infinity is the Reference because Infinity is the fundamental essence of Reality. Remembering that whole dimensions are of infinite measure, all other "references" are shown as arbitrary fractions of dimension, or dimensionals, along whatever whole dimension the measurement is made.
Recognizing the need for a reference, scientists have sought reliable standards to be used as reference measurements. The ideal, or most reliable, standard would be a finite measurement that is absolute, or that never changes. Because no such measurement exists, scientists have settled for standards that rarely change rather than one that never changes. However, choosing any measurement greater than zero, yet less than Infinity is, in truth, an arbitrary choice. Infinity, while Absolute, is not a measurable quantity, so it cannot be recognized as a standard in the realm of finite measurements.
But Infinity is the only true, or absolute, Reference, so what does this mean in terms of finite standards and measurements? What this means is that any measurable quantity, when compared to Infinity, is so small that a true (absolute) measure cannot be assigned to it, only a symbolic or relative one. Furthermore, any measure which, by definition, is a limit, cannot have any meaning unless compared to another limit. Thus, the true meaning of measurements is relative, not absolute, and this understanding opens the door to a new conception of numbers.
This also means that all finite measurements meet the definition of infinitesimal when compared to the true Reference, Infinity. Thus, the quality of the infinitesimal, rather than being of mere esoteric interest to mathematicians, is of vital importance in understanding Reality.
So what of numbers, such as 2, 11, -1028, or 15i? While the conventional understanding of numbers is that they can be arranged in a series such that at one end you have the smallest and at the other end you have the largest, this concept can be true only in a relative sense, as in the comparison of one finite quantity to another finite quantity. In terms of Reality, all finite numbers, including so-called whole numbers, are fractional. What this means is that 2 symbolizes the 1 divided into two parts, each of which is smaller than the 1, and 11 symbolizes the 1 divided into 11 parts. Contrast this with the conventional understanding that 2 is twice as many as 1, and 11 is eleven times as many as 1. The observation that 2 is twice as many as 1, for example, is a comparison between two fractions of the One, and that is what yields the illusion of greater and lesser. This is easier to grasp when you recognize that the symbols 2 and 1 are actually 2 ⁄ ∞ and 1 ⁄ ∞. So, 2 ⁄ ∞ is greater than 1 ⁄ ∞, but this is a relative comparison only because so long as Infinity (∞) remains a part of each fraction, the actual values are infinitesimal. Leaving Infinity in each fraction references the Reference (i.e. ∞ ⁄ ∞, the One), so in order to make space/time sense of these numbers, 1 ⁄ ∞ and 2 ⁄ ∞ must each be multiplied by ∞ in order to remove it from the equation. You are then left with 1 and 2 as relative quantities.
Consider an example using a group of apples. To evaluate a group or quantity of apples, the universe of apples must first be referenced by default in order to establish a relative measure and thereby give meaning to it. This universe of apples is a finite quantity, so it can be used as a relative reference without yielding infinitesimals. Within the universe of apples, a portion of it, such as eleven apples, is a fraction of, not more than, the universe. The group called "eleven apples" represents a subset of the universe of apples, so to measure eleven apples you must divide that universe such that you have a quantity you can call "eleven".
Furthermore, you cannot reference the universe of apples without first referencing the universe of fruits in order to establish what an apple is, relative to all other fruits! This referencing must go on until you, at long last, reference the Reference, Infinity. However, since a measured quantity becomes infinitesimal when compared to Infinity, those working with finite quantities will stop short of any referencing that involves Infinity.
So, in this example the universe of apples is symbolic of Infinity, and any subset, group, or quantity which is less than the entire universe is a fraction of that universe, symbolic of finite numbers.
Since the infinitesimal cannot exist without relative measurements, the infinitesimal is an artifact of the illusion of separation, and is dependent upon the concept of limitation, which is also illusory.
The primary reason why I am proposing that numbers greater than one are in fact fractional is the basic mystical Truth that All is One. Thus, anything that does not equal All is less than, or smaller, than One. One, therefore, must be the center, and all numbers we are used to thinking of as less than one as well as all numbers we are used to thinking of as greater than one are fractional in nature, and represent dimensional iteration of the One. Therefore, this One, or center, has to equal Infinity in order to be greater than any finite number.
With this in mind, we find that values as a series can be represented on a polaric axis with One (1) at the center, all numbers "smaller" than 1 on the left side of the center and all numbers "larger" than 1 on the right side of the center. Since this center is actually the One, or ∞ ⁄ ∞ , we see that all numbers, whether on the left or the right of center, are infinitesimals, having Infinity as their denominators.
