#awarenesslogic

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Awareness as Conscience

My regrets for the delay in the write up for Awareness Logic. Just when I thought I had it all figured out, I hit upon an unexpected implication which actually depend what I originally intended to present, but it’s taken me some time to work out. The next section should be up shortly. In the meantime, this goes along with  The parable of “The Three Roles of Humanity” present in section 2.  I’d been racking my brain literally for years to come up with a better word for Awareness in the social context, and talking with a friend the other day, it finally occurred to me: conscience. The diagram below neatly encapsulates the modern state of the three roles:

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8 –Awareness Logic Waveforms

Converting the Awareness Logic square grid to a circular grid is fairly simple. Here the awareness point (أ) essentially expands with the circle to form the basis of a reference line (with the intersection at the radial still equal to 0∞1). The halfway points of 2 or 1/2 in each quadrant of the square grid become the 90° (1/2 π) radial points at the top and bottom halves of two separate circles. One circle is for values less than one (<1), and the other for those greater than one (>1). The radius is still derived from the base count value. The examples below show a circle with a radius of 4 with same appropriate fractional values from the square grid:

This works naturally for the <1 values, but is less intuitive for those >1. The ideal use for this arrangement actually is for wave forms in which the <1 values stand in for the period (T), and the >1 reciprocal values for frequency (f):

In this example, instead of fractions, we’re using decimal values (truncated at 3 places) for easier reference. The same fundamental relationships of Awareness Logic apply, but just using the fractions can force us into some unnatural or unintuitive numbers, particularly for the frequency.

In this diagram, we’re using the +/- variations of the frequency to represent the positive and negative peaks of each wave in the cycle, but this is not the only possible use of this variable (spin or orientation are examples of other possibilities.)

We’re using basic sine waves to represent our waveforms here, but keep in mind we’re just establishing a grid. How to represent more complex waveforms can be inferred from this.

Every number can be broken down in Awareness Logic into the three elements of base count, unit, and context . But generally speaking, the base count corresponds best to the amplitude of a wave, the unit to its period/frequency, and the context can best be used to represent the phase of a wave.

We can see from the diagram above how the amplitude and period/frequency are derived with the base count and unit, but in order to see how to define phase values, we need to first get a more thorough sense of exactly what we’re measuring with context…

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7 – The Awareness Logic Matrix

In order to fully utilize the potential of Awareness Logic we need to be able to clearly express how the base count, context, and unit values relate to each other, and intersect at the aware point (أ). One way to begin to do this is with a handy mnemonic diagram called the Awareness Logic Circle.

If we recall back to Section 4 (Counting with Awareness Logic), we established that the best way to express base count values was a (more or less) standard numbers line with an origin of 0; the best way for unit values was a numbers line with an origin of 1; and for the context values a circle was better than a line. With the Awareness Logic circle, all we’re doing is combing the three.

The expanded diagram below shows how the Awareness Logic Circle works, and how it logical radiates outward:

For the base count we have a vertical axis centered on the radius of the circle that shows positive and negative numbers extending up and down respectively from 0.

The context value can be expressed around the circumference of the circle centered on the radius, and the cycle from 0 to 1 is completed by rotating (or revolving) around the circumference the full distance from any point along the circumference from 0 to 1 (or back). There are a potentially infinite number of fractional points anywhere around the circle.

We can picture (n) ∞ as creating circles that expand away from the radius of the circle, and (∞) n as shrinking towards it. However, the actual radius is not determined by the context, but by the base count, so this is more a handy visual reference.

For the unit values, we have a horizontal axis centered on the radius of the circle with an origin of 1. Values along the line greater than one (>1) are on the left, and values less than one (<1) on the right.

Note that we’ve replaced the x/1 and 1/x we’ve used previously with 2/1 and 1/2. There are a few reasons for this. The first is that 2 is the next natural whole number after 1, and logically denotes that progression.

The second reason has to do with the nature of the ratio of a circle to its circumference (π). 2π creates a complete circle. The 0 axis here is always at a 90° angle to the 1 axis centered at the radius of the circle. This is defined by the unit value 1/2, and 1/2π is equal to 90° around the circumference of a circle, and a full circle (360°) is 2π.

What this means is that along the numbers line that forms the one axis, the value on the left is always equal to 2/1, and the value on the right is always equal to 1/2. As the radius expands we can multiply the radius by 2 and 1 for the numerator and denominator for the 2/1 values, and by 1 for the numerator and 2 for the denominator for 1/2 values. This creates and expanding line, but all the fractions still simplify to 2 and 1/2 respectively.

