Introduction[In this article, only the most significant aspects will be discussed. My intent is to present enough basic material so that the actual relation between modern mathematics as a discipline, the physical sciences and certain theological concepts can be discussed properly.]
It is a great event in the life of a graduate student when he/she is awarded a Ph. D. in Mathematics. Does such an exalted professional degree make one a mathematician? This depends upon your definition of mathematics, but, in actuality, a Ph. D. is not a necessary element in being a mathematician where mathematics includes all the areas that are defined as such. What should be, but is often not, the most significant aspect of a Ph. D. is its research aspect. If one does not do mathematical research, such a degree is often not necessary. But it’s estimated that only a small percentage of those that earn a Ph. D. in mathematics actually contribute significantly to mathematical research in its purest sense. In this regard, the Ph. D. only indicates a minimal research capability. [The same holds for such academic awards given in all areas of physical science.] The educational requirements necessary to become a mathematician are, however, somewhat nebulous.
In order for one to understand my comments as well as other significant factors relative to the subject area termed Mathematics, it’s necessary to given an explanation of the basic mathematical areas, what a mathematician does, and how mathematics affects certain important philosophical questions.
First, however, a definition for the discipline called “Mathematics” seems appropriate. The strangest and most worthless definition I’ve heard is when one is informed that “Mathematics is what mathematics do.” Of course, this is a circular definition that says nothing for what does the word “do” signify? Actually, I’m not going to define the discipline called Mathematics except to say that almost all the a stuff housed in the mathematics section of the local library is mathematics. But the discipline of Mathematics includes numerous volumes hidden in other library sections. You can find them in the philosophy area, the science area, and even the theology area. Librarians often don’t know where to house a volume that appears to contain what they perceive is mathematics. This means, however, that at the least mathematics is something disseminated in written form or its equivalent. Yes, notwithstanding the subjective processes used to obtain the written forms, it’s totally objective in nature. This discipline requires that, whatever mathematics is, it must be disseminated to others. To say, “I’ve a great mathematical result, but can’t describe it to you” has no meaning within the discipline of mathematics. Notice that “I’ve a great result, but can’t describe it to you” may have meaning for other disciplines.
Mathematical ActivitiesThe serious investigation of the foundational aspects of mathematics as a discipline only began about one hundred and fifty years ago. Thus all of the material presented here is relative to what would be termed as “Modern Mathematics.” It is probably not possible to determine all of the intuitive notions that led to mathematical concepts in the ancient past since many such ideas actually lead to contradictions. Modern mathematics attempts to avoid as many of the recognized contradictions as possible. Yet, the most fundamental aspects of modern mathematics depend heavily upon human experiences, especially with written languages. Concepts such as writing symbols in succession next to one another [Bourbaki, 1968, p. 15], writing symbols to the right or left of another symbol [Bourbaki, 1968, p. 17], being able to follow the directions of “replacing” one symbol xwherever it occurs in a collection of symbols A by another symbol B, recognizing when strings of symbols from a language are different or similar intuitively [Bourbaki, 1968, p. 17], knowing what it means to write symbols in columns or rows, and many other common and not mathematically defined notions such as orientation. These intuitive notions and, usually, being able to apply a few procedures from classical logic are fundamental and required to obtain definitions and mathematical proofs.
It is entirely false to say that mathematics is something absolute or pure, and exists without considerable human invention and personal experiences.
There exists three broad areas of “mathematical activity” (i.e. areas relative to the contents of those library books) where teaching is excluded from these areas. Teaching should be considered as a special mathematical activity distinct from these three.