Using this new coordinate axis, what is the difference between two points labeled 0.5 and 2 on this axis? The answer is that they are equivalent in value, because 0.5 is 1/2 of the One and 2 is the One divided into two equal parts. The symbols simply distinguish which "side" of the axis the measurement is being made along. Of course, Reality is that the One cannot be divided at all, so the idea of dividing the One is really the dividing of a unit, or 1, not Infinity.
Next, what of negative numbers, and so-called imaginary numbers, such as the square root of -1? These numbers are found along a second and third axis, respectively, defining coordinates in a space existing along three dimensions when taken together with the first axis. The first dimensional coordinate axis is the axis I described with "1" at the center, the numbers less than 1 on one side and the numbers more than 1 on the opposite side. The second axis has -1 at the center, but the two sides are the numbers larger than -1 (like -0.5) and the numbers smaller than -1 (like -2, etc.).
The third axis has the root of -1 at the center, and the parts of the axis on each "side" of the center are like the parts on the first axis I described, with the exception that each of the values are multiplied by the square root of -1, or the symbol "i", as designated by mathematicians. For example, the number on the third axis corresponding to the number "2" on the first axis is the number 2i, or 2 multiplied by the square root of -1. The opposite side example would be a number such as 0.5i, or 0.5 multiplied by the square root of -1.
While we are on the subject of so-called imaginary numbers, I must point out that there are no imaginary numbers. This illusion has arisen in part because of the unfortunate selection of the negative symbol to distinguish some numbers from others, plus the erroneous idea that a negative number is the opposite of a positive one. In this new model, a negative number is simply a number on a coordinate axis at right angles to the so-called positive axis. The problem arose when positive numbers and negative numbers were placed on opposite sides of zero on the same axis. Consider that negativity should be understood as a dimensional, or directional, attribute, not a value attribute.
The truth, then, is that adding positive and negative numbers is like adding west and south; it can only be done in the sense of fixing the location of a point using west and south coordinates relative to a reference point, or center. The relative distance from the point described by the coordinates to the center would, in essence, be the sum of the two directional attributes, or vectors. However, this is not the same as a sum defined as the result of adding two numbers.
Using this new coordinate axis model, a point may be located anywhere in this complex space by using three coordinates, one from each axis given. The center coordinates are (1, -1, 1i); contrast this observation with the conventional model, in which the coordinates of the center are (0, 0, 0). The idea that 0 is the center is congruent with the concept that such a thing as true "nothingness" exists, and encompasses the idea that from nothing came all that we know of as the physical universe. Likewise, the possibility of having "nothing" (0) is an artifact of the concept of limitation and separation which, as I have previously explained, is not Reality. To "have nothing", in Truth, would mean that you could have no consciousness of the fact, by definition.
The apparent difficulty here is how 1, -1, and 1i can be equivalent, or identical in value. This difficulty is due to our use of symbols to represent dimensionality, and is not an insoluble mathematical problem. You see, while we are using different symbols to represent coordinates in a space along three dimensions; the center of one axis is the same as the center of the other two axes, but with each axis itself oriented at right angles to the others. In order to describe coordinates for points on the different axes and distinguish the axes one from another, we use a different symbol (the minus sign, "-", or the letter "i") but the "three centers" are all, in fact, the one center, or One.
Another way of describing the equivalency of 1, -1 and 1i is to state that all three have an absolute value that is the same. This absolute value is actually Infinity (∞ ⁄ ∞, the One). This also shows the center to be the Infinite Point.
This apparent difficulty can be eliminated entirely by designing a completely new coordinate system requiring no negative or imaginary numbers. This can be done simply by starting with the center, which is Infinity, and constructing the three dimensional axes as before. However, the values for the first axis would be as follows: all numbers "left" of the center would be given the suffix "w", while all numbers "right" of the center would have the suffix "e". The second coordinate axis would have all numbers "above" center to have the suffix "n", and all numbers "below" center, the suffix "s". Coordinate axis number three would have all numbers "behind" the center given the suffix "d" and all numbers "in front" of the center given the suffix, "u". All numbers on each axis are positive because the location of the number relative to center is given by the suffix rather than by a "-" or a "+" sign. The first whole number in any direction from the center would be 2 plus the relevant suffix. Two examples are 2e and 2u. Locating a point in three dimensions is done easily with coordinates such as (5e, 15u, 21n). Using this new model, one can easily understand why all three axes have the same center, and why the center has the same value for all axes.