On the Awareness Logic circle, we can then create additional fractions that occur between 0 and 1/2 with either positive or negative values (+/-) around the circumference on the right, along with their corresponding reciprocal values on the left. In the diagram of the expanded Awareness Logic circle above, we can see this expressed as (+/-) 1/4 on the right and (+/-) 4/1 on the left.

One of the most important aspects that the Awareness Logic circle demonstrates is the one to one relationship between the three elements of Awareness Logic; base count, context, and unit. The diagram above (and the subsequent examples) uses whole number fractions because this is the easiest way to illustrate this one to one relationship. However, it should be easy to infer how to use decimal (or any other number system) from these.

For example, on the expanded circle above, we show the positions for (+/-) 1/4 and 4/1 on the outer line with 4/2 and 2/4. For the inner circle with 2/1 and 1/2 we can infer that the same points along its radius would occur at (+/-) 2/0.5 0.5/2 respectively.

The purpose of the basic Awareness Logic circle is mainly to illustrate the relationships of 0, ∞, and 1 to أ. From this, we can begin to see how the Awareness Logic circle might be used as a circular or spherical special coordinate system. However, a few deficiencies with this basic configuration soon become apparent. Rather than detail the flaws now, let’s just skip over to demonstrating what actually does work.

There are two ways we can properly represent the full potential of an Awareness Logic matrix. The first is on a modified Cartesian grid square that expands upon the relationships of the Awareness Logic circle in a linear fashion, but doesn’t represent the correct radial angles and distances. The second is an expanded circular diagram that does include the correct angles and distances, and actually works perfectly as a grid for plotting wave functions.

Each representation has its advantages. The main advantage of the square matrix is that it more easily displays the relationship of all three elements together, and is more similar to what most people are accustomed to using. It is accurate enough for most digital type applications, whereas the circular matrix is more accurate for analog applications. As we’ll see, both are also fully compatible with each other, and can be used together or interchangeably. Which is preferable in any situation really just depends on the application.

Let’s start with the square matrix:

The key difference between this and a standard Cartesian grid is that what would normally be the x axis has an origin of 1 for the unit values, and the standard progression occurs as either the numerator (right) or denominator (left). The y axis for the base count values is more standard with an origin of 0 and positive (upper) and negative (lower) values.

The reference numbers for this grid have been displayed on the outside instead of in the center for easier reference of the 0 axis (vertical) values, and also because we can see the nonstandard progression for the 1 axis (horizontal).

The combined values for x and y are expressed horizontally along each 1 axis as fractions instead of as individual coordinates. This is carried out in the same way as with the expanded Awareness Logic circle above.

This matrix is also differs from a Cartesian grid in that the outermost right and left unit values are fixed so that the always simplify to 1/2 and 2/1.  This in turn fixes the horizontal radius of the line from the origin of this axis to the unit value itself which starts off on each side with double on the right, or the inverse of double the unit value on the left.

Because of this doubling, the figure presented here displays only even numbered unit values, but odd values occur at the halfway point between each whole number on the vertical grid. (For example, 3/1 and 1/3 occur at the 1.5 point vertically).

You may notice that for the unit values on display in each quadrant, the table above leaves off fractional values between 1/2 and 1 along with their reciprocal values of fractions between 1 and 2. These can easily be added.  

Notice the nice neat diagonal 45° line formed by the 2/1 to 1/2 horizontal lines on the grid. By in effect rotating the reference lines 90° (clockwise) for the additional range of values, we can make use of the empty space on the grid to continue the progression of these fractions vertically. This creates squares in each quadrant of the grid. The angles from the radius for this essentially work out too (with the same accuracy limits as the rest of the square grid).

One of the most important aspects of this matrix is how shifting through these four quadrants relates to the three elements of Awareness Logic. The center of the entire matrix is always equal the awareness point (أ) which is the intersection 0, ∞, 1. Anytime we shift values vertically, the horizontal reference line intersecting the center represents 0. For horizontal shifts, the vertical reference line represents 1. (Note, this is basically flipped for the second halves of the progressions, but the same principles apply.)

 When working with base count values, we use addition (and subtraction). So if we shift horizontally from any value to its opposite across the vertical reference line, the sum of the two is always equal to 0.

When working with unit values, we use multiplication (and division). So if we shift vertically from any value to its reciprocal across the horizontal reference line, the product of the two is always equal to 1.

As for context, if we shift diagonally between any two opposing values across the center of the grid (أ), we reverse both the (+/-) value, and the reciprocal (1/x). Multiplied together, these always equal -1.

So what does this tell us about the context value?

First of all this means that context occurs around the radius of أ. Every possible radial line around أ indicates context, but unlike the horizontal and vertical axes that cross the 0 and 1 lines, the value for context in each quadrant remains constant.