The mathematics activity used the most is the activity called calculation. This is what the general public considers as mathematics. The “schoolmath,” as it is being called, is the basis for this type of mathematics. But also included under this heading is any activity where an individual applies known mathematical procedures to calculate numerical quantities or obtain any rational conclusion based upon numerical quantities. This area also includes basic geometry not as an application of logical discourse but as it approximates the physical world. The known procedures also include the meanings – the interpretations – given to the conclusions. It often requires great knowledge and a strong intuitive understanding of a particular mathematical theory and its relation to the entities being calculated. I’ve not stated what is being calculated. It can be used to calculate mathematical entities like large prime numbers or, more usually, nonmathematical entities where a nonmathematical entity has a name taken from another discipline. Of course, certain mathematical terms often take on multiple meanings, a mathematical meaning within a mathematical discipline as well as distinct meanings in other disciplines. The concept of “volume” can be associated with the pure mathematical subject called “measure theory,” refer to a concept from the mathematical discipline of geometry, or refer to brick lying. The words that surround the term in a written expression – the context – determine its meaning.
Prior to categorizing the two other areas of mathematics, it’s necessary that the concept some termed as mathematical “truth” as related to “fact” be discussed.
Mathematical TruthDealing as I do with many philosophical questions, I dislike using the term “truth” within the discipline of mathematics. Within mathematics, mathematical “truth” is not a significant “truth.” But on the other hand, it’s a concept which I, sincerely, believe is as close as humankind can come to a perceived physical fact when properly applied. It is customary for mathematicians to use the terms “true,” “holds” or “established” but what is meant by this term is an “explicit” fact. It means that it is an explicit fact that a certain fixed finite set of words – a set of statements, the hypotheses – have been written down. [The mathematician often uses certain terms when the hypotheses are written down such as “Suppose that these hold.”] It means that it is an explicit fact that other statements using words from the first fixed set of statements has been written down. It means that it is an explicit fact that a small group of individuals within the mathematical community have concluded that the last statement written down was obtained from the first statements by the methods they allow. In other words, “truth” for this form of mathematics requires something to be acceptably and explicitly written down – a “proof.”
It’s only after such a proof, that the term “true” might be assigned to the statement that has been “proved.” To say that the statement “every positive integer not equal to one can be expressed in a unique way as the product of one or more prime numbers” is (mathematically) “true” simply means that there exists an acceptable informal (or sometimes formal) proof for this statement. There is nothing mysterious or hidden or deep being communicated. It doesn’t correspond to some philosophic notion of “truth” in a broader sense. For those that agree that the “proof” is correct, this ends any further consideration as to its “truth.” The case is closed. Thus, in this context, mathematical truth is an explicitly demonstrated fact plus expert witnesses. [In abstract model theory, there is also a notion of “truth” relative to an explicit set-theoretic definition.]
In mathematics, the term “satisfies” is employed when specific methods of substitution are applied. These methods allow for numerical or other terms to be substituted into expressions that contain “variable” type expressions. The term “satisfies,” “holds” or “true” applies to the cases when these substitutes do not lead to a contradiction. For example, one can write x + y = z. For x = 1, y = 3, z = 4, one has 1 +3 = 4. The 4 on the right hand side does not contradict the definition for these symbols and their addition. This is not so for y = 6. Hence, x =1, y = 3, z= 4 “satisfies” this equation or the equation “holds for” x =1, y = 3, z= 4.
In mathematical logic, the term “satisfies” is used when a special substitution process is used so that a formal set of formula, such as {A&B, (A&B)&C}, yields a “T” for each of these. In abstract model theory, one takes “language constants.” a,b,c etc. from a formal language and assigns each to specific members a’,b’, c’ etc. of a set taken from set-theory. Say a’ = 1, b’ = 2, c’ = 3, etc. Considers the predicates “x is going to y.” For this, the x and y are ordered, so we let them be represented by an ordered pairs. [These are the same objects one usually sees in basic Cartesian graphing.] For this predicate, in the set theory, the a’, b’, c’, . . . are assigned to a set order pair {(a’,b’), (b’,c’) etc.} = P. But P does not contain (a’, c’). The a’,b’,c’ etc. can be specific mathematical objects like the natural numbers.
Now suppose that “a” denotes John, “b” denotes Bob, “c” denotes Pete. Then upon substituting (a,b), via changing of notation, if (a’,b’) is a member of P, then one states that P “satisfies” (a,b). Notice that this is for the particular predicate “x is going to y.” Also notice that (a,c) does not satisfy P. So, this is a rather specific substitution processes. Then P is said to (formally) model (a,b) and not (formally) model (a,c).
