Simplifying the Technical Aspects of the D-world and GGU-models
Robert A. Herrmann Ph. D. Mathematics Department U. S. Naval Academy 572 Holloway Rd. Annapolis, MD 21402-5002
1. Introduction, Language Usage and Other Stuff.
Before I start this informally presented article, I must make one very, very important disclosure. All of the D-world and GGU-model results are PREDICTED from but a few common everyday human experiences that can be verified within a laboratory setting. Except for the most basic mathematical axioms used throughout all of science, NO additional mathematical or physical axioms or presuppositions are assumed. This must always be kept in mind, for there will arise the myth that the results have somehow or other been included within the axioms or presuppositions. This myth is entirely and utterly false.The discussion presented here does not contain the most recent signifiant results and simplifications that appear in my book "Science Declares Our Universe IS Intelligently Designed" (Xulon Press)
From Nobel prize winner Louis deBroglie, comes the statement:. . . the structure of the material universe has something in common with the laws that govern the workings of the human mind. [March, 1963, p. 143] Then from the viewpoint not of a scientist but a philosopher, recall what C. S. Lewis writes:
. . . that events in the remotest parts of space appear to obey the laws of rational thought....According to it what is behind the universe is more like a mind than it is anything else we know. [Lewis, 1978] In this small article, I'll present the intuitive and foundational concepts needed to establish scientifically these two conjectures, but this will be done in a simplified manner. Much of the seemingly incomprehensible material that appears in various technical papers will be removed by showing that, in reality, the foundations are easily understood. The great difficulty only comes when the modern methods of mathematical modeling are employed in an attempt to predict new results. First, we need a definition as to what is the most common feature of all of scientific discourse.
Science, "at the least," is composed of systematized and reasoned descriptions. By the way, one term in this definition has been altered. The usual term "knowledge" has been replaced by the less innocuous term "descriptions" since a "description" need not mean this illusive thing we call "fact." The term "reasoned" means a specific type of reasoning called scientific reasoning. Such reasoning is a combination of classical logic and the somewhat weak process of induction.
Is there a common feature associated with my basic definition of a science that's so obvious that it may have been over looked? Having systematized and reasoned descriptions is somewhat worthless if such descriptions can't be communicated to others. "Obviously," it's assumed that such descriptions will be communicated, somehow or other, to other individuals working within a particular discipline. The ability to communicate ideas is one of the crowning achievements of the human mind and shouldn't be underestimated. Let's consider some of the more basic aspects of this common feature -- basic aspects that apply to all disciplines, in general.
In order to transmit information, oral communication is translated into written symbolism. Such communication is comprised of words, phrases, sentences, paragraphs, books, and libraries of written expressions. Along with these written expressions are the common rules - the rules of grammar - that guide us in combining such expressions into collections so that they have meaning to a particular object with which you wish to communicate. Notice that I just used the word "object" since, today, you do tend to communicate with, what is still classified as, nonhuman objects. You may have been taught to use a special language to communicate with a computer or an automatic teller machine. You discover quickly the trouble that occurs if these special languages aren't used. Such experiences have certainly increased our understanding of what exact communication means.
Clearly, all of the symbolic means employed to communicate haven't been discussed as yet. There are diagrams, figures, and drawings that one might classify as symbols for something or other. But then what would you do with the great visual communicators that are photographs, motion pictures, television, television tape and the like? First, since photographs, motion pictures and television pictures can be stored on television tape, only such tape needs to be considered. Would you consider the information contained on TV tapes as expressible in a written form?
Modern technology is used to produce an exact string of symbols that will reproduce, with great clarity, any visual image or sound impression stored on a TV tape. Today, information is "digitized." Each small fluorescent region on a TV screen is given a coded location within a computer program. What electron beam turns on or off, the intensity of the beam and the like is encoded in a series of binary digits. The computer software then decodes this information and the beam sweeps out a glowing picture on your TV tube. At the next sweep, a different decoded series of digits yields a slightly different picture. And, after many hundreds of these sweeps, the human brain coupled with the eye's persistence of vision yields a faithful mental motion picture. Each of these single glowing pictures can also be faithfully reproduced in a written language. All one needs to do is to state that at a particular moment of atomic-clock time a picture element (pixel) has a particular intensity and a particular "color." The entire description composed of such statements for every pixel will correspond to a single TV image and when "translated" by the electronic hardware, that can also be completely described, becomes the desired visual image. Please notice that digitizing is not necessary for such a pixel description.
Did you know that when you make a telephone call that the major telephone communication networks digitize the sounds and then faithfully reassemble the numerical codes into a reproduction of what the listener believes is "your voice"? Record companies digitize music in order to improve upon the reproduction quality. Schematics for the construction of equipment can be faithfully described in words and phrases if a fine enough map type grid is used. Thus, the complete computer software expressed in a computer language, the digitized or other types of inputs along with schematics of how to build the equipment to encode and decode digitized information, taken together, can be considered as an enormous symbol string the exact content of which will be what you perceive on "the tube," hear from your CD player, or other such devices.
Today scientists use these mechanical devices in the place of human observation whenever possible since human observation is often not considered a trusted observational procedure. Thus, in all that follows, the notion of a "string of symbols" as a representation for scientific information will always include the idea that portions of such a string also communicate visual or audio information to the human brain. But, since it's communicated to the human brain, then the notion of the content also reveals itself.
All of the effects that such a string of symbols has upon the human brain is the content contained within that string of symbols.Now the exact content of a string of symbols, certainly, depends upon an individual's personal experiences and technical training for the same written term or visual image often conveys a different content relative to different disciplines. Although it may present some content problems, much of what science considers to be perception may be replaced by a long exact string of symbols (i.e. symbol strings). This leads to what is truly the common feature shared by all sciences.The common feature is communication by means of strings of symbols of one sort or another. Members of a specific discipline create what is termed as a technical dictionary that contains strings of symbols that are accepted by its members. [Recall that this will always include visual images, audio impressions and, if you wish, tactile or olfactory descriptive information.] For the present discussion, you don't need to consider this dictionary as containing any basic meanings for the individual strings of symbols. What's required is a relatively fixed set of rules for combining different symbol strings into longer more complex collections that all members of a discipline can use. The technical symbol dictionary and all the accepted combinations dictated by these rules forms what is termed a discipline or technical language.
Although the content of a particular set of symbols may be described as all of the effects it has upon an individuals's brain, the simplicity of this definition stops at the point where you analyze its properties. Content has many diverse characteristics that often lead to the difficulties humans experience in communicating ideas. Content is exquisitely related to human experience, and it's particularly significant for what is called a description.
Individuals experience the results of mental impressions, whether such impressions are internally or externally generated.Why is a description only an approximate reproduction of the original mental impressions? As a child, I had a great deal of difficulty communicating orally. I was very frustrated for I didn't have a large enough vocabulary to communicate my ideas. This difficulty in communicating mental impressions continues to plague any scientist who discovers a "truly" new concept. The new concept requires new terms. And, when these new terms are created, they need to be defined by descriptions using previous dictionary terms. The combinations of these old dictionary terms, if possible, should not occur in previous definitions so that these new terms will definitely have content different from all the other technical terms. Unfortunately, even if I'm careful in term definitions, my experience has been that a description using these new terms never seems to convey all of the content I intend. This becomes more obvious when I ask other scientists to re-describe the content of my description in their own words, so to speak. Their partial descriptions often fall far short of what, I perceive to be, is the true content of the new terms. Of course, in actuality, there's no absolute way to determine that the content a description conveys to one individual is the exact same content that it conveys to another individual since one would need a form of communication to make such a determination. This leads to a kind of spiraling processes that may never end.A description is a combination of symbol strings (including those representing visual or audio information etc.) that when considered by an individual evokes mental impressions.