This makes perfect sense if we think about how the number lines for Awareness Logic work. The lines for the base count and unit extend from an origin with a fixed value (0 and 1 respectively) that potentially progress to infinity in both directions. However, the line (or circle) for context has fixed points on either end, 0 and 1, with a potentially infinite progression between the two. 

We can best see this using the examples of the 45° lines that form the halfway points of 2/1 and 1/2 in the grid diagrams above.  Flattening these out, we get:

Notice that when the constant equivalent values that occur at all points on the lines in every quadrant are multiplied against their diagonal opposites, the product equals -1. The same is true of every radial line that cuts across the awareness point (أ). (This includes both reference lines too.)

This same principal is true for every line that can be drawn across the radius. For example, we can draw a 4/1 to 1/4 line across the radius (at 22.5°) angles with the following values:

Also notice how ∞ is used at the center of all these lines. Since the values are effectively constant equivalents on each side, the entire range between the two occurs at the center point in between.  

With this we can now clearly see how all three elements of Awareness Logic, 0, ∞, and 1, intersect to form the awareness point (أ). This square grid shows how every element and every potential value of Awareness Logic is related.

For the purposes of understanding basic digital logic applications, this arrangement is sufficient. However, the exact angles and distances of the radial lines that are better served on a circular grid. And as it so happens, such an arrangement of Awareness Logic is not only useful for digital application, it’s ideal for expressing analog waveforms as base count, unit, and context, can be translated to amplitude, and phase…

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6 –Basic Awareness Logic Operations and Symmetry

Before we get to the Awareness Logic matrix, we need to first establish a little more about basic Awareness Logic operations. So far, we haven’t really done much that’s truly exceptional in terms of defining how to count and use balanced ternary to express Awareness Logic. The real unique worth of this system is in how it all comes together. In order to see how that happens, let’s begin by once again going back to the axiom of awareness. All values simultaneously reference zero (0), infinity (∞), and one (1), the unique intersection of which is equal to the Awareness Point (أ).

 0∞1 = أ

For Awareness Logic to function as a unique logic system, we need to look at the axiom of awareness as an operational equation. Since we are essentially using the ∞ symbol to represent specific context values, we can treat it for the purposes of Awareness Logic system as a number. So, as an equation the axiom of awareness becomes:

 (0 ∙ ∞ ∙ 1) = أ

This simple formula represents the actual value at the intersection of the awareness point itself (أ). This is always the starting point for any Awareness Logic operation.

To distinguish the range of potential values (variables) that can occur for the base count, context, and unit from the specifically stated values of 0, ∞, and 1, let’s introduce naming convention for each variable with the أ followed by 0, ∞, and 1 respectively in subscript. So the more general equation becomes:

(أ = (1أ ∙ ∞أ ∙ 0أ

Unless otherwise stated, for any Awareness Logic set, the base count is always assumed to be 0, the context is always assumed to be ∞, and the unit value is always assumed to be 1.

The numbers 0, 1, and ∞ (which again, for the purposes of Awareness Logic, we treat a number) each have certain unique multiplication properties.

- The product of any number (n) multiplied by 0 always equals 0.  (n ∙ 0 = 0)

- The product of any number (n) multiplied by 1 always equals that number. (n ∙ 1 = n)

- The product of any number (n) multiplied by ∞ equals ∞.  (n ∙ ∞ = ∞)

 For the function of the system of Awareness Logic, we are going to add three unique rules.

The first rule is established by the axiom of awareness itself, and the logical result of converting it into a formula.  0 multiplied by ∞ and by 1 together always equals the awareness point. (0 ∙ ∞ ∙ 1 = أ)

As per the axiom of awareness, أ is the unique result of the intersection of these three values together (in this case expressed as a product), and only these three values, equals أ. If any other number is plugged into the equation, normal operational rules generally apply.

However, within the specific system of Awareness Logic, there are a couple important caveats to this, and these form the basis of the second and third rules.

 The second rule is For the purpose of arriving at the product of any awareness logic set, any context value equal to ∞ is effectively canceled out UNLESS the base count is equal to 0 AND the unit value is equal to 1. In this case, as per the first rule, the product of that set is equal to أ.  

This simply means that anytime we encounter a context value of ∞, we can effectively assume that it is simply a place holder used in defining the order of sets. This, however, does not apply if the base count or unit value is equal to ∞, in which case, the normal operations for ∞ are carried out.

The third caveat is that In any series of awareness logic sets, anytime the product of an individual set is 0, that 0 value is not multiplied against the final overall product of all the sets together, but carried over as a separate set equal to 0 with its position in the series maintained by all applicable set parentheses.

This one sounds a bit more complicated than it actually is. The purpose here is to identify a set that has no relative value (0) but that is still specifically identified as a set in which operations are carried out. This is in contrast to a set that has a context of ∞, which effectively implies its existence is effectively ignored for the purpose of any final product of the Awareness Logic equation. This all becomes much clearer with a few practical examples.

