Solutions to the "General Grand Unification Problem,"
and the Questions "How Did Our Universe Come Into Being?"
and "Of What is Empty Space Composed?"

Robert A. Herrmann
Mathematics Department
U. S. Naval Academy
572 Holloway Rd.
Annapolis, MD 21402-5002

Original paper presented before the MAA, Western Maryland College, 12 NOV 1994.

[Note: Part 2 below is a highly modified version of the original article. The original article still appears at 9903110 but with changes on section 11.1.2. The new material discussed in 2 is found in the appendix of this arxiv paper.]

1. History.

[Note: References for this section appear at the end of the secton.] In August 1979, I was to give a paper before a gathering of mathematicians at the Summer Meeting of The American Mathematical Society. [1] This summer meeting is a joint meeting with the Mathematical Association of America and was being held at the University of Minnesota at Duluth. I boarded an airplane at Chicago for the last leg of my trip, and who should be setting next to me at the window seat but John Wheeler, the Joseph Henry Professor of Physics emeritus at Princeton. Professor Wheeler was to deliver an invited talk before the Mathematical Association of America on what he believed was a fundamental difficulty with the philosophy of science associated with physical theories.

We discussed various things as we passed over some magnificent Minnesota thunderheads. Looking out of the window Professor Wheeler made the curious but simple statement, "We can't do that." What does this statement mean and who are the "we"?

After more discussion, the meaning of his comment was clear. First, the "we" are physicists and what they can't do was illustrated by the thunderhead. The thunderhead is assumed to be a collection of Natural systems. You have the water droplets, or crystals. Then their paths of motion caused by internal forces. The various electric potential differences produced by such motion and hundreds of other factors that scientists claim contribute to the overall behavior we observed from our window.

A Natural system is a set or arrangement of physical entities that are so related or connected as to form an identifiable whole. Many Natural systems have an associated physical theory. These theories are used to predict behavior for each Natural system. BUT many of these theories use different methods to predict behavior. Indeed, one often has to restrict a particular theory to specific areas of application. When this is done you often have a boundary between two theories - an overlapping region -- where one or the other but not both apply for the methods used in one of the theories are inconsistent with the methods used in the other theory. The hundreds of Natural systems (now called natural subsystems) that comprise the Natural system called a thunderhead are then put together in some manner to obtain the entire thunderhead one observes. With respect to the universe as a whole, the differences between various predicting theories are profound. What Professor Wheeler meant by the ". . . can't do that" with respect to our universe is the acknowledgment that physicists had not found a general unifying theory that predicts all of the behavior of all natural subsystems that comprise the entire Natural system called our universe. Science needed a theory that would be totally consistent, and a theory that would replace the piecemeal approach. Of course, such a unifying theory did not even exist for the behavior of the thunderhead we observed.

I restated this problem in the following manner.

Does Nature really combine natural subsystems together, in a way that we can understand, to produce an entire Natural system or does it use an entirely different method so that inconsistencies are somehow avoided? Is there something else required, something more basic than science has yet described and that's needed to combine all the Natural systems together that comprise our universe? Indeed, how can a universe that's perceived by humans to have order and harmony really be a product of chaos?
By the way, I added that last question to this problem since it is certainly relevant.

After our discussion, it occurred to me that something discovered in October 1978, and now called ultralogics [2], might possibly lead to an answer to these questions. But I also knew that the answer might be rather startling.

The standard approach is a piecemeal "bottom-up" approach beginning with the "bottom," so to speak, of the scientific hierarchy (or "domain of explanation"). Supposedly, if you can find a unification of the fundamental forces (or interactions), then this would lead to an "upward" process that would eventually unify all physical theories. As discussed above, this is a doubtful assumption. Moreover, many biologists and philosophers of science believe that there are emergent properties of organisms that cannot be fully understood as products of DNA or chemistry and, thus, do not follow directly from the four fundamental forces. Furthermore, the concept of "time" as described within some quantum physical models specifically contradicts the concept as it is described within certain cosmological theories.

On the other hand, there may be a top-down approach that answers the Wheeler questions. This means to find some scientific theory that gives processes which yield cosmologies, cosmologies that contain all of the processes that control the behavior of all Natural systems.