First, an individual's mental impressions lead to the formation of a description. Secondly, a description is intended to evoke within the same individual, and others, approximately the same mental impressions which originally determined the description.
What does the display of descriptive information have to do with the deBroglie and Lewis statements? It's what comes after such a description that's the bases of modern science. Suppose that a description conveys information about the behavior of a natural object at a particular time or place. The modern scientist takes such a description, couples it with additional statements called natural laws, and deduces scientifically a new description that is claimed to represent the behavior of the natural object at another time or place. It's the notion of scientific deduction that ties together descriptions and the deBroglie and Lewis statements.
2. Predictions. First, consider the following very useful definition.
A Natural-system is an arrangement of physical objects that are so related as to form an identifiable unity. Usually, scientists will declare that a particular arrangement forms a Natural-system and give such a system a name.Thus the sun, the earth, the solar system and a virus are Natural-systems. Suppose that you're conducting an experiment with a named Natural-system. Let's call a description for an observed or assumed real physical phenomenon associated with such a Natural-system a cause. Notice that in this case, the cause is a description, a linguistic-like object. Let the observed or assumed real physical phenomenon itself, the one assumed to exist in objective reality (outside of the brain of the observer), be called an event. Another description for an observed or assumed real physical phenomenon is written down and suppose that your experience and training leads you to conclude that this second observation is closely related to the first observation. You might conclude that the second described observation is the effect of the cause. This leads to the following "cause and effect" patternCause produces (produced) the effect (1) It's rather obvious that the concept of "produces" takes on a dual meaning.
(i) From a symbol string or language point of view, you have one set of symbols, called the cause, followed by the word "produces" or "produced," then followed immediately by another set of symbols, the effect(s).These dual characteristics associated with scientific communication are very significant for the concept called predictions. Why did the second event occur? Suppose that you now couple the first description with what you claim is a set of natural laws or additional general hypotheses, and apply scientific logic to the combination. Further, suppose that you have deduced logically the exact same second description that was obtained from your laboratory observation. The term "produces" should have a content that includes these additional general statements, the applicable natural laws or processes, and logical procedures. Further, these laws or processes should apply to many different descriptions for original events. So suppose that you claim that certain natural laws or processes have produced the described effects. You wish to establish your claim. You set up hundreds of other similar experiments and using the same natural laws predict, from descriptions for different original events, descriptions for new events that are actually observed within the laboratory. These new events seem to follow from each specific original event. The natural laws are the fixed part of the "produces" content. It appears that for humankind to comprehend the idea of "produces," we assume that Nature is "putting together" the cause description with the natural laws to produce an effect in the same logical manner that the human brain puts together symbol strings and deduces new symbol strings.(ii) From the physical point of view, you have one real physical phenomenon, somehow or other, physically yielding another physical phenomenon.
Let's discuss and illustrate more fully exactly what is actually being done. There are certain laws and procedures that have come down to us from Aristotle and others as to how we should put strings of symbols together and arrive at another string of symbols. These are called the classical laws of human logic. These laws correspond to our everyday linguistic experiences where strings of symbols are used to describe those natural events that are perceived by our five senses. These logical laws are also those accepted by the majority of humanity as the proper way "to think" since these logical laws are used to communicate and even construct the material things we use in everyday life. If we can't perceive a natural event, we simply assume that the same logical laws hold. However, the laws of human logic are more general in character and can be applied to any string of symbols not just a string that appears to describe a natural event. It's the pen and paper process - a game to some - of writing down strings of symbols and combining them together by use of certain accepted laws of human logic that yields a scientific theory.
Let's discuss one of the simplest accepted patterns for human thought as applied to strings of symbols. Since this law of logic applies to any string of symbols, one probably shouldn't write done a specific string of symbols to illustrate this pattern of human thought. This pattern will work for whatever string you choose. So, to simplify matters, let the bold face symbols X, Y, and Z denote any written statements (i.e. any written string of symbols) considered as a description. Let NL denote the string of symbols termed natural laws. Let & denote the word "and." Now consider a new string of symbols which, in two equivalent ways, is expressed explicitly as follows:
implies, (or) implies logically where the concept of "implies" is obtained intuitively by considering its properties as expressed by the laws of human logic. The idea of understanding intuitively a new explicitly expressed string of symbols shouldn't be new to you if you ever had a class in high school geometry. After all, to what does a straight line correspond? It's anything to which the properties stated in a geometry text for a "line" can be associated.
The above prediction could be written symbolically as
X & NL implies Y & NL (2) Now let E1 be the event described by X and E2 the event described by Y. Then we have the pattern
E1 => E2 (3) You can associate the => with any physical term you wish, our interest is in the event pattern (3).
Let's put the three patterns together for reference purposes.
Cause produces (produced) the effect (1) X & NL implies Y & NL. (2) E1 => E2 (3) Note the following important fact. The natural laws are themselves not part of an actual "observed" event, they are part of the "produces" content. The event is simply described linguistically. Thus the natural laws are like what is called an axiom system that is always present and can be used at any moment. The actual descriptions for the observed events have the pattern
X implies Y (4) It matters not whether the natural laws are considered as part of the description for a natural event, or suppressed entirely and considered as part of a natural law axiom system. Expression (4) uses actual "observations" for Natural-system behavior and is the bases for the actual logical patterns. Now one of the most basic properties for "implies" is illustrated by the following set of directions. Write down a set of statements:
X implies Y Y implies Z (5) The classical laws of human logic now allow you to write down the statement:
X implies Z (6) (You can "sort of" erase the Y.)
Letting E3 be the event described by Z, we have the event patterns
E1 => E2 E2 => E3 E1 => E3 (7) Such patterns are accepted within science as correct in that it's assumed that this represents how humanly comprehensible natural events are actually being combined together within a Natural-system. Of course, this is a very elementary illustration for the deBroglie and Lewis statements. Such patterns exist for all of classical logic. By the way, if you want an immediate "proof" for a portion of this last remark, consider the classical propositional logic. These are the exact patterns used to construct each and every modern computer. However, not all Natural-systems follow all the patterns for classical logic. But, all Natural-systems do follow a certain weak portion of the rules for classical logic and this weak portion is the actual set of rules used to obtain many of the D-world, and GGU-model results. What can be gleaned from observing the absolute similarity between the linguistic patterns (5) and (6) and the event patterns (7) and the exact same similarity between more complex linguistic patterns associated with human thought and event patterns is that
the behavior patterns for certain Natural mechanisms is being mirrored or "modeled" by the linguistic patterns associated with mental processes.
3. What does the term model mean? I once wrote a long article discussing the different meanings of this term but in the present context it means one and only one thing. A "model" need have no "resemblance" to the actual physical objects being modeled. After we have assigned to each physical object its corresponding object within the model, all that a model does is to "behave in a similar manner" as the physical objects. This correspondence is called an interpretation. One somewhat strange example should suffice. In the Big Bang theory for how our universe behaves, we have the concept of the expanding universe. One basic way to model this is by the balloon analogy. Take a rubber balloon. Using small spots of glue, glue pennies on this balloon while it's somewhat inflated. The correspondence is that the pennies correspond to galaxies. Now begin blowing up this balloon. What do you see? Well, one of the things you see is that the pennies are moving slowly away from each other. This behavior is suppose to illustrate, or be a model for, the expanding universe. But, it doesn't do so in any fashion except one.