If we are presented with a series of Awareness Logic values, the first thing we would do is start with our basic Awareness Logic formula of  0 ∙ ∞ ∙ 1 = أ (unless otherwise stated). We would then take each number defined as either a base count (أ0) or unit (أ1) value, and plug them into their respective places in the formula.

 Since the operation is included with the base count and unit values, we would immediately apply the stated operation and numeric value to the base count and unit positions of the formula. In effect this creates a parenthetical operation that establishes that base count and unit are potentially sets themselves. This is true because according to Awareness Logic every number and value can be broken down into constituent base count, context, and unit values.  

 As we’ll see with our examples below, anytime we are presented with a context (أ∞) value, we shift the set position to the concentric location defined by the ∞ symbol away from the current set position (which contains the current base count and unit values) implied by n. All we’re doing here is changing which set we’re applying the base count and unit values.

As an example, Let’s use the following series of Awareness Logic values which we’ll label A - I:

 A)     +2

B)      4/1

C)      -1

D)     ½

E)      (n)∞

F)      -2

G)     (((∞)))n

H)     ¼

I)        +3

Note that because the operation is included with each, we can always easily tell the difference between base count, unit, and context values. So, before we start plugging these values into the formula, we would start, as we always do, with awareness point:

(0 ∙ ∞ ∙ 1) = أ

Then we would carry out the operations for each value from the examples above in succession, so that the equation would change with each step as follows:

 A)     ( (0 +2= +2) ∙ ∞ ∙ 1) = 2

B)      (+2 ∙ ∞ ∙ (1 ∙ 4/1)) = 8

C)      ((+2 -1 = +1) ∙ ∞ ∙ 4/1) = 4

D)     (+1 ∙ ∞ ∙ (4/1 ∙ 1/2 = 2/1)) = 2

E)      (+1 ∙ ∞ ∙ 2/1) ∞ = 2

F)      ((0 -2 = -2) (+1 ∙ ∞ ∙ 2/1) 1) = 1

G)     (-2 (+1 ((∞)) 2/1) 1) = 1

H)     (-2 (+1 ((0(0 ∙ ∞ ∙ 1/4)1/4)) 2/1) 1) =  ((0)) 1

I)        (-2 (+1 (((0 +3 = 3) ∙ ∞ ∙ 1/4)1/4)) 2/1) 1) =  3/16 = 0.1875

Note that starting with example A, the context values of ∞ are canceled out for the final product. 

With examples E and G, we shifted the focus of the equation by adding an ∞ to a new set, but the final product did not change because as with each example, we continue to effectively ignore any context value = ∞. Then, in the subsequent examples of each, we applied the base count and unit values to the new set positions.

In example F, we applied a base count value of -2, and because there was no specifically stated unit value, we assumed أ1 was equal to 1.

In example H, we applied the unit value of 1/4, but because there was no specifically stated base count value yet for that set, we assumed base count was equal to 0. The product of that specific set was then 0, and we carried this 0 over with all the set specific parenthesis to the final product of the equation itself. 

With example I, the set with the value of 0 in H gains a numeric value other than 0 (in this case 3/4 or 0.75) for أ0, so all the attending set parentheses were canceled out as well.

The purpose of identifying the location of sets equal to 0 is that in Awareness Logic, a set equal to zero specifically calls attention to itself as an empty set that we are aware of. This is in contrast to a set with a context of infinity that we may be potential aware of, but because we can’t affix any finite value to it, is effectively ignored within our (finite) awareness. This is not to say that we cannot be aware of a potentially infinite value, however. This is why in Awareness Logic we apply the normal rules of using ∞ if it is applied to a base count or unit value.

This differentiation between 0 and ∞, and how these operate depending on if they are applied to the three elements of أ is an essential concept to the most fundamental principles of Awareness Logic. It’s an important part of what makes this system unique. Fully understanding the real value of this distinction depends a great deal on the specific practical application in which Awareness Logic is used. This will become clearer later as we explore some of the theoretical scientific applications of Awareness Logic.

For now, there is another general concept of Awareness Logic that is essential to grasp when considering basic operations. That is the principle of symmetry inherently implied by the existence of the awareness point (أ) itself. 

0, ∞, and 1 together are always equal to the awareness point (أ). This implies that أ itself has an inherent value that can be expressed relative to these. If the axiom of awareness holds true in that “all values simultaneously reference zero (0), infinity (∞), and one (1)” than this means that أ has a value that can relative to all numbers, and by logical extension anything that can be expressed by numbers.

So then, how do we define the value of أ against numbers other than 0, ∞, and 1?