Having established that the centers of all axes are equal in value, let us go further and point out that the nature of Infinity is that no matter where you are in Infinity, you are at the center. Infinity, by definition, is the Center. So if you select any point, that point is the center because the ultimate distance from "it" in every direction equals Infinity. With this in mind, the ancients were correct in asserting that the Earth was the center of all, but they would have been just as correct to assert that the Sun was the center, or the Moon, your heart, or any other point in space (or out of space)!
Mathematical functions using numbers with suffixes or prefixes are necessary when working with multidimensional constructs because without them you'd have a difficult, if not impossible, time specifying values in more than one dimension. With this in mind, imagine the fun of working with all 11 space/time dimensions in a mathematical way!
The apparent complexity increases once you recognize that in this space we have defined with the three axes, there are infinite possible alternate sets of axes with set orientations differing by degrees (or fractions thereof) relative to the set of axes defining our space! This model provides the basis for the existence of alternate universes; these are not the same as the different realities resulting from the dimensional threshold maxima and minima that I described in Fractalic Awakening - A Seeker's Guide. The alternate universes I am describing here would have the same dimensional threshold maxima and minima defining what we know of as space/time, but the three fundamental spatial dimensions would be shifted axially relative to our universe. This shift would be along a dimension at right angles to the three axes I illustrated above (corresponding to the 4th dimension, time), and this shift could be represented by coordinates measured in degrees around a sphere relative to the "location" of our universe. For example, a one degree shift would mean that the net shift of all three axes in the alternate universe would equal a one degree temporal difference relative to the three axes that describe our universe. This alternate would be one degree out of phase with our space/time universe no matter how much time passes.
This shift along a dimension equivalent to time in nature makes these alternate universes physically invisible to us, but they can be perceived in our imagination because consciousness/Awareness can transcend our space/time dimensional limitations. This "imagination" can be developed so keenly that the perceiver actually "sees" the alternate in minute detail. In fact, it is the perception of these alternates (in varying degrees) that is the source of ideas for our "fiction" literature and movies.
Alternate universes are not to be confused with alternate realities. Alternate realities owe their existence to dimensional variances that are not universal, or equivalent for all dimensions in a universe. For example, while an alternate universe may have what I call a universal circumferential shift of 1 degree relative to our universe, an alternate reality may have a local circumferential shift of a fraction of a degree along only one dimension. This enables each of us to experience our own alternate realities without these alternates differing enough to separate us individually into alternate universes. This means that all of our alternate realities intersect together in our universe, enabling us to share lives with each other, yet still have distinctively different experiences. Alternate realities are experienced whenever we make individual choices as we live our lives. This is because each choice made that differs from the choices of others is equivalent to a fractional phase shift, making the chooser's reality different from others by that fraction. By contrast, an alternate universe is created by incorporating one or more dimensional differences, relative to a reference state, at the moment of universal creation. The reference state can be either a previously created universe of a given type or a parametric set that defines universes of a specific type. In the case of our universe, the given type would be "space/time", or the specific parametric set that defines a space/time universe. Variances in dimensional values relative to the reference state set the new parameters for the alternate universe.
Incidentally, conventional mathematics is actually a mathematics of infinitesimals. This is the logical deduction resulting from the recognition that all values that are finite, or less than Infinity, are infinitesimal. As I have demonstrated in this essay, practical use of conventional numbers is possible only when Infinity is removed from the equation. However, removing Infinity from a mathematical formula does not eliminate it in Reality, but rather merely gives the illusion that Infinity does not exist. This observation also reveals that conventional mathematics is the mathematics of illusion. One has to wonder, then, what the mathematics of Reality is all about. The mathematics of Reality is what is required to develop the kind of technology that will enable intergalactic travel and access to unlimited energy, to give but two examples.
The ability to interact with both infinitesimals and Infinity is one of the reasons why fractals, as generated on a computer screen and manipulated with programs that allow exploration and magnification of the constructs, are so useful on Path and in understanding of Reality.
- Paradox and Path
- The Eleven Dimensions of Space/Time (Spatial, or linear, dimensions in a temporal domain) Important Update 2/20/07
- The Eleven Dimensions of Time/Space (Temporal, or nonlinear, dimensions in a spatial domain) New for 2/8/07
- The Myth of Zero, or nothingness Updated 11/23/05
- The Kronospheric Important Update 12/14/06
- Infinite Dimension and dimensionlessness New 9/19/06