In this model, you aren't suppose to notice the balloon. You are to suppress the fact that it's the rubber that's stretching. You aren't suppose to notice that it's being blown up by a lot of hot air. You aren't suppose to notice that you are observing the entire event from "outside" of the so-called universe as a god might. You are only suppose to "see" the pennies (galaxies) moving away from one another, and you are also to conclude that this motion isn't related to any force acting between the individual galaxies themselves nor is it produced by the galaxies themselves nor produced by any other perceivable material within the universe. This is a model for this and only this behavior and has NO correspondence to reality. Indeed, what is the reality? The reality is that the balloon represents a "nothingness" or a complete emptiness from the stand point of any known perceived substitute for the rubber of the balloon. But, from this nothingness, it's also claimed by many that certain real things called virtual particles simply pop into existence. Although there have been various ad hoc attempts to describe this nothingness, only the GGU-model predicts from actual very simple physical considerations the entities that may actually comprise this nothingness. But, it still can't be perceived by means of any machines constructed by any "intelligent" life form.
Let's expand slightly upon the concept of "behavior." In the context of this strict definition for a model, the behavior means a description for a relationship between objects within a model where you substitute for the modeling objects the corresponding physical objects. Thus, in the balloon case, we have that the pennies = galaxies are moving away from each other not due to any forces or properties of the individual pennies = galaxies themselves. Although this might lead one to ask a question such as how is this possible, one should suppress this desire. Why? Well, just because we can ask such questions, doesn't mean that there's an answer that can be expressed in any human language or that we could comprehend the answer if it were expressed by means of some string of symbols. After all, our comprehension is based upon how our minds function.
4. Modeling Thinking. Patterns such as those that appear in expressions (5) and (6) and much more complex ones have been studied by philosophers for thousands of years. Mathematicians began studying such patterns in the late 1800s and have created the applied mathematics discipline called Mathematical Logic. In the 1930s, the mathematician Tarski abstracted three of the most basic properties relative to how the human mind puts together various strings of symbols and logically deduces new strings of symbols. The mathematical "thing" or operator that's used to express these three relations is called a (finitary) consequence operator. Various terms are used for the objects that this operator "operators" upon. Tarski calls them sentences. They are also called readable sentences as well as words or readable words. Whatever they are intuitively, they are sets of finitely long strings of symbols. To be consistent, from this point on and when no confusion will result, the term word will be used for a finitely long string of symbols. Some are very long words such as the string of symbols that started at the S in the title of this article and ends with the period symbol . that appears at the end of the last sentence just after the word "symbols". Due to our modern language constructions, you can also use a special symbol for the spaces used within this somewhat long word. The consequence operator "models" the actual process the brain goes through when it deduces a conclusion by considering words as inputs and what words one produces mentally as outputs. But, is this the same concept as the balloon model? It's almost the same.
You know that a door is unlocked. So you push on it, but it doesn't open; so you immediately pull the handle. Why did you do this? Your brain has gone through a very rapid process and deduced that this is what you should immediately attempt. The same thing happens thousands of times a day. In reality, science doesn't actually know what your brain has done internally although there are various theories. Now there are some individuals who would continue to push the door. For some reason, their brain doesn't process the information in the same way as, say, 99.9% of humankind processes the information. But our human society is built upon the ability for each of us to process this information so that we all, well almost all, arrive at this same simple conclusion. Individuals who can't do this can't survive very long within our society and are usually confined to an institution where they will not harm themselves or others.
Suppose you write down a statement X. You also have learned many so-called natural laws. For some reason, your brain processes all this information and you conjecture that statement X implies Y. Scientists, usually, will not accept your conjecture in this fashion, they want a derivation or proof that what you conjecture does follow "logically." What you must do is to show how the natural laws can be combined together using accepted logical laws and at the end of the informal deduction the final thing you write down is the Y. But, in informal work, you don't actually write down any logical law as such. You might not even know that there are such "laws." Since each step in the derivation is small enough, you hope that other scientists will simply "feel" that everything is OK logically. Indeed, it may be impossible to convince all individuals that what you have derived is logically correct since you express the derivation at the logical and experiential level of the audience you are intending to convince AND, usually, everyone in your audience assumes the same unstated presuppositions. That is, you don't express everything. In mathematical logic, mathematicians attempt to model many aspects of how the brain functions but without knowing how it functions biologically. That is, they model, among other things, the logical procedures that don't appear in the informal argument the scientist gives, the procedures that would simply appear to be correct to another individual without any further justification. But, this modeling process isn't exactly the same as the balloon model.
Besides the general consequence operator properties, certain fixed and very simple rules have been discovered that will allow anyone, who can apply the rules, to take the X and the natural laws and, in theory anyway, deduce the Y. Indeed, in many cases, a machine can actually apply the rules and determine whether the X logically implies Y. Different types of logical deduction correspond to different sets of rules. Of course, a certain amount of low level "thinking" must be done in order to apply the rules. But, the rules are statements written in words and sentences; they aren't biological processes that the brain might use. Thus there's no correspondence between these rules and the processes that the brain actually uses to combine all this stuff together. This is the great difference between modeling logical processes that take a word from a language and produces another string of symbols. All we know is that we get the same result by application of these rules. What is important in this model is the pattern of the information to which the logical process is applied and the pattern of the results, among other interesting patterns. Suppressing the actual rules for the logical process is, in this case, equivalent to suppressing the rubber as the stuff for the expanding universe since we don't know how the brain actual does the deduction. Of course, you can spend your entire life deducing predictions without ever knowing of the existence of these sets of strict logical rules.
Two examples should suffice as to how this model is used to investigate thinking. Suppose that you have two individuals Bob and Ray. Now Bob takes a word X, an hypothesis. Supposedly both Bob and Ray have studied the same set of natural laws. Now Bob deduces, in one hour, 35 consequences, conclusions or predictions; the Ys. But Ray deduces, in one hour, 135 consequences, conclusions, etc. and these include Bob's 35. Indeed, every time this mental experiment is done Ray always deduces Bob's conclusions and many, many more. One might conclude that Ray is a more powerful thinker than Bob. This is exactly the definition that is applied to Ray's deductive processes, Ray's deductive processes are called great than, or better than, or more powerful than Bob's. We don't know anything else about Bob and Ray's actual thinking processes. But, we model these unknown processes by the symbol string results.
In the first example, we simply looked at the mental inputs and the outputs as words, but we also have the additional refined process of looking at specific rules of logic. Let's restrict Bob and Ray to the specific written rules for their logical approach. When the rules are applied, one makes a list of the actual rules used. Each time a rule is used, a step must be written down on a piece of paper and the specific rule used must be stated. You give both Bob and Ray the same hypothesis X and you tell each of them that it's possible to deduce, using the rule approach, a conclusion Y. After one hour, Ray has written down 36 steps, with reasons for each step, and his last step is Y. Bob, after one hour, hasn't reached, by application of the rules, the Y. He is "stuck" and has only 8 steps. Indeed, it's known that no one can apply less than 30 steps and obtain the result Y. Hence, the fact that Ray could do the much longer application of the rules and Bob couldn't, within the allotted time, would indicate that Ray has "stronger" mental powers.
By the way, although I can "think" in pictures any time I wish to do so, most of the time, when I'm conscious of my thinking, I think in words. These words are spoken by what I call my own mental voice. I assume this is the case with most individuals. If so, then language is, indeed, of exceptional significance.
How do we model the above examples? There are various ways to do this modeling. The most basic is to model only a few words that appear in the symbol strings, such as the words "and, or, not, implies" by means of special symbols and use letters, such as P, Q, R, X, Y, etc. to represent or correspond to the other portions of the symbol strings. We study the patterns produced by these new symbols when the logical rules are applied and relations between these new symbols. For consequence operators, the basic approach is a weak set-theory where we correspond a specific set to a specific set of words and relations between these sets of words are studied. Of course, to count things such as the number of steps in a logical derivation, the number of specific symbols in a given word and things like that, basic number theory is used. Number theory is the most empirically consistent and has the longest history of any of the mathematical theories used today. But, for the D-world and GGU-models, an additional step is used, a step conceived of originally by Gödel although Tarski may have thought of it prior to Gödel.