If we consider the examples above, all we’re doing with Awareness Logic is taking the potential of that infinite, universal series endless 0s ∞s and 1s, breaking it down into concentric sets, and then collapsing these sets further into usable chunks by applying finite numeric values. 

For the purpose of Awareness Logic, every set can be assigned an order of operation (implied by the use of the parentheses) regardless of it is defined by finite or potentially infinite values. No matter how we define with numbers or break down these set with operations, the implication of the axiom of awareness is that at the heart of it all is the awareness point (أ).

So, if we apply أ to the operations that we have so far defined for Awareness Logic, what we can logically infer is that its value and function of أ is symmetrical based on how the formula (0 ∙ ∞ ∙ 1) operates.

So far we’ve established the concept that 0 and 1 are the complimentary (opposite) values that define all finite values. We’ve also established that the concepts of finite and infinite (∞) are complimentary opposites essential to defining each other in the same way.  On the numbers lines we’ve created, we’ve seen that if the center value is either 0 or 1, the line goes off in a potentially boundless progression (∞) in both directions. If the boundaries of the line are defined as 0 and 1, however, then the boundless range (∞) is between the two.

So essentially, we can say that on a numbers line, infinity (∞) is the transition or pivot point between two complimentary finite values (0 and 1, for example); and that any finite value ( again 0 or 1 for example) is the transition or pivot point between two potentially infinite (∞) or boundless ranges of values.

The finite relative limits of 0 and 1 contrast each other against infinity (∞), and infinity is at the boundary of every 0 and 1. This is a fundamental symmetry.

Therefore, as the awareness point (أ) is the intersection of 0, ∞, 1, أ logically applies as a transition or pivot point between any given number or value and a symmetrically defined complimentary value based on the relationship of the awareness point’s essential elements of 0, ∞, and 1.

To fully understand how these symmetrical complimentary values can logically be determined, we need to plot everything we’ve so far defined onto a common grid: the Awareness Logic matrix… Note: there is a problem on this page with html reversing a subscript combined with the Arabic letter  أ at seemingly random times. The correct annotations for base count, context, and unit are found in this formula:

(أ = (1أ ∙ ∞أ ∙ 0أ

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5 –Awareness Logic Trits

Now it’s time to forget about binary.

Awareness Logic is essentially a form of balanced ternary that generally conforms to Kleene’s Logic.  Balanced Ternary is typically expressed in terms of trits (trinary digits) which have three possible values:  -1, 0, and 1. Balanced ternary can also be used to express the logical statements of false (-1), unknown or indeterminate (0), and true (1).

Awareness Logic makes use of these standard balanced ternary values (relative to the base count as we’ll see), but the basic elements of Awareness Logic are 0, ∞, and 1 (which correspond to -1, 0, and 1 respectively). From this point forward, we’ll use 0, ∞, and 1 to distinguish Awareness Logic from other systems of balanced ternary.

Okay, so let’s create some trit tables.

Awareness Logic is a base 3 system which means that a single trit yields 31 possible values (3). As this represents the most fundamental range of values, the basic 0, ∞, and 1, we’ll denote the first column of any trit table with the symbol for the Awareness point (أ) which, of course, this also conveniently represents the intersection of these three elements.

Two trits yields 32 possible values (9). With this we begin to add the range of possible base count (0), unit (1), and context (∞) values. Trit tables are read from smallest numbers to largest from right to left, so we’ll add the second column to the left and denote it with 1.

For the base count, the values we gain with the 1 column are -1 and +1. It’s worth noting that the operators – and + are both explicated shown to illustrate that, for base count values, the addition (and subtraction) function is always carried out in Awareness Logic (as we’ll see later).

For the unit values, the 1 column creates 1/x and x/1, with x here representing any number. This follows the precedent set by the modified numbers line for the unit values we created in the last section. 

Once again, whole number values are stated as improper fractions to make the relationship to 1 more explicit, and also because multiplication (and division) operations are always carried out with unit values. To prevent redundancy, we can understand that 1 means the value is 1. 1/x is the option to create a reciprocal value, and x/1 simply means dividing x by 1 (which normally leaves the value the same). This brings us to the context values.  If we think back to our Awareness Logic stream, the way we utilized the ∞ symbols was to replace them with parentheses to denote concentric sets. So, what we’re doing here is essentially the same. First we establish where in the stream the numeric values (n) are. These are defined by the base count and unit. Then we establish where the context value (∞) being defined is placed relative to n.

(∞) n indicates that ∞ occupies a “smaller” set that is concentrically inside of n. 

(n) ∞ indicates that ∞ is in a “larger” set that is concentrically outside of n.