The "deductive-world model," (i.e. the D-world model) was the first constructed (1978) and is interpreted in a linguistic sense. D-world model properties will not be discussed directly within the following pages. Many of these properties can be found in sections 1 - 5 of the book The Theory of Ultralogics [2] which is available free. (See Free Books). However, some of what appears in the next sections does have the requisite linguistic interpretation associated with the D-world model. The actual model used to solve these problems is generally termed the General Grand Unification Model (GGU-model). (For other more specific applications of the GGU-model, a portion of this model the metamorphic-anamorphosis model (MA-model) is often employed.) Further, the intuitive concepts and mathematical methods that determine the GGU-model properties are very similar to those used for the D-world model conclusions. Thus becoming familiar with the GGU-model conclusions will aid in ones comprehension of the D-world model.

The basic construction of the GGU-model uses the following philosophical stance as described by deBroglie.

. . . the structure of the material universe has something in common with the laws that govern the workings of the human mind. [3, p. 143]
It was determined that an appropriate cosmogony would depend upon the properties of a logic-like operator and subsidiary concepts associated with the nonstandard physical world (i.e. NSP-world) [4, part III]. The construction of the GGU-model began in August 1979 and a series of announcements [5 - 10] relative to its mathematical construction appeared in the Abstracts of papers presented to the American Mathematical Society. In particular, [5] mentions the logic-like operators.

In 1982, papers by Bastin [11], and Wheeler and Patton [12] were discovered in The Encyclopedia of Ignorance. I mention that [12] does not contain an important appendix that appears in the original 1975 paper. The paper by Bastin describes the discreteness paradox and the paper [13], using a few GGU-model procedures, presents a solution to this paradox. Next the requirements for an acceptable cosmogony stated in [12] were compared with GGU-model properties. A major difficulty in showing that the GGU-model meets all of these requirements is in the language used to give physical-like meaning to the abstract entities. After considerable reflection, the prefix "ultra" was decided upon. The use of this prefix is consistent with its use in the construction of the nonstandard structure. A relatively explicit construction of such a structure uses the concepts of the ultrafilter, the ultraproduct and the ultralimit. In 1987, a paper justifying the fundamental mathematical theory of nonstandard consequence operators was published [14].

Although a few technical aspects associated with the GGU-model were yet to be fully justified, two papers were written in 1986 [15], [16] announcing the solution to the general grand unification problem and the pregeometry problem of Wheeler and Patton. To properly prepare paper [16], the original Wheeler and Patton paper [17] was utilized. An appendix in [17] describes how Wheeler and his colleagues at Princeton tried to construct a pregeometry from the statistics of very long propositions and very many propositions, where the term "proposition" refers to the propositional (i.e. the sentence) logical calculus. They failed to achieve a solution, but Wheeler left open the possibility that a solution could be obtained using concepts from the area of Mathematical Logic. This is exactly what has occurred.

Beginning in about 1989, a project was instituted to justify fully all of the GGU-model concepts, among other aspects of the NSP-world, and to present them in monograph form. The basic monograph [2] was completed in 1991 and the final result that completes the justification process was published in 1993 [18].

In the next section is given a very brief, general and mostly nonmathematical description of how the GGU-model and the language of the NSP-world solve the problems mentioned in the title. Presented in the third and technical section titled "The Mathematics," with additional comments, is most of the actual mathematics that, when interpreted, describes the properties of the GGU-model [2] that correspond to the nonmathematical descriptions. [This appears in the complete zip. file as listed on the original abstract page.]

The following quotations are taken from [17] and are all relative to the Patton and Wheeler cosmogony requirements.

(1) Five bits of evidence argue that geometry is as far from giving an understanding of space as elasticity is from giving a understanding of a solid. [17, p. 539]

(2) They also suggest that the basic structure is something deeper than geometry that underlies both geometry and particles ("pregeometry"). [17, p. 539]

(3) For someday revealing this structure no perspective seems more promising than the view that it must provide the universe with a way to come into being. [17, p. 539]

(4) It brings us into closer confrontation than ever with the greatest questions on the book of physics: How did our universe come into being? And of what is it made? [17, p. 540]

(5) Tied to the paradox of the big bang and collapse is the question, what is the substance out of which the universe is made? [17, p. 543]