First, each and every distinct word is given a unique natural number name. This is done in such a way that this process models the usual method of expressing a symbol string one portion of a string at a time, say by writing it from left-to-right using single symbols or combinations of symbols. A given word is thus composed of all of the possible strings of symbols that can be put together in this fashion to produce the given string. The use of number theory yields what is technically called a numerical encoding. This allows the modeling to be structured by entities from a very empirically consistent mathematical theory. Further, the process allows for the counting of all the symbols that make up a specific word and a specific mathematical entity allows the mathematician to study the behavior of a particular symbol that is located at a particular place within a word. Of course, computers do the same thing for the single members of a symbolic alphabet. This is the ACSII code numbers. The logical processes are modeled by relations between the corresponding numerical code names. For example, suppose that an hypothesis X has the numerical name of 2431 and a consequence operator conclusion has the name 98043. Then these two symbols are related. One could write them in say "order pair" form (2431,98043) as a representation of such a relation. Of course, you could go backwards, in this case, and correspond each numerical code with the actual words that are so related.
The above processes are the methods used to obtain the standard model for the D-world model. The standard model includes all of the additional properties that are obtained by means of mathematical reasoning. There's, however, one more very important concept. If we stayed with language and only with language concepts, then what we have is a model for linguistic behavior, a model for how human beings think with strings of symbols and an immediate correspondence of a symbol string to the behavior that the string may be describing. But, suppose that a word is a description for an event. Are there physical notions that can be modeled by relations between corresponding symbol strings?
5. Nature and Logic. The meaning of the last statement in section 2 should now be somewhat more easily understood. We don't really know how the brain biologically puts information together in order to proceed from X to Y. However, we do know of a set of written rules that will yield the same result. From my view point, we also don't know how Nature puts together just the right natural laws and only those natural laws to take the event described by X and yield the event described by Y. Does Nature simply say, "Event E1 you will, indeed, you must follow this and only this collection of natural laws and nothing else and event E2 must be the result"? It will serve no further purpose to discuss this aspect of Nature, at present.
Most scientists simply assume that whatever "forces" entities to follow certain specific natural laws, is whatever "forces" entities to follow certain specific natural laws. Let's illustrate what comes next by the art of cartoon animation. An animator has decided that he wants to make a cartoon of flying pigs. In our immediate natural environment, pigs don't seem to fly. The animator has some mental ideas as to how he would adjust natural laws so that pigs could fly. He puts his plan into action and starts to draw hundreds and hundreds of images of how he believes, using these ideas, the pigs would appear as they fly over a lake. Flipping through the drawings at just the correct speed, due to his persistence of vision, the pigs actually appear to fly and their flight doesn't seem chaotic but is rather smooth or, should we say, logical in character. If he wants to reproduce these drawings on motion picture film, using the 30 frames per second mode, he needs 300 drawings for 10 seconds of pig flight. Technically each drawing might be called a frozen frame, freeze frame, (like the "still" or "pulse" control on a VCR), or simply a frame.
Now, as discussed previously, each drawing can be made to correspond to a specific string of symbols, a word, that is called a frozen segment in order to correlate it to the freeze frame idea. The entire set of all such symbols can be made to correspond to a special single string of symbols, a special standard word. What is interesting is that there's a very simple logical process S, a set of actual rules, that when these rules are applied to this one word each and every single pig flight "image" (i.e. the symbol string that corresponds to an image) appears in the exact same order needed for the animation process to make the pigs appear to fly. In other words, one could say that this logical process S "forces" the pigs to appear to fly when the process is applied to this single static word that contains enough of the images for pig flight or the flight plan, say. The animator now shows his animation to another animator who will continue the flying pigs cartoon and asks, "Can you figure out how I've altered the natural laws so that pigs now fly?" Determining this might take a great deal of scientific experimentation with the animator's flying pigs. It would have been better if the animator simply wrote down his alterations to the natural laws and gave a copy of this to the second animator. This would be the easy way out and isn't what is usually encountered within physical science.
The most basic aspect of the scientific method is that there's a describable cause for each alteration in the behavior of a Natural-system. Secular science attempts to discover the natural processes that will lead to such alterations and using such processes the scientist can create a series of freeze frames that will yield the moment-by-moment behavioral changes where each change is an actual event. Once again, in many cases, this leads to a word w that when S is applied to w, the S will also force the Natural-system to follow the same moment-by-moment alteration. Of course, one must also apply the fixed correspondence between each frozen segment and a specific event. Since this correspondence is always assumed as part of the process, the mentioning of it is often suppressed. But, there's an additional difficulty. How refined, how many freeze frames, are needed to express properly the actual alterations in Natural-system behavior? For a flying pig and motion picture film reproduction, you only need one drawing per film frame. Is this all you need for a refined description for how Natural-systems alter their behavior?
Relative to the method used by Richard Feynman to describe how electromagnetic radiation interacts with natural objects, he writes, ". . . while I am describing to you how Nature works, you won't understand why Nature works that way. But you see, nobody understands that. I can't explain why Nature behaves in this peculiar way." [Feynman, 1985, p. 10] Relative to one implication, I must disagree with Feynman. Scientists don't know how Nature works, in this case. Feynman has only modeled Nature by little probability arrows on a piece of paper, stop watches, rules for combining arrows, rules for calculating probabilities, and the so-called paths taken by photons. I respectfully submit that Nature probably doesn't do geometric diagramming with probability arrows, Nature probably doesn't do the mathematics and other such human paper and pencil activities. His geometric approach is, I submit, but a model for how Nature works and nothing more AND as with models like the balloon model, except for the behavior between entities, many aspects of the Feynman approach need not correspond directly to anything in objective reality. Indeed, certain predicted aspects of the GGU-model can replace various aspects of these Feynman diagrams. The science of the invisible microscopic world is the science of models as I have defined them. Scientists don't really know how or why Nature does something in many, many cases. Because of all of these uses of models, it shouldn't be difficult to comprehend the GGU-model. But what about the paths of motion for photons? Considering the mathematics used, you can't use a finite collection of freeze frames as a complete refined description for such behavior. Thus, in such cases, there wouldn't exist an actual standard word w such that S applied to w would yield the same refined alterations in photon behavior that produces its path of motion.
When they exist, words such as w might be described as the images, or building plans for how a Natural-system will evolve since it's the force-like process S that would put them together to form the actual development. After the concepts of the word and the force-like S were discovered, there was also discovered the following quotation attributed to the exceptionally brilliant scientist Hermann Weyl.
Is it conceivable that immaterial factors having the nature of images, ideas, "building plans" also intervene in the evolution of the world as a whole? As will be seen in the remainder of this monograph, such immaterial factors are indeed conceivable.
Finally, it's a simple matter to show that Tarski's three properties for the consequence operator, of which S is an example, expressed in event language and hundreds of other properties seem to "model the behavior" of any evolving Natural-system. As stated above, this is the entire idea behind theoretical science, and much of science in general, that "logical" behavior governs the development of any Natural-system. Whether or not such logical behavior is simply a requirement for human perception and comprehension, or Nature actually behaves in this fashion is a philosophical question, the discussion of which may depend upon considerations that need not be scientific in character.