I have used the word “informal.” This means that the words used to relate the beginning and ending statements in a proof include a natural native language such as English, French, Russian and the like that has content. [Content means all of the impressions that a native language evokes within the mind of the reader.] Within many native languages you have words or their equivalent such as and, or, not and a few others. The meaning of these words is intuitive and comes from their use in everyday conversation. A description in words evokes within the mind images or impressions that, in many cases, take the place of a perceived event that can occur exterior to the mind – in objective reality. In this case, you can associate the term “truth” with the occurrence of a perceived physical event. Using this operational definition, a majority of individuals would agree, I think, that the following is linguistically correct. In a laboratory experiment, you describe a set of initial conditions H (the hypotheses). You then make two observations. A physical event described by D_1 occurs. A physical event described by D_2 occurs. As you prepare your notes you write, “Based upon the conditions H, physical event D_1 occurs and physical event D_2 occurs” to convey the results of the experiment.
Explicitly, using different modes of expression, this idea is translated to mathematical “truth.” From a given a finite set of hypotheses H, a statement “E is proved” is explicitly demonstrated. From the same set of hypotheses, a statement “S is proved” is explicitly demonstrated. Putting the two proofs together and applying the accepted rules of informal classical logic, the statement “E and S is proved” is fact via an acceptable collection of written symbols, where “E and S” is the last step of the “proof.” “Thus it has been demonstrated.”
One can then apply other accepted rules for classical logic and obtain the informal word-forms, E or S; S or E; if E, then S; if S, then E; E if and only if S; S if and only if E explicitly as the last step in a proof using H. I point out that, technically, each member of H can be considered as “proved” since, even if not used for any further demonstration, you may write each member of H as a step in the proof. Then each of the statements “E is proved,” S is proved,” E and S is proved,” “S and E is proved,” “E or S is proved,” “S or E is proved,” “if E, then S is proved,” “if S, then E is proved,” “E if and only S is proved,” “ S if and only E is proved” is a demonstrated fact. These ten statements are not actually part of the informal mathematical proof itself but are external observations. Using classical logic external to the proofs themselves, these results can be stated as follows: (L) if H is given, then (substitute any one of these 10 statements).
No matter what the word “true” might entail, there is a pattern of how this word applies to classical logic. If one substitutes for the word “proved” in (L) the word “true,” then the classical “truth-table” pattern is produced. Thus classical truth is being modeled by explicit procedures involving allowed linguistic processes. Personally, I almost never use the word “true” in constructing an informal proof. I do use the word “holds,” meaning that you are in fact “given” or have explicitly “proved” some statement and, hence, the statement can be used in a proof.
The important word here is “modeled” where this has the informal meaning “behaves in a similar manner or has a similar pattern.” At this point in this discussion, this translation of classical truth to mathematical truth has nothing else to say about the application of this classical truth concept to any other discipline.
There is one very important aspect to this modeling procedure. Assuming classical consistency, and I stress consistency is often assumed from empirical evidence only, if a statement “E is proved” holds for a consistent set of hypotheses, then the statement “not E cannot be proved” or, for our purposes, “not E is not proved” holds for the same set of hypotheses. Under the same consistency requirement, if “not E is proved”, then “E is not proved.” Assuming classical logic a pencil-and-paper activity can yield a proof or cannot yield a proof, and not both can occur. That is, either “E is proved” or “not E is proved” and not both can be so proved. Notice that if you substitute, in the case of consistency, for “proved” the word “true” and for the words “not proved” the word “false” you get the classical truth-table pattern for “true” and “false.” [You can also substitute the symbols “1” and “0,” respectively, and this yields the binary-tables.] [It is possible to show mathematically that with respect to certain conditions neither a statement E nor its negation can be established by the methods allowed. For example, the famous axiom of choice cannot be established from the other usual axioms of set-theory with respect to abstract model theory.] Thus classical mathematics is said to be two-valued in terms of proving a theorem. Only two things are possible, one or the other but not both. For this aspect of “mathematics,” the aspect that yields the Fields Medal, if you’re young enough, “truth” must be explicitly displayed by an explicit proof.
