Chapter 2: Mathematics and Computers
Mathematics is a powerful tool for our exploration of the universe. But what does it represent? Thousands of years ago humans developed the mathematical counting board. The abacus, still in use today, remains the prominent example.
Most counting boards use ten symbols. It has a series of columns, each column using nine beads for nine of its ten symbols. The tenth symbol is represented by the column when it is empty. Combinations of beads and no beads in successive columns allow the user to achieve rapid computations.
For example, the number 306 uses the first three columns of the accounting board. The first column represents three beads, the second column no beads, and the third column six beads. Numbers can be added and subtracted rapidly by simply moving the corresponding numbers of beads to their corresponding positions.
The ancient counting board performed simple calculations on the same principle as today's electronic calculator. But there was a major limitation upon the counting board's application to advanced computation.
No one had converted the counting board's method of positional counting to the written symbols of positional counting that are necessary for complex computations. If you were a citizen of ancient Rome, imagine attempting to add, subtract, multiply and divide large numbers of papyrus using the Roman numeral system.
To illustrate, use Roman numeral notation to square CCCVI with pen and paper. The question is, what prevented people from translating the counting board principle into writing? The answer is that no one grasped the idea of creating a written symbol for the empty column of no beads.
It is incredible that after thousands of years of hundreds of thousands of people counting millions of empty columns, no one thought of making a written equivalent of positional notation for the empty column.
This was finally accomplished through the invention of the zero, which represents the empty column. Euclid, Pythagoras, Archimedes and all the other intellectuals of Greece failed to see it. The failure to see nothing and its crucial relationship to something remains a central reason we are at an impasse in our thinking on discovering fundamental explanations of reality.
This is not to say that our understanding and use of the physical is unimportant. The problem is, we have failed to recognize that it is only through an understanding of the non-physical that we can make great strides in understanding the physical. Each depends upon the other.
The non-physical nothing is and will continue to be an equal partner with the physical something in shaping the cosmic events of the universe. Finally, a long since forgotten Hindu decided to use the symbol of a dot to represent the empty column. This Hindu religion may have prepared him to do this because Hinduism recognizes the value of nothing as spatial organization created and formed by the physical.
Much later the dot was changed into the modern zero. The introduction of the zero transformed mathematics into its modern form and gave it its pervasive status. The counting board deserves a closer examination because the principle behind it has a profound application not only to humans and their structured civilization but to the entire natural world.
The counting board uses the physical to construct the framework for its columns and beads. The physical rod of the column on which the beads slide is an example of how the physical outside surface of an object organizes the space next to it.
The purpose of the columns is to give a geometrical form to the nothing around the columns to provide a place of occupation, a hole to be filled for beads. Note that a hole was placed in the beads for occupation by the physical part of the column.
The process of emission and absorption could then be used for the entering and the leaving of the beads. Each column of nothing is designed to hold 1 to 9 beads. The challenge is, if there are only 9 beads, how can we then indicate the quantity of 10?
Let's start with using only the first column. As we add more columns, we add them to the right. To use only the first column, we begin with that column in an empty state. There is nothing to fill without a previous empty. Now we place one bead after another into the first column until we reach the ninth bead.
At this point, we want to show 10 entities that could be 10 of anything. To do this, we add another column to the right. To show 10 of something, we place only one bead in the first column instead of nine beads, and the second column to the right is left empty.
Now the one bead in the first column indicates a column completely filled one time. We know 1 means a one-time fill of 10 because of the emptiness of the second column to the right. This emptiness states that the first empty of the first column has now been filled.
The 1 of the first column and the 0 of the second column is a salient example of the dependency of opposites. An empty 0 is necessary to determine a full 1. In this example, an adjacent nothing empty of the second column is necessary to identify an adjacent something which is one bead of the first column showing a column completely filled one time.
If we did not use an empty second column, we would know that one bead meant just one of something in the first column and more beads could be added to the first column. Thanks to the anonymous Hindu, we can transform the principle of the counting board to pencil and paper by expressing 10 with a 1 in combination with a 0 to the right of it.
Without the combination of the 1 and the 0, the 1 alone would just represent 1 of something, a single integer. This illustrates the necessity of introducing nothing to understand something. The zero to the right is the one, is saying, I am here to let you know that I am ready to be filled again because the first zero, the prior empty condition of the first column, is now completely filled.
The written symbols showing the completely filled condition of 10 of the first column is 10. If we use two columns with two beads in the first and no beads in the second column, we can represent twenty. This can be transferred to written form with the symbols two zero.
These two means that two zeros are filled with ten each. This information is understood because the new zero is ready to be filled to the right of the two. the number of beads to the left counts how many zeros have been displaced by being filled.
We know this because an empty second column has been introduced to the right. This is an example of something requiring nothing next to it to gain existence and recognition.
The one with a zero to its right is usually thought of as 10 in written form.