6. The Ultra-stuff. The D-world model is the standard model for thinking as discussed in section 4 plus the ultra-stuff. The GGU-model is the logical model for how Nature behaves as discussed in 5 plus the ultra-stuff. But, what is this ultra-stuff?
The mathematical theory that correlates directly to the model for thinking in section 4 is also called a mathematical structure. In the early 1960s, Abraham Robinson discovered a general method to expand such mathematical structures so that the new structure would essentially include the old structure and often a lot more. Now this is a general approach. No new mathematical axioms are used, and no logical or physical presuppositions are added. The new structure obtained is usually one that can be simply termed as an enlargement. It's within this new structure that the ultra-stuff is predicted to exist mathematically. The word mathematically is very important here. It simply means that we have mathematical proofs using mathematical reasoning that state that "For each consequence operator S, there exists an ultra-object *S with such and such properties." Now recall that the properties of the original S are really properties about collections of natural numbers. BUT, when describing the linguistic properties of *S we suppress the numerical encodings, just like the rubber in the balloon model, and go back to the original symbol string correspondence and linguistic concepts for the D-world model based upon the material in section 4.
The same suppression of the numerical encodings is used for the ultra-stuff associated with the material discussed in section 5 with an additional suppression. The linguistic correspondence is also suppressed in the GGU-model and we go to the physical-like behavior being described, the "event, the force-like and object" interpretation. Each altered frozen frame is interpreted as a (physical) event. The consequence operator is a force-like physical process that yields the event. There are a few more physical-like interpretations associated with the GGU-model that correspond to other suppressed linguistic concepts. Thus, for the GGU-model, the actual process starts with mathematical relations between "numerically" encoded things that behave like strings of symbols. Then the linguistic interpretation is suppressed and the physical-like interpretation brought to the forefront. Such an interpretation scheme isn't foreign to theoretical science since this is done within the subject of quantum logic and was done, in the early 1970s, in an attempt to answer the pre-geometry cosmological problem.
Often the same terms are used for the linguistic and physical-like interpretations. But, the context of a discussion will always imply which of these two interpretations is to be used.
It's a remarkable fact that mathematicians are able to "see" mentally the relations between both the original S and the new *S. Indeed, the entire idea of Nonstandard Analysis is that the mathematician is exterior to these processes and can make such observations. Unfortunately, however, there were no linguistic like terms that could be used to discuss such a relation for the D-world comparisons and no physical-like terms for the GGU-model comparisons. These new terms had to be invented and this was difficult to do.
The difficulty in selecting terms was that they should be mentally related to the previous terms, yet different from these terms and, if possible, not be the same as any other scientific terms. The prefix "super" was chosen first. But, it was discovered that another scientist had also chosen this term for an entirely different purpose in discussing logical operators. After literally years of trying, the term "ultra" was decided upon and you will see exactly how this prefix is used. It should be pointed out that within the actual mathematics itself, the "ultra" term need not be the actual general term used. Recall that we have the original interpretation for such linguistic terms as a string of symbols, the word w, the consequence operator, a set of logical rules or laws, etc. What we now have is an additional interpretation for similar but different objects within the enlargement that carry an "ultra" prefix. But what do I mean by similar?
One of the most significance properties of this mathematical approach is that if the properties of the consequence operator S are characterized mathematically, then *S has these same properties with respect to standard and other types of words if one is careful how one expresses these properties. But, *S has new and startling properties that S doesn't have under the new interpretation. Moreover, these new properties are predicted by the enlargement all by itself. All one has to do is to find them. Nothing is ever added by the mathematician except for the names given to these new objects.
It's also very important, as these relations are discussed, that the concept of the model be always kept in mind. The discussion is only about behavior and not about the stuff of which the model is composed. The discussion is about new objects that are predicted logically by mathematical means. At no time does this ever imply that these new objects exist in objective reality. But it's scientifically rational to assume their existence. Such an assumption is much more rational in character than many of the assumptions within theoretical science since this assumption is based entirely upon a mathematical model. Within subatomic physics and cosmology, assumptions may only be partially based upon mathematical structures. In these sciences, the belief that certain objects or processes exist in object reality is either a result of indirect verification or is based upon what is perceived to be as a better description for Natural-system behavior, a description that doesn't contradict certain philosophical presuppositions. The exact same philosophy of science is applied to the D-world and GGU-models.
7. The D-world Model. What is an intelligent life-form? In this work, an intelligent life-form is defined as any entity within the natural universe and that (1) did, does or will communicate by strings of symbols in the manners discussed previously in this monograph and that (2) uses mental deductive processes that can be modeled mathematically by methods similar to those previously discussed. In order to correspond to later work, intelligent life-forms, as defined, will also be called standard intelligent life-forms. Note that the methods used in this analysis are methods that can be applied at any time and to any such intelligent life-form. This should always be understood. What has been developed uses the simple methods used to compared the mental power and other such concepts for various intelligent life-forms; but, as will be seen, the conclusions refer to ANY standard intelligent life-form at any moment during its development, not just humankind.
The D-world (deductive-world) model models mathematically linguistic processes which are themselves models for mental processes. Such processes are associated with a standard mathematical structure and this structure becomes the standard model. Then everything is embedded into an enlargement and we get an enlarged or nonstandard model. The word "nonstandard" doesn't mean that there's something wrong with the model. This is but a technical term.
Remember that, technically, all entities in the D-world and GGU-models are mathematical objects associated with the natural numbers. But, the relations and terms are translated consistently in terms of linguistic concepts for the D-world model and physical-like concepts for the GGU-model. It's the relations between such translated terms that gives a rational description for behavior. The simplest way to illustrate this is with the simplest example. Let P denote the set of all possible standard words, as defined previously, that exist anywhere within our universe at the very moment you read this sentence. Thus P is the standard language used by the standard intelligent life-forms. The mathematical model states that since P exists then (it's rational to assume) that there exists another set of stuff *P, generally called a nonstandard language, with certain properties.
First, members of P are members of *P. This is also stated as "P is a subset of *P." But what properties does *P have? Well, members of *P have many of the same properties as do members of P. Indeed, if certain properties for P are stated in a fixed but special way called a first-order statement, then certain members of *P have these "same" properties. For example, given any natural number, say n = 1,000, there's in P a word with n = 1,000 symbols counting left to right, with repetition. This property can be more generally stated as follows: given any natural number, there exists in P a word with length equal to that natural number. Further, there's an intuitive relation called the "length" relation that assigns to every word the number of symbols, with repetition, contained in that string of symbols as you might count them from left to right. The above statement gives a property for this length relation.
Mathematically, the length relation is also termed as a function or map or mapping. What this length function does is to take a given member of P, call it p, and assign to p its length n. This might be denoted as L(p) = n. More formally the above property says that (A) given any natural number n, there's some p in P such that L(p) = n. To understand properly how the length function behaves, you might write down a few statements about counting the number of symbols from left to right such as if there's a word of length 1,000, then adjoining a symbol at the right most end will give a word of length 1,001. Now does *P have these same properties? The answer is yes if you translate these properties correctly. For example, one translation would formally look like the following:
(*A) given any *natural number n, there's some p in *P such that *L(p) = n. In mathematics, it's said that (*A) "holds" in our enlargement. This means that if (A) is assumed, then there's an acceptable mathematical proof where the (*A) statement is the conclusion. Now what do the "stars" mean and how would this formal statement be translated into a statement about symbol strings, about words?
In technical work, the "star" takes on various names such as "hyper" or simply "star." But if the starred object had a name prior to its being starred, then that name is also included. [The reason for this will be illustrated below.] Thus "*natural number" might be read "hypernatural number." In this example, there are two new entities to examine, the hypernatural numbers and the *L.