We can picture the concentric progression of Awareness Logic sets like ripples growing infinitely “larger” away from the center point of 0 towards an eventual 1. We could say that with (∞) n, the set (∞) is “smaller” than the set that contains n. The 0 trit value that falls in the 1 column for the context values indicates the sets are infinitely regressing towards the 0 point, implying smaller and smaller sets. With (n) ∞, the ∞ is in a “larger” than the concentric set that contains (n).

The 0 trit value that falls in the 1 column for the context values indicates the sets are infinitely regressing towards the 0 point, implying “smaller” and “smaller” sets. The 1 trit value in the 1 column for context indicates the sets are infinitely progressing towards some eventual 1 point, in effect creating increasingly “larger” sets.

However, physical size and even numeric value can be largely independent of this concentric progression in Awareness Logic. This is why instead of assigning a numeric value to context, we’re illustrating it here with just parentheses, because the actual values of the sets themselves may be independent of the concentric order in which they occur. We’re mainly just concerned with the relative position of the sets here. We’ll look at how to value these sets in more detail in the next section.

Three trits yields 33 possible values (27). As with the 1 and 2 trit tables, the first column establishes the Awareness Logic element is being defined. The second is the 1 column, and the third is 3.

There are a number of possible ways we could designate this range of values depending on the application, but the most intuitive way is to simply continue the progression of values for each Awareness Logic element.

For the base count, this creates a range with a low of -4 through 0 to a high of +4.

For the unit values, we have a range to 1/4 through 1 to 4/1. 

For the context values, we can see that the progression places the ∞ position further and further away from the n (defined by the base count and unit) into “smaller” or “larger” sets depending on if the value in the second (left-most) column is 0 or 1 respectively.

It’s easy to deduce the progression of additional trit values from here. Adding more trits yields 3n possible values, so the additional columns added after 1 and 3 would be 9, 27, 81, 243, etc. 

We can also calculate the range of possible values for each of the three Awareness Logic elements for any number of trits by calculating 3n divided by 3. We can also determine the numeric range on either side of the 0, ∞, or 1 by subtracting one from that, and dividing the remainder by half. So for example, we’ve seen that 3 trits (33) creates a numeric range of 4 on each side of the 0, ∞, and 1 values. 4 trits (34) would make for a range of 13 values on each side; 5 trits (35) creates 40 on each side, 6 trits (36) creates 122, etc.  

Looking at these tables, it soon becomes apparent that there is a range of potential digital logic applications for Awareness Logic. But, before we can get to those, we need to see how these values all function together. We can do this by constructing an Awareness Logic matrix that takes full advantage of the system’s unique features…

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4 – Counting with Awareness Logic

So far, you might have observed that the stream of numbers for Awareness Logic basically looks just like a binary series, but with ∞ inserted between the 1s and 0s. “What’s the big deal?” you may ask.

 To properly answer that question, we need to spend a little more time considering not just the potential value of infinity (∞), but also how we tend to be aware of it.

With the Awareness Logic stream, we’re basically establishing that between each 0 and 1, there is a potentially infinite number of additional 0s and 1s that are undefined. Now when our brains encounter ∞ in this type of context, we don’t normally stop what we’re doing, and try to calculate the entire infinite progression like some poor robot from a bad 20th from a bad century sci-fi show. We accept that this potential exists as a concept, and simply move on from there. 

In essence what we’re doing is deciding that ∞ represents a finite transition that occurs at some potential point in the infinite series of 0s and 1s that allows us to move forward. Again, we can intellectually grasp that there’s infinite potential to some degree, but what we’re generally more concerned with overall is that, somewhere in that infinite progression, there’s a transition that we can understand in practical terms.

That transition provides the context necessary to differentiate 0 values from 1 values. Otherwise it would all just be an infinite indefinable blob. Or, alternatively, we might see it as an endless series of 0s and 1s with no common context or transition point between them, thus making it forever impossible to get from one to the other. So, Awareness Logic begins by actually representing that context as ∞.

“Great,” you might reply, “so how do we count with it?”

Well, that depends on where we start. If we imagine our endless stream of  0, ∞, and 1, we can just designate any 0, ∞, or 1 as the point to begin counting. To illustrate this, let’s keep it simple, and begin our first example of counting by starting with 0.  

The way we can indicate the chosen 0 is by replacing the ∞ symbols on either side of it with brackets [0], which denotes that we are defining it as the basis of a set. We would then continue to replace all the surrounding ∞ symbols with brackets so that the initial 0 set is the center of a series of concentric sets radiating outward. For example:  

In essence all we’ve done is simply replace the ∞ symbols with defined transitions between sets that center on a particular 0. What we end up with looks like a binary series that goes off in two directions. In other words, we’ve just (more or less) created a basic numbers line:

This number line illustrates how we derive the 0 (“base count”) values of Awareness Logic.