In contrast, what if the column of the counting board was designed for six beads instead of 10? When using two columns where the first column has one bead and the second column has no beads, this means the zero of the first column is completely filled with one time with only six.
Now the symbols 10 in the written form would mean 6 of something instead of the usual 10 of something. The arbitrary use of a base of 10 in the Arabic numeral system probably evolved from humans having 10 fingers. The 1 of the 10 in the written form can also represent any full container as in the full container of the counting board column.
The zero of the 10 can represent any empty zero condition that always surrounds, identifies and is next to all full containers. As an example, the container column of the counting board may be designed for 10 beads, while a container pack for beer may only be designed for 6 beers.
The zero empty column of the counting board container takes 10 and the zero of the beer container takes 6. The filled one six pack of beer, and the nothing, zero, which is next to and surrounds this filled condition, can also be referred to in writing as 10. Now instead of 10 of something, the symbol 10 means 6 of something, beers.
All full containers in nature can be represented by 10.
It is important to emphasize that the zero represents an empty container. If the zero is designed for 8 of something and we write the symbol 4 by itself, we know the zero is only half full and has room for 4 more of something.
The value of the invention of the 0 and 1 designations is to fill 1 and to empty holes 0 on a writing surface. Now we can add, subtract, multiply and divide physical objects with a pencil or pen without getting involved with their actual physical manipulation.
Learning to count with numerical symbols on paper led us to a great success in filling and emptying real objects represented by those symbols. The use of nothing is crucial in mathematics for the same reason that nothing is the frame of reference, corollary number 2, for not only our observations but also for the transportation of physical objects.
Nothing is crucial for all universal action. The zero in written form records the nothing holes used by nature's physical entities. The zero in the numerical ten describes the empty condition next to all completely filled conditions.
The 10 is written to represent containers that are completely filled. However, to interpret nature through our senses, we always require an empty, zero, in contrast to the physical, one, in understanding any object. Even on the printed page, a 1 written by itself needs space, 0, around and next to it to observe its identity.
The central premise here is that the zero in mathematics is a hole used in transporting numerals representing anything we want from one place to another.
The zero enables us to represent emission and absorption on paper. We move things represented by numbers in and out of a hole indicated by a zero. This symbolic representation mimics nature.
The power of mathematics is in its use of the same emission and absorption principles that form the basis for all natural processes and events. More complex forms of mathematics merely use different symbols and methods for departure and arrival process that are also found throughout nature.
The Romans used the symbol X to represent ten somethings, but with the introduction of the zero, the Roman numeral system was rendered obsolete. The ten notation represents ten of something. Whatever the size the zero is designated for, the 10 is saying that we have a one-time filling of the first hole because of the emergence of a new hole next to it.
If the base size of zero is 6, then the zero is designed for 6 of something and 10 would then represent 6 of something. The arbitrary decision on the size of the whole, the size of zero, is infinite.
The important point is that the underlying foundation of the Arabic numeral system and mathematics involves the use of the 1 and the 0, that is, the physical 1 and the non-physical 0.
The numerals 2, 3, 4, 5, 6, 7, 8, 9 are single integers that each express how many ones are being used. The principle is the same with whatever base we choose and all base systems have the 1 and the 0 as their foundation.
We can even create a base of 1. To use a base of 1, the 0 is designed to be filled only by a 1 of something. In this example, when we write 10, it would mean the container 0 can only accommodate a single 1. With a base 1, however, it would be unnecessary to use a 0 to the right because a 1 is all there is within the base 1. There is no need to place a 0 next to the 1 because the 1 by itself says it all.
By design, we already know this is all we can use. It can be argued, isn't this an example of not having to use the zero because the one of something has no necessity of the zero of nothing? Is this a case where binary zero and one relations do not apply and the one can stand alone?
To show that all numeral-based systems use the zero and one as their underlying principle, the zero must somehow be used. The purpose of any piece of writing paper is to recognize the zero of its surface for filling with words, numbers, pictures or whatever is desired.
When writing a one, the physical ink, the one, does not occupy the physical paper. It displaces the place zero on its surface. To place a physical entity anywhere requires a hole.
Similarly, alphabetic letters need spaces and holes between them so we can distinguish them. Words need even more space to distinguish them. Writing the 1 by itself is not possible if by itself means, without the 0.
The vertical slash mark of a 1 not only fills a container whole, the 0 of the writing surface, but it is also identified by the nothing that surrounds it. A useful instructional challenge is to try to identify anything involved with your observations or actions that does not require the use of both the 1, physical, and the 0, non-physical.
To use the 1 of anything means accomplishing an emission to absorption phase that requires the use of the 0.
To write a 1 on a sheet of paper is an emission process for the ink being absorbed by the hole that is organized by the surface of the paper. The base of 1 is a natural conclusion where there is the in-depth analysis of the zero. Is the base 1 revealed in mathematical texts? It seems simple by itself, but it is not simple to recognize its relationship with everything else.