How do the hypernatural numbers behave? Well, the same translation method is used with the properties of the natural numbers. Please note that the interest is in behavior patterns. Also, to look at some interesting properties, we adjoin the concepts of the addition of two numbers and < (less than) to the natural number basic properties. Thus we know that given any natural number n, then n + 1 is a natural number and further n < n+1. This property translates to the following: given any hypernatural number n, then n *+ 1 is a hypernatural number and further n *< n *+ 1. Note that the exact same behavior pattern is being displayed and holds mathematically for the hypernatural numbers. This is the reason the standard name is retained as part of the nonstandard name. Of course, we also have the apparently new objects *+ and *<. But to learn how these new objects behave, one writes down in the required special way the properties for + and < and then interprets these properties in terms of the new object *+ and *<. Again the exact same behavior patterns appear. By the way, the process of translating into the star notation is called "star transfer" or simply "transfer." Now if the hypernatural numbers behave exactly like the natural numbers in all respects, then hypernatural numbers would be of no additional interest. But, the facts are that in certain respects the hypernatural numbers don't behave exactly like the natural numbers and it's only because the mathematician can stand "outside" or "external" to this transfer process and make comparisons that it's known that the behavior is somewhat different or that the hypernatural numbers behave in a nonstandard way. That is, they have properties that the natural numbers don't have.
It's not the purpose of this monograph to discuss in-depth the mathematical differences between the hypernatural numbers and the natural numbers. These few mentioned differences are just being used as illustrations. However, when a comparison is made, it's discovered that the relation < and the relation *< have different properties. For any set of natural numbers, < has a special property. But, for certain sets of hypernatural numbers, *< doesn't have this same special property. Now what about the more interesting, at the least for this investigation, length function *L?
It's customary to use a positive language when discussing objects that exist mathematically in the sense that the statement being made is a conclusion of a mathematical proof. This customary procedure will be followed throughout the remainder of this monograph. This is illustrated by the next statement. First, its very important to note that there are (i.e. there exist mathematically) pure nonstandard words in *P and these pure nonstandard words are not members of P. Since this application of these mathematical techniques is relative to linguistic concepts, some of the more significant of these pure nonstandard words are called ultrawords. Finite sets of these nonstandard words have the same basic properties as the words that appear in P. But, they have other properties that they don't share with any member of P. A simple analysis shows that some of these nonstandard words have lengths that are greater than, in the usual sense, any standard word. A deeper analysis shows the remarkable fact that some of the ultrawords actually contain objects that would correspond to an ultra-alphabet. That is, objects that behave like the alphabet from which the words in P were constructed but which can't be members of any standard alphabet. If you take from P only those strings of symbols that, intuitively, "have meaning" and characterize what one might define by the phrase "have meaning" for a standard intelligent life-form, then there are ultrawords that have ultra-meaning and they would, in general, behave in a similar fashion. Although this monograph is designed only to give a basic illustration of how these models behave, it's useful to illustrate one of the greatest differences between ultrawords and standard words.
As mentioned in section 6, symbols such as S and C, the one used next, can be used as part of a mathematical model for how standard intelligent life-forms think. Further, for a given consequence operator C, the ultralogic *C has the exact same *-transferred behavior patterns as does C. But, *C actually differs greatly in its overall behavior. Just a few examples will suffice. Let w be a standard meaningful word upon which the logical process C is applied. The logical process being modeled by C is the simplest one that human beings apply thousands of times a day, the propositional logic. These are the same logical patterns used to construct modern computer microprocessors.
The *C also applies to w. Comparing the two results of these processes, we find that every standard conclusion obtained by the logical process C is also obtained when *C is applied. But, numerously many pure nonstandard words are also obtained. Thus it's ALWAYS the case that *C is more powerful than C under our definition of such a concept. Indeed, analyzing any of our measures for the strengths of a logical process leads to the conclusion that *C is stronger than C in each and every case. Further, there are ultrawords to which *C can be applied, but C can't be applied to any of these ultrawords. It's also interesting that when *C is applied to certain ultrawords, than standard words may be part of the "logical" conclusions so obtained. These results and many more generate the theory called the deductive-world (D-world) model. Finally, its very important to note again that these D-world model results are all PREDICTED and they can be applied to any discipline that might need such scientifically obtained conclusions.
8. The GGU-model. Approximately 90 years ago, Einstein introduced the idea of the "photon" as an "imaginary" physical entity that carried energy and momentum. It carries the light (electromagnetic radiation) energy among other properties. How does this object behave? Can we truly relate this to anything within our human experience? Is a photon a particle or a manifestation of a wave, a wave produced within a medium? Many scientists still believe that photons are manifestations of waves produced within a medium. But, Nobel Prize winner Feynman states, "The first important feature about light is that it appears to be particles. . ." [Feynman, 1985, p. 36] Indeed, if one doesn't assume that it behaves let a particle, one would probably have a difficult time passing a modern physics course. But, does a photon actually behave like a particle?
What is a particle? Usually, it's a very small material object. If this is the case, then a photon doesn't behave like anything that we have ever experienced within our material world. Feynman's major contribution to physical science is in the area of quantum electrodynamics and this theory uses as its basic entity the photon. But Feynman states, "The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense." [Feynman, 1985, p. 10] My dictionary states that absurd means "clearly untrue or unreasonable." I suppose that Feynman means "unreasonable" in the sense that the behavior being described is actually counter to our experiences. When a photon interacts with an assumed material object such as an electron we are sometimes told that, "The laws of conservation of momentum and conservation of energy are then applied to this interaction as they would be to the collision of one billiard ball with another." But a photon doesn't behave like a billiard ball since the first billiard ball doesn't vanish after interaction while the specific photon ceases to exist after interaction. Further, the photon doesn't have a mass. The actual energy and momentum appear to depend upon the frequency and wave-length of the photon, respectively. Indeed, the velocity component of the photon isn't transferred to the electron except that the product of the frequency and wave-length is the velocity of the photon, and the frequency and wave-length are transferred to the electron in the form of energy and momentum.
Since the entity described as a photon seems to vanish, there's another way to describe this interaction. Duff states that the electron "absorbs" the photon and, when an electron accelerates, it also "emits" photons [Duff, 1986, p. 26]. But an electron is supposed to have no internal structure. How does something with no internal structure do such things as absorb or emit a photon? (Of course, maybe all "fundamental entities" are actually bundles of photon-like objects.) I mentioned that a photon has no mass, but it's affected by a gravitational field as if it had some mass. Every particle of matter known that has ever been observed to be ejected from some source, will add its velocity to the velocity of the source from which it was ejected. But, this isn't the case with the photon. Feynman calls the entire theory he uses as "crazy" and "strange" and it's clearly distinct from anything that humans have ever experienced within their observed environment. But, Feynman was able to introduce "picture thinking" into this theory, the Feynman diagrams. So, something that doesn't follow any of our human experiences does follow, according to Feynman, a set of geometric diagrams. This means that individuals who wish to apply this theory to predict behavior must learn all the rules and procedures as specifically stated and not gave any further thought as to the obvious "strange" behavior presented or the complete lack of any experiential intuition. Further, don't ask certain questions. Just do as you are told to do. But, why have I brought up this "absurd" theory as a major illustration?