This number line is also annotated with the binary value for each numeral, but since binary is base 2, an additional binary digit must be added to differentiate between the -/+ values, which we haven’t done in the example above. As Awareness Logic uses balanced ternary, it’s base 3, so this extra position is not necessary. We’ll illustrate how the coding for that works later.

But first, let’s consider what happens if we pick a 1 in our stream as the initial point to start counting:

This modified number line illustrates the basic concept of how we derive the 1 (“unit”) values of Awareness Logic. You’ll notice that there are a few differences here from the line for the base count (0) values number line. First of all, instead of brackets, we’re using parenthesis to indicate a mathematical order of operation for these sets. This is the standard for Awareness Logic for here on out.

There are also numbers on both the top and bottom of the number line. This is done to indicate fractions. The top numbers are the numerators, and the bottom the denominators. If we establish 1 as our basic unit for counting, instead of having numbers less than 0 (-n) on one side, and greater than zero on the other (+n), we have numbers less than one (n<1) and greater than one on the other (n>1). Since numbers are normally read from lowest to highest from right to left, the numbers less than one (<1) are on the right, and the numbers greater than one (>1) are on the left.

So from right to left, starting with the numbers on the right of the line, we have 1/7, 1/6, 1/5, 1/4, 1/3, 1/2. One (1) is in the center. Then the numbers to the left are 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1.

Once again equivalent binary values for the numerals are also displayed. But as with the -/+ values of the previous line, we would again need additional binary digits to denote the fractional operations. This is where binary really starts to fail us, and balanced ternary becomes an even more attractive option.

If we multiply the values from the right and left of 1 from any set, the product of all these sets equals 1. (For example, 1/2 ∙ 2/1 = 1). As we’ll soon see, this is standard operation for Awareness Logic.

It’s important to note, however, that the example above of the number line based on 1 is less than ideal because it displays a one to one symmetry for the distance and order of the numbers on both the right and left of 1 that doesn’t actually exist. This is for illustration purposes only. There is a legitimate way to maintain a similar symmetry that’s useful for Awareness Logic which we’ll see later, but for right now, this is sufficient to get the basic idea across.

So what happens if we pick an ∞ symbol from the stream as the point to start counting? 

Once again, we’re using parentheses instead of brackets. However, as the central value that starts our counting is ∞, replacing the next two immediate ∞ values to the right and left creates an innermost set of (0 ∞ 1). Note that the rest of the sets are then contrasting 0s and 1s indicating that there are potentially two different types of progressions: the first starting from 0, and the second starting from 1.

Also note that instead of using an arrow to designate the starting point, this time we used أ, the symbol for the awareness point.  What this means is that the designated starting point for counting on this stream is not simply ∞, or 0 or 1, but 0 ∞ 1 together: the awareness point. This represents a common point where all three intersect together. This is the foundation of Awareness Logic (0∞1 = أ).

“But,” you may ask, “How does that work exactly? How can we conceive of all three values, 0, ∞, and 1, as occupying the same point?”

If we picture these three elemental values as separate points on a number line (∞←0-∞-1→∞), it is difficult to understand them as simultaneously occupying a single point together. However, the trick to Awareness Logic is that it’s not just about lines, it’s also about circles.

   This circle illustrates how we define the ∞ (“context”) values of Awareness Logic. With a circle, we start at 0, and then make one full revolution around the circle to 1. We could then continue around in an infinite number of revolutions (in either direction). Also, any starting point we pick is one of an infinite potential number of such points around the circumference (regardless of the diameter). So now we’ve established that in Awareness Logic we have a more or less standard number line for our 0 (base count) values; a modified numbers line for our 1 (unit) values; and a circle of infinity for our ∞ (context) values. What we’ll soon see is that when combined, these form a unique matrix that allows for the triangulation of all three values. This can create some surprising insights into the very foundation of numbers theory, and opens the door to a number of exciting practical applications. But, in order to see how these three elements work together, we need to first continue our exploration of counting by defining how to code Awareness Logic using balance ternary…

[awarenesslogic]

3 - The Trouble with Binary

The genesis in my head of what eventually became Awareness Logic started as a thought experiment. I was trying to imagine how to define in the most fundamental terms a data matrix that could theoretically express all the information in the universe. A tall order to be sure!

The first logical choice was binary, but I quickly realized there were some real problems with this. Now, there’s a whole group of people (many of them programmers) operating on the more or less unexamined assumption that the entire universe can be expressed in binary terms. Binary is simply a two position number system consisting of 0 and 1, and, of course, has proven perfectly useful for countless digital applications. However, when the challenge is to map everything into a singular database, there are at least two important concepts that cannot be expressed exclusively in binary terms. The first is infinity, and the second is the role of the observer to any given observation expressed in binary.