(6) But is it really imaginable that this deeper structure of physics should govern how the universe came into being? Is it not more reasonable to believe the converse, that the requirement that the universe should come into being governs the structure of physics? [17, p. 558]

(7) It is difficult to avoid the impression that every law of physics is "mutable" under conditions sufficiently extreme, . . . . [17, p. 568]

(8) It is difficult to believe that we can uncover this pregeometry except as we come to understand at the same time the necessity of the quantum principle, with its "observer-participator," in the construction of the world. [17, p. 575]

(9) . . . . a guiding principle, is what we seek. [17, p. 575]

As the GGU-model pregeometry is discussed, I will refer to these quotations where applicable. Relative to (9), the first half of the guiding principle is the deBroglie statement viewed from the NSP-world. The second half of the guiding principle is relative to the philosophy of realism and an observation made by the originator of nonstandard analysis Abraham Robinson. First recall that Newton believed that infinitesimal measures were real measures associated with objectively real entities. Berkeley and Leibniz did not accept Newton's belief. Robinson at the end of his very first published paper on nonstandard analysis made the following statement relative to the modern concepts of how mathematical models are used to predict indirectly Natural system behavior.

For phenomena on a different scale, such as are considered in Modern Physics, the dimensions of a particular body or process may not be observable directly. Accordingly, the question whether or not the scale of non-standard analysis is appropriate to the physical world really amounts to asking whether or not such a system provides a better explanation of certain observable phenomena than the standard system. . . . The possibility that this is the case should be borne in mind.
Fine Hall,
Princeton University [19, p. 440]
Robinson is referring to infinitesimal measures in this quotation. Since the publication of [19], nonstandard analysis has been applied to entities that are not infinitesimal in character. The second half of the guiding principle is an extension of this Robinson statement to the NSP-world. Thus the acceptance of the NSP-world as a viable realism depends upon whether or not it provides a "better" rational explanation of certain observable phenomena than the standard world model. The philosophy of what constitutes "a better explanation" is left to individual choice.


[1] Herrmann, Robert. A. (1979). Perfect maps on convergence spaces I, Notices of the AMS, (26)(5): A-475.

[2] Herrmann, Robert. A. (1991). The Theory of Ultralogics. (See Free Books for the most recent updated version. There is an version at Part I and Part II.

[3] deBroglie, L. (1963). In March, Arthur and Ira M. Freeman, The New World of Physics, Vintage Books, New York.

[4] Herrmann, Robert A. (1989). Fractals and ultrasmooth microeffects, J. Math. Physics, 30(4) (April):805-808.

[5] Herrmann, Robert A. (1981). Mathematical philosophy, Abstracts of papers presented to the AMS, Vol. 2(6). #81T-03-529:527.

[6] Herrmann, Robert A. (1983). A useful *-real valued function, Abstracts of papers presented to the AMS, Vol. 4 (4), #83T-26-280:318.

[7] Herrmann, Robert A. (1984). Nonstandard consequence operators I, Abstracts of papers presented to the AMS, Vol. 5 (1), #84T-03-61:129.

[8] Herrmann, Robert A. (1984). Nonstandard consequence operators II, Abstracts of papers presented to the AMS, Vol. 5 (2), #84T-03-93:195.

[9] Herrmann, Robert A. (1984). D-world alphabets I, Abstracts of papers presented to the AMS, Vol. 5 (4), #84T-03-320:269.

[10] Herrmann, Robert A. (1984). D-world alphabets II, Abstracts of papers presented to the AMS, Vol. 5 (5), #84T-03-374:328.

[11] Bastin, T., (1977). A clash of paradigms, in The Encyclopaedia of Ignorance, Duncan and Westen-Smith, eds. Pergamon, New York.

[12] Patton, C. M and J. A. Wheeler, (1975). Is physics legislated by cosmogony? i>The Encyclopaedia of Ignorance, Duncan and Westen-Smith, eds. Pergamon, New York.

[13] Herrmann, Robert A. (1983). Mathematical philosophical and developmental processes, Nature and System, 5(1/2):17-36.

[14] Herrmann, Robert A. (1987). Nonstandard consequence operators, Kobe J. Math. 4(1):1-14.