(Note: The metamorphic-anamorphosis model (MA-model) forms a portion of the GGU-model in that it represent the special ``sudden'' appearance of a physical entities or alterations in Natural-system behavior.) The GGU-model has a few things in common with the basic philosophy of science used in the theory of quantum electrodynamics. The fact is that the GGU-model describes the behavior of new physical-like entities, entities that don't follow human experiential intuition as physical entities but do follow human linguistic experiences. If one can learn how photons are supposed to behave within quantum electrodynamics, then it shouldn't be difficult to learn how the GGU-model behaves. If one accepts the strange, absurd, or crazy behavior of photons as described within quantum electrodynamics, then it shouldn't be difficult to accept the behavior of ultrawords, ultralogics and subparticles within the GGU-model without having any further in-depth or additional comprehension. Indeed, we have more knowledge about such things as an ultraword than we have about photons since the internal structure of an ultraword can be analyzed. But, of course, it would be nonsense, would it not, to consider a photon as having an internal structure or that it's composed of yet other fundamental stuff? Well, the GGU-model predicts that a photon is composed of other fundamental stuff and that everything within our universe is composed of the exact same fundamental stuff. How this new stuff is put together to produce such things as photons, electrons, quarks, etc. is much, much simpler than putting together the words found in this particular sentence. For example, all one needs to do is to bundle together a collect of ultrasubparticles, look closely at the bundle and what you'll "see" is an electron.
Remember that the GGU-model uses many of the same terms as used within the D-world model; but, like a very, very intelligent animator, the GGU-model assigns to these entities physical-like behavior. But, first and foremost, the GGU-model is a scientific cosmogony. Such a cosmogony gives processes and entities that can produce many different cosmologies, where a cosmology is a theory for how a particular universe is formed or evolves over time. On the other hand, such a cosmogony can't be what is called a tautological description. It must be what is termed as "logically falsifiable." This is the case with the GGU-model. There are certain statements about GGU-model processes that are speculation. On the other hand, simply due to the existence of the GGU-model, there are statements that are general conclusions about the GGU-model and that are fact. Before presenting some of the GGU-model speculations, these facts are stated.
(1) Any scientific description that describes a developing Natural-system and that includes any statements relative to mechanisms that at any time T, in the past, could produce or sustain such a development has the GGU-model as an absolute alternative.How are all of the above conclusions obtained? In this brief monograph, only the most basic and least technical aspects of the GGU-model will be discussed. As mentioned in section 5, we have descriptions (i.e. symbol strings) that yield information on how a Natural-system evolves over a period of time. Such a sequence of descriptions, like the animator's drawings, follow from the natural laws that the Natural-system is somehow or other required to follow. Further, the generation of these descriptions from known natural laws follows the basic rules of logic. The descriptions exist even if the appropriate natural laws haven't as yet been applied properly and specifically to such a development by a standard intelligent life-form. Of course, the actual development might vary somewhat depending upon the intervention of other processes. But, we, at the least, have a sequence of descriptions that yields an ideal behavior. Each member of this sequence of descriptions corresponds to a natural event. Thus we also have an actual event sequence as well. A particular sequence of such descriptions is called a developmental paradigm. Recall that such a sequence includes representations for visual, audio, etc. impressions. Hence, from a standpoint of what is allowed within science, the developmental paradigm represents all that science can "know" about the actual appearance and development of a Natural-system.(2) For a given scientific theory that you might select as consistent with your personal philosophy, whether it be a Big Bang, biological evolution or otherwise, the GGU-model can reproduce the same scientific theory for all time periods after T.
(3) There are infinitely many scientific descriptions for how a specific Natural-system may have developed prior to T including all of the standard theories. Further, there may never be a scientific language that can describe in any detail how a specific Natural-system developed prior to T.
(4) Scientifically, the GGU-model predicts that there are infinitely many different scenarios, different "beginnings," and the like that could have produced the universe in which we dwell, the earth on which we live and everything about it, and even you yourself. This includes an actual beginning for a possible cyclic universe that may appear to "infinitely repeat itself." But each of these infinitely many scenarios could produce the exact same results, the exact same images on photographic plates, the exact same readings on most data gathering machines as we observe these machines, today, within every laboratory. On the other hand, among these scenarios there may be one that gives a more logical explanation for evidence found today within our solar system and, especially, for seemingly anomalous evidence.
(5) It's possible logically for a Natural-system to appear suddenly in complex and structured form.
In what follows, recall the Richard Feynman statement. Often science is interested in patterns that illustrate how Nature behaves and such patterns will not tell us why Nature behaves as illustrated by such patterns. Indeed, there may be no possible standard way to know why Nature behaves in such a way. Also recall the Hermann Weyl description for a possible physical-like object that gives "images, ideas or building plans" for an evolving Natural-system. The GGU-model predicts that for any Natural-system that is described by a developmental paradigm there exists a physical-like ultraword w that can be described in the same general manner as described by Weyl. [Indeed, it contains a "super" amount of information, something like an informational "superball."] Then when another physical-like process *S is applied to w, the entire developmental paradigm is produced in the exact moment-by-moment sequence that corresponds to the exact moment-by-moment sequence of the actual events. Now the physical-like process *S has all of the basic behavioral patterns associated with an ultralogic. As mentioned, we often drop such an intermediate linguistic-like phrase as "developmental paradigm." When this is done, the mathematically generated interpretation becomes "The physical-like process *S when applied to the physical-like object w produces each physical event associated with an evolving Natural-system." But, there's a great deal more than just this or similar statements.
An ultraword exists for ANY such Natural-system including the one that yields the photon path mentioned previously. Further, the internal composition of w can be analyzed. Using the Weyl idea, when this is done, it's discovered that w contains nonstandard "images" or nonstandard "animator drawings." Under application of *S, these nonstandard "images" would correspond to other events that were not present within the original event sequence. But, what type of events are these? They have any of the *-transferred properties one might associate with standard physical events. However, they can't correspond to any standard physical event. Indeed, it can be shown that the actual symbols that appear in a description for these nonstandard events use nonstandard alphabet symbols. This means that no standard intelligent like-form can describe or comprehend a description for such a nonstandard event. It just wouldn't "register," so to speak, on the brain as an event. For these reasons, such events that might be associated with these nonstandard images are called ultranatural events. What good are these ultranatural events? Well, the GGU-model wouldn't exist without these ultranatural events and within the GGU-model every physical event sequence has associated with it these ultranatural events. Further, there can be ultranatural events even without there being any standard natural events. For what behavioral purpose could they be used? They seem to represent additional and necessary physical-like "stuff" needed to sustain and hold together, so to speak, all of the physical events. Indeed, if they were not present throughout the development of a Natural-system, then the Natural-system, from the GGU-model viewpoint, wouldn't exist. I note further that statements (1) -- (5) are conclusions obtained from analyzing the composition of ultrawords and various developmental paradigms.
There's one more rather nontechnical aspect of the GGU-model that is somewhat startling. The universe in which we dwell is composed of billions and billions and billions . . . of Natural-systems. Each of these Natural-systems has it's own ultraword. Do these ultrawords and the application of *S to each one work independently from one another or is it possible that there's an underlying logical correspondence between all of them? It's a remarkable fact that within the GGU-model there does exist an object w', termed an ultimate ultraword, with the following property. The process *S applied to w' yields all of the ultrawords that produce the event sequences for each of these many Natural-systems. Thus, there's an ultralogic correspondence between all such Natural-system event sequences. In very simple terms, one can say that not only does *S applied to w' produce the universe in which we dwell, but, w' and *S also hold everything together as the universe evolves. Further, they do so by processes that follow the same behavior patterns as would be displayed by an infinitely powerful mind.