First of all, how do you express infinity in binary? As all the 1s and 0s possible in the universe? I can write the word and conceive the concept of infinity, but how does infinity as a value get expressed using only 1s and 0s?

This brings us to the problem of expressing the role of an observer with binary. Suppose you establish as an observation a digital binary switch that has an output of 1 or 0. What is the value of the switch itself? How do we establish its basic function? You need a set of more 1s and 0s to explain that the switch is part of a lager set, and to define how it works.

 Okay, so then what is the value of an observer or operator in this equation? Why, another set of 1s and 0s of course! And the world around me is more 1s and 0s, etc.

 But how do we ultimately make the distinction between all those 1s and 0s? In order to differentiate this multitude of sets and functions, the concept of set and subset itself is essential. Without a higher order language to translate these values into distinct sets, it’s all just a meaningless endless series of 1s and 0s.

With Awareness Logic we have three fundamental values to work with instead of two. It is a system of balanced ternary. However, instead of using the standard values of  -1, 0, and 1 for balanced ternary (which will come into play later),  in  Awareness Logic 0 and 1 are the complimentary values instead of -1 an 1 (as in binary), and both are related to ∞ as the third value (instead of the standard balanced ternary 0). This structure not only creates an automatic distinction between finite and infinite values, it also establishes a basic set structure with the set of 0 and 1 as a subset of ∞.

Now before we go any further, let’s establish some clear definitions for the terms of the Axiom of Awareness.

 All values simultaneously reference zero (0), infinity (∞), and one (1), the unique intersection of which is equal to the Awareness Point (أ).

 0∞1 = أ

So, we have four values here: 0, 1, and ∞ plus  أ (the Arabic letter alif) as the symbol for the awareness point. The fourth one is a new concept that will take some explaining, but the first three are relatively simple to define.

 Zero (0) here is simply a unit of no relative value.

 One (1) is a unit of relative value.

Establishing the relative nature of both 0 and 1 is essential because in Awareness Logic these are contrasted to infinity (∞) as distinctly finite values, not absolutes.

Infinity (∞) is understood simply as a value without bounds; a value that is not finite.

 Some mathematicians hate the very concept of infinity itself because by its very nature, it does not fit neatly into any finitely definable system. So, clearly defining ∞ in this, its most basically accepted mathematical terms is essential.

∞ is not properly considered a number, but in Awareness Logic it is mostly used like a number. When working with the value of ∞ in Awareness Logic it is important to generally disregard any philosophical or cosmological implications inherent to the concept of infinity, and stick to its most commonly accepted usage as a value without bounds.

One of the most routine usages of ∞ is to represent an increase without boundaries, and understanding exactly what progression of values ∞ represents is generally context dependent. We can note how this works in three common examples:

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, → ∞

 333.3333→ ∞

 0→ ∞ →1

In the first example, we understand that ∞ represents the endless continuation of the series of numbers from 0 to 9 with 10, 11, 12, etc. In the second example, we understand that the decimal value is an endless series of 3s. The third example is a bit different because in this case, the 0 and 1 represent clear boundaries, however we can infer from this that there is a potentially infinite series of values between the two. This usage is particularly relevant to Awareness Logic.

0 and 1 define each other’s existence. As is expressed with binary, these are the two essential finite values. As 0 and 1 work together establish the range of finite values, 0 and 1 as a pair contrast with ∞ to establish the dynamic of finite and infinite. The differentiation between finite values and infinite potential is necessary to express any set function.

The fourth essential to define from the axiom of awareness is the awareness point (أ). 

The first letter of the Arabic alphabet, alif, is used here as the symbol for the awareness point. This letter was chosen to represent the awareness point because not only is it associated with some appropriate symbolism in Islamic culture, it also implies an association with its linguistic cousin in Hebrew, the aleph (א), which is used to represent the transfinite numbers of set theory. The mathematical concepts behind أ and א are similarly related by this usage as well.

 (Note: The alif-hamza form (أ) was chosen instead of the stand-alone form (ا) because of the latter’s similar appearance to the character “1”.)

The awareness point (أ) is a unique value and essentially a new number that expresses an implied but previously undefined value. Because it represents the intersection of both infinite and finite characteristics, أ can be understood as the essential reference for our awareness of all numbers and values, hence the use of the word “awareness” in Awareness Logic.

 In order to define exactly what the awareness point (أ) is, and how it’s used, the first thing we need to establish the absolute basics. That is how to count using 0, 1 and ∞.  And by adding ∞, we’ll transcend the limitations imposed by binary, and begin to define that data matrix I imagined that can potentially express all the information in the universe...