[15] Herrmann, Robert A. (1986). A solution to the grand unification problem, Abstracts of papers presented to the AMS, Vol. 7 (2), #86T-85-41:238.

[16] Herrmann, Robert A. (1988) Physics is legislated by a cosmogony, Speculations in Science and Technology, 11(1):17-24.

[17] Patton, C. M and J. A. Wheeler, (1975). Is physics legislated by cosmogony? in Quantum Gravity: an Oxford Symposium, Ishan, C., Penrose, R., and D. Sciama, eds, Clarendon, Oxford.

[18] Herrmann, Robert A. (1993). A special isomorphism between superstructures, Kobe J. Math. 10(2):125-129.

[19] Robinson, A., (1961). Non-standard analysis, Nederl. Akad. Weimsch Proc. Ser. A64 and Indiag. Math. 23:432 - 440.

Important Reference

The above reference [2], from which the mathematical portions of the monograph are taken, contains the fundamental concepts associated with the methods and processes used within the discipline of nonstandard analysis as they would apply to all aspects of the GGU-model and D-world model.There are refinements to the methods presented there. In addition to the link 9903110 mentioned above, these two present additional material. 0605120 and 0603573

2. Modified on 1/5/13 for the New Approach.

[Note: In this modified version, the scheme that appears in section 7 of the appendix in 9903110 is interpreted.] Let me point out that as long as scientists use mathematics to obtain their theories and use written symbols, diagrams, photographs and the like to communicate their concepts, then the GGU-model can't be eliminated. It will always be there lurking in the background.

Now to answer the question "How was our universe created?" Let's start by considering a single geometric point a few feet in front of you. A geometric point in this sense is a position in our universe and, for the present, has no other meaning. Now, I'm able to magnify this point for you by using a mathematical microscope with a power that's greater than any power that can ever be achieved by human means.

Suddenly, you see the point open up, like the iris of your eyes. What's revealed to you is a background universe, a substratum, or whatever you might like to call it. Now, the physical-world is the world we can scientifically perceive, and this physical-world point is still in your field of view with a small portion of the background universe surrounding it. You can't make out much detail, but there's definitely something there. The detail you see is sharper and clearer near to the single physical-world point. Then clarity slowly fads as you proceed further from that one solitary physical-world position within our universe. You can find no clear outer edge within your field of view. This background universe forms a portion of what I call the nonstandard physical world - the NSP-world. [Note: The entire collection of all possible standard world points coupled with all of the NSP-world points that are infinitesimally near to them is called the finite or bounded portion of the NSP-world.] The term the NSP-world is also used for other applications. However, for our purposes I'll discuss a portion of this NSP-world, the GGU-model.

Our universe is inside and "just as near to" the NSP-world as I have described it. One might say that this background universe is scientifically omnipresent. Now the GGU-model portion of this background universe specifically states that there will never be a human language that can give a completely detailed description for the mechanisms that produced our universe, but there do exist such mechanisms within this background universe. Such mechanisms exist but, no matter how hard we try, the human mind can't comprehend all of the details. [Note: I have switched, as many scientists do when discussing a theory, to a positive language. This means that although what I describe can be qualified with such terms as "might be," "can be," etc. such terms are dropped.] Well then, what is known about how universes, such as ours, are created by NSP-world mechanisms? We can know general properties.

To begin with, there exists within the NSP-world a set of "things." These "things" need not be considered as being within our physical universe. But why do I call them "things"? I do know a lot about these "things." But, unfortunately, if I were to give you a basic description for their contents, I would only use various mathematical terms from the fields of mathematical logic and set-theory. The most difficult task I've faced is to give some "physical" meaning to these "things." I know that these "things" behave like informational "superballs."

After literally years of reflection, it was determined that these "things" can be described as containing all of the building plans, the instructions, that lead to the step-by-step production of an ideal universe. They also contain a great deal of information that is incomprehensible to the human mind. It's interesting to note that, after I came to these conclusions, I came across a quotation from one of the greatest scientists of our time. Hermann Weyl is quoted as saying the following:

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?
We have "things" in the background universe that are ready, when the conditions are just right, to aid in the production of universes, ours included. And different predicted "things" aid in the production of different universes. Actually, these "things," which I now call ultra-work-like objects, don't work alone. A particular ultra-word-like object X is just one piece of an entire process. All pieces of the "puzzle" must be put together before a universe is produced. Now the existence of a particlar ultra-word-like object X that determines all aspects of our universe may not seem startling, but a lot more is yet to come.