The more useful a principle is, the simpler it is. We add to knowledge by making less explain more. The complexity of nature is structural, how things are put together, but the underlying binary 0 and 1 principle governing the building process is simple.
The base one has limited usage in mathematics. It is used here to demonstrate the principle that a physical existence, a place for an object to occupy, depends on the prior existence of the non-physical hole and the recognition of something, one, comes from being surrounded by nothing, zero.
The zero's circular shape is ideal for a hole's description. It is the physical nature of ink, chalk and pencil surrounding nothing, zero, that defines nothing. Something defines nothing and nothing defines something.
The physical circle of the zero encloses the non-physical space inside. The physical surrounds the non-physical, creating the circular shape and is complete in organizing the empty space into a hole represented by a circle.
In this example, the zero, in three dimensions, more physical boundaries will better organize the non-physical. Within the limits of writing paper and its two dimensions, it is difficult to imagine a more appropriate symbol for this information than the zero. We point to nothing by completely enclosing it with something.
The physical slash symbolizing the numeral one is opposite the zero. The non-physical completely surrounds and defines the physical. Both the zero and one are defined by the boundaries of each other, corollary number one.
Consider the example of a tree standing in a forest compared to a tree standing in a clearing. In which of these scenes does a tree stand out and command our attention? Not surprisingly, the tree in the clearing is the standout. The lone tree stands in sharp contrast to the emptiness of no trees, the clearing.
Similarly, think of a meadow, 0, surrounded by a forest, 1, compared to a meadow surrounded by other meadows. The physical part of the forest defines these spaces better because it surrounds it.
Understanding the 0 and 1 frames of reference for our observations sensitizes us to the importance of contrasting opposites. Realizing this binary principle will add greatly to the meaning behind our observations.
We use mathematics to solve problems. Problems always reveal ignorance, lack of information and knowledge that represent empty holes to be filled. Once aware of these holes of ignorance, we can seek to fill them with useful knowledge.
The mathematician begins the solution of his problems through the emission of equation combinations that comprise previous states of knowledge. This motional development of the mathematical equation is analogous to the necessity of finding unknown pathways for the arrival of solutions, answers.
The question, zero, leads to the answer, one, of absorption, through the phase of emission. Any question is a zero waiting to be answered with a one. The mental absorption of a solution can provide the knowledge, the emission and absorption for us.
When our body successfully occupies space, this indicates a prior successful occupation of space in the mind. As children, we were taught that when we subtract two somethings from four somethings, we are left with two somethings.
However, when we remove a 1 of something, we always leave a 0 behind. It is important to understand that removal, subtraction, of a filled, one, condition leaves behind an empty 0 condition. When we subtract something, we add nothing.
To advance our understanding of something, we must be cognizant of the position of the organized nothing, zero, as we are cognizant of the organized something. To learn this early in life can give us a great advantage in how we both approach and solve problems.
When the 2 of something is subtracted from the 4 of something, we are left with the 2 of something and the 2 of nothing. As we subtract objects, we add the 0 of no objects and as we add objects, we subtract the 0 of no objects.
In the following chapters, the symbol 1 will be used to indicate the physical and in contrast, the symbol 0 will be used to indicate the non-physical. The zero indicates nothing and the one indicates something.
Computers
In only half a century, the digital computer has become a prime mover and organizer of human civilization.
The modern computer is built upon the ancient principle of the counting board. The counting board manipulates a series of zeros defined by the columns that are created to emit and absorb the ones of beads.
The computer manipulates a series of zeros, semi-conductors, defined by the ones of atomic structure that are created to emit and absorb the ones of electrons. The organized emission of electrons by the cathode ray tube and absorption of those electrons by the computer screen forms alphanumeric characters and pictures.
These electron combinations give us the physical appearance of data which is seen in contrast to the non-physical surrounding the physical.
The digital computer is based upon the binary system of zero and one that is manipulated by an ingeniously designed series of switches. Like all switches, they can be switched to either the one, on, or off, 0, position.
Computer electronics manipulates the mathematical base of 2, which requires only the 0 and 1 for its numerical designations.
In the base 2 system, the 0's size is for the 2 of something. In binary notation of base 2, the symbol 10, one/zero, means 2 instead of 10. The basis of the computer, however, is not the two of something. It follows the binary principle of zero and one behind all base systems, mathematics and natural phenomena.
The digital computer is a binary tool because it uses two opposite concepts, the hole and the electron which fills 1 and empties 0, the hole. The binary 0 and 1 act as pairs to create the absorber, hole, for the absorption and emission of electrons.
The computer is commonly thought of as unique in its binary applications. However, a purpose of this work is to reveal the universality of binary, zero and one, relationships and applications.
The off is the emergence of the zero and the on is the emergence of the one. All information can be represented with this power on, 1, and power off, 0, principle.
The computer follows the same rules of nature that govern all emission and absorption events. Only the structures are different.
As will be shown, humans and the civilizations they build must also have this binary zero and one base to attain viability, stability and durability.
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