9. The Nothingness. In this section, it's shown how the GGU-model actual predicts the composition of the nothingness, the quantum physical vacuum, discussed previously relative to the expanding universe concept. Unfortunately, this may be the most difficult aspect for a practicing physical scientist to comprehend. Physical scientists often, but not always, apply a certain sequence of mental processes when mathematical models are constructed. When a student first begins describing the behavior of Natural-systems, the technical dictionary terms are combined together. For example, a photon with a certain amount of energy interacts with an electron and the electron is ejected from a piece of metal. After this statement is made, a mathematical model is chosen from which additional predictions can be made. For example, what would be the relative velocity of the ejected electron? The model says the if f is the frequency of the photon, h is Planck's constant, v is the electron's maximum relative velocity, m is the apparent electron mass and W is something called the work function that represents the amount of energy needed to remove the electron from the metal, then the mathematical model states, assuming that this event actually occurs, that these numbers are all related by the equation hf = (1/2)mv^2 + W Thus knowing everything but, v, the relative velocity can be calculated. In an important way, the GGU-model reverses this process. The GGU-model can actually "talk," so to speak, and create its own physical theory stated in terms of a technical dictionary of its own.
One of the statements associated with electromagnetic radiation is a statement about its energy spectrum. In order to express this concept in words, one can write down a large collection of actual symbol strings that state the various energies that such radiation can acquire. The set of all these statements yields what is called a general paradigm. Each of these "words" can be encoded with natural number relations as discussed previously and embedded into an enlargement. After this, it's observed that there are nonstandard words that are members of the embedded general paradigm. These aren't the same things as the previous ultrawords and these new nonstandard words can be analyzed carefully. When this is done, it's discovered that they describe the exact same behavior as for such electromagnetic radiation but they have one symbol missing from each description. Having this one symbol missing doesn't detract from our comprehension of the behavior of these possible new entities. A new or different symbol is selected and inserted into the place where a symbol is missing. This inserted symbol is used to represent a new mathematical measure called an infinitesimal and would have the same strange behavior as the set of all mathematical infinitesimals. It's from considering these general paradigms and what the model is actually "saying" that the theory of subparticles has evolved. In other words, the theory of subparticles is NOT an ad hoc construction; but, it's based entirely upon what the GGU-model is trying to communicate to us.
Before discussing a very simple illustration for subparticle behavior, I mention the quantum physical concept of the virtual particle. According to many scientists, these are real particles; but, they can't be detected by any laboratory process. [Feynman, l983, p. 95] Some of, or often all of, their characteristics just pop into existence from the nothingness and very quickly these characteristics disappear into the nothingness. As a theory, quantum electrodynamics can't exist without them and since this theory predicts actual verifiable laboratory results so well, then it's claimed that they must exist. But, where do virtual particles come from and where do they go?
The GGU-model predicts the existence of a dense field of ultrasubparticles. Each of these subparticles is identical in character. Based entirely upon the experiential process of choosing a finite set of objects from a bag of objects, the GGU-model gives an actual refined mechanism that easily combines these identical ultrasubparticles together in such a way that the end result is a specific subatomic entity. The final step in this process is the application of the well known mathematical operator called the standard part operator. This operator takes such a combination of ultrasubparticles and activates them within the natural world. In quantum physics, you also have such operators but they are applied to the nothingness and not to a describable and predicted object. Further, due to this subparticle mechanism, the virtual particles of quantum physics can be eliminated from the subject as real entities and replaced by this subparticle mechanism. Of course, the virtual particle notion can still be used as a model for behavior. There's a rather simplistic illustration of a portion of this subparticle mechanism.
Suppose that you buy a large box of very, very thin spaghetti noodles. Each noodle is the same length and even looks exactly the same. You reach into the box and take a hand full. Now you take the approximately cylindrical column of noodles and you tie a string around them. At the very moment you make a knot in the string in order to hold the noodles together, they "morph" (i.e. smoothly, but suddenly, change) into an a specific subatomic particle. But, you know where the string is relative to the particle. So a few moments later, you cut the string and just as suddenly the column of noodles returns and the subatomic particle disappears. The continuous standard part operator mentioned previously behaves like the string. A combination of ultrasubparticles that behavior just like a finite set of objects is gathered together. The standard part operator is turned on. A subatomic particle appears. As long as the standard part operator is applied to that particular set of ultrasubparticles the subatomic particle continues to appear within the natural world. But, if the standard part operator is removed or turned off by a particular law of Nature, then the subatomic particle returns to its non-natural world state, a simple combination of ultrasubparticles. But, at that very moment a re-gathering of the ultrasubparticles can take place, the standard part operator can be turned on and new subatomic particles may appear within the natural world. From the natural world point of view, the original subatomic particle is changed into other stuff just like what occurs when radioactive decay takes place. By the way, there's an ultraword such that when *S is applied to that ultraword, an ultrasubparticle is produced as a ultranatural object.
10. The Infinitely Powerful Mind. In this section, I expand upon the remark made at the end of section 8. The GGU-model is a re-interpretation of the D-world model. Hence, the GGU-model can be discussed in linguistic terms. This gives an enhanced view of what the concept of a model truly signifies. All of the standard aspects from which the D-world model is obtained are finite in character. The term finite is to be understood intuitively. Indeed, the dictionary definition that a finite thing has measurable or definable limits is quite adequate for our purposes. Mathematically, it's possible to model such "measurable limits." But, discussing such a model within this monograph is unnecessary. For our purposes, the term infinite simply means "not finite."
A standard word has finite characteristics such as its length and the number of alphabet symbols used within its construction. In GGU-model and for all cases discussed, the ultrawords, when compared externally, are not finite with respect to these characteristics. A difficulty occurs when this external view is not used. In this case, these ultrawords have the same behavior patterns with respect to the finite concept as do the standard words. The term hyperfinite is used to indicate this mathematical fact. The ultrawords used in the GGU-model are hyperfinite in character; they behave as if they were finite but, in comparison to the finite character of a standard word, they are infinite. The ultralogic *S also behaves, in certain respects, as a hyperfinite object. Certain properties of the standard S are finite in character. These same properties for *S are hyperfinite in character. For certain nonstandard words to which *S applies, when a comparison is made, these hyperfinite properties have an infinite character.
The mathematical fact that *S is more powerful, from the point of view of logic, than S has been discussed. Coupling this with the above information on finite versus infinite, there seems to be no other general linguistic way to describe the GGU-model than to say that it represents, as a model, an infinitely powerful mind. Thus, the deBroglie and Lewis quotations given on page one are justified scientifically from the viewpoint of the GGU-model.
In this article, I have attempted to explain in the simplest possible terms the basic intuitive concepts needed to more easily comprehend the various aspects of the D-world and GGU-models. It's sincerely hoped that the basic notion of a "model" for behavior is understood fully and that the behavior being exhibited by these models is the primary predictions to be considered. For the GGU-model, linguistic entities are replaced consistently by physical-like entities. I repeat that such replacements aren't foreign to theoretical science since this has been done within the subject of quantum logic and was done, in the early 1970s, in an attempt to answer the pre-geometry cosmological problem. These models have yielded answers to many perplexing questions within science and philosophy. As illustrated, the methods used are the exact same methods used within theoretical science and the intuitive foundations for these models follow from our every day experiences. You should now be fully prepared to undertake an in-depth study of the more complex aspects of these models as they are discussed in various, but more formal, journal publications.
REFERENCES
- Duff, B. G., 1986. Fundamental Particles: an introduction to quarks and leptons, Taylor and Francis, London.
- Feynman, R., 1985. QED, the Strange Theory of Light and Matter, Princeton University Press, Princeton.
- Lewis, C. S., 1978. Miracles, Macmillian Paperback, NY, p. 32: 1960. Mere Christianity, Macmillian Paperback, NY, p. 32.
- March, A. and I. M. Freeman, 1963. The New World of Physics, Vintage Books, NY.
-1996-
Mathematics Department, U. S. Naval Academy, 572 Holloway Rd, Annapolis MD, 21402-5002.
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