What conditions exist in the NSP-world that trigger the creation of a universe such as ours? At present, there is no human secular scientific language that can detail these conditions, no human understanding of what these conditions are or were, only that such conditions exist.

I remind you that this cosmogony comes from a mathematical model, a model that can't be eliminated from modern science. This cosmogony satisfies all of the basic theoretical requirements of the scientific method. The universe in which we dwell, our solar system, the Earth, a virus, or an atom are physical-systems. Physical-systems are studied by the physical scientist in a piecemeal fashion. They apply distinct procedures that seem to describe the moment-to-moment behavior of each distinct physical-system. Now within the NSP-world there's a special relation denoted here by "*S" that relates the ultra-word-like object X to its members. It yields a scheme for removing each member from X. It is an intrinsic (hidden) ultranatural process. (The word "intrinsic" means that it has a signature that mimics a thought process,) The *S-scheme is predicted and is one of the entities termed as an ultra-logic-system.

When the conditions are just right, a force-like process *A' is applied to the ultra-word-like object X and using, *S, *A' separates the X into its constituents. These constituents are yet relations that generate, in its entirety, a universe via the step-by-step development of universe-wide frozen-frames. Each of these "frames" is a complete slice of a universe at a moment during its development. From the set of constituents of X, a relation *V(a) is selected that is relative to the complexity of a universe when it is presented as a collection of physical-systems. Each member *F(i,j) of *V(a) is an ultra-logic-system scheme.

A force-like process *A, almost identical to *A', is applied. There is a single "starting" member *F(b,c) to which *A is applied. Application of *A to the relation *F(b,c) predicts logically each set of *instructions that generate, by properton combinations, each physical-system, if any, for the "first" universe-wide frozen-frame.

When the complete set of *instruction sets for the (b,c) frozen-frame slice is rationally obtained, the gathering operator G is applied. This predicted operator through processes relative to *instruction-information gathers the proper collections of propertons that will yield every physical-system that is contained in the (b,c) frozen-frame. It can also yield other entities that produce physical-like systems or unidentified other-systems. These events are atemporal and sequential. Thus, from an observer time viewpoint, no observer time would lapse during the application of G. The result is an info-field.

Info-fields can exist without activation. They behave as does the surface of a DVD that contains the information that reproduces an image of a universe-wide frozen frame. Each info-field is activated by a final operator, the St operator. This operator is a well-defined mathematical object that yields the physical characteristics of each physical member of a universe-wide frozen-frame. It is called the realization operator for the GGU-model.

Now the logical operator *A *deduces each universe-wide frozen-frame in sequential order and each necessary set of *instructions. The *instruction are, at least, partially comprehensible. The processes combine together, controls and coordinates all of the distinctly different physical-systems that comprise a universe. The processes produce each physical-system and yield the moment-to-moment alterations in the behavior of each and every physical-system. Further, each of the operators and processes has predicted behavior characteristics or signatures. These, as done in quantum logic, can be ignored. However, they can be interpreted as revealing the presence of a higher-intelligence that designs each physical-system, and that, via procedures that mimic thought processes, produces the step-by-step development of our universe and the alterations in the behavior of each physical-system as it develops.

The General Intelligent Design (GID) Model is the interpretation of the GGU-model processes in terms of the higher-intelligence signatures. The GID-model can be characterized as behaving in its entirety like a super, super, super to an infinite degree, mind. The processes are similar to how an almost inconceivably powerful mind or maybe a computer behaves.

[Relative to the deBroglie statement and to all the areas that appear to be humanly comprehensible, the GGU-model is based upon one and only one general notion. The notion is that how nature combines together physical laws and processes to bring into being a physical event or a change in a physical-system is modeled (mirrored, mimicked, imitated) by the mental processes human beings use to describe the produced events or changes. This hypothesis is verified by the largest amount of empirical evidence that could ever exist since whenever a scientist discusses or predicts physical-system behavior human mental processes are applied. This includes all of the methods used to gather and analyze evidence that would tend to verify any scientific hypothesis. 3 JAN 1999]

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