For basic two-valued mathematical truth, in the sense of proof, the concept of what is “mathematically true (holds)” or “mathematically false (does not hold)” is not a vague physical or philosophic concept but, rather, it is very explicit and absolute in character.
Aspects of Proving Statements MathematicallyThe most mysterious and myth prone area of mathematics is the area of acceptably “proving” a theorem. I don’t mean by this category, re-proving a statement although this is done a great deal of the time. I mean proving new things. This means “writing down” a statement that has never appeared before at the end of an informal proof. Then attempting to write an acceptable proof which has this statement as its conclusion. What is most misunderstood about this process is how it begins.
There are two seemingly distinct types of beginnings to this proof procedure. You have a beginning that is very concrete in nature and another beginning which is vague and highly experiential. We are sometimes told that the true mathematics is “the pure mathematics, the abstract mathematics” where the written statements are empty of content or nonsense symbols. This is one of the great myths. For the very definition of an abstraction says that you start with concrete processes, physical processes or human processes of writing down pretty patterns of symbols.
In the beginning when one learns to abstract, one seeks a feature that is common to more than one of these processes. For example, there are many different patters for the symbols we use to represent a positive integer. Consider string of symbols 2 = 2^1, 3 = 3^1, 4 = 2^2, 5 = 5^1, 6 = 2^1 x 3^1, 7 = 7^1, 8 = 2^3. Each displayed symbol 2, 3, 4, 5, 6, 7, 8 has what appears are common characteristics. The numbers 2, 3, 5 and 7 are prime numbers. Each of these 7 positive integers is written in a form where the prime numbers, if there is more than one, are written in an increasing form as one proceeds from left-to-right. (Notice you must know your left from your right.) Using these forms, I cannot find a different way to express the number that involves a different set of exponents. In the minds of some individuals, the idea is that maybe such characteristics persist for all the symbols one might use for a positive integer. So, one “abstracts” the characteristics by definition using, at the least, the informal axioms for the positive integers.
Using classical logic one “proves” that an arbitrary positive integer satisfies the characteristics. Finally, using the logical notion of “generalization” either explicitly stated or implicitly, the “Fundamental Theorem for Arithmetic” is established. However, to establish the “uniqueness” of this form, since it is a mathematical form, one needs intuitive knowledge as to what it means to “write symbols next to each other from left-to-right” so as to reproduce the symbolic forms. In general, one uses various definitions that yield common features. This, along with a set of axioms, yields the basic hypotheses from which a mathematical theory is created.
For the another beginning, while building a theory or just for the fun of it, an individual might postulate, from vast experience, that other patterns of behavior or relations between the defined entities may present themselves, where the behavior or the relation is the important concept. These other patterns may not have been the originally conceived common feature but from “intuition” a theory builder simply might “feel” that “interesting” mathematical results could be produced. Among these interesting theorems are different ways of expressing equivalent notions, ways that help mathematicians comprehend mathematical ideas.
The ability to give an acceptable “proof” is the great art of mathematics. It’s learned by first reading thousands of proofs produced by others within the mathematical community. Then one practices thousands of proofs for theorems that it’s claim can be proved since they are problems in textbooks. Usually, some other individual checks your proofs, states why they are NOT acceptable and sends you back to start over again. Finally, your proofs are accepted, at least on the student level. All of this work has somehow or other altered your patterns of thought. It’s this that leads to the “intuition.” But it’s usually impossible to describe in words a fixed set of rules that would always lead to the “feeling” that pursuing a certain avenue of thought will produce an “interesting” mathematical result. Thousands of new mathematical truths are produced everyday, they are probably interesting to one or two individuals, but only a few are judged publishable and, hence, interesting to the mathematics community.
Three facts are important. First, there are infinitely may different and new statements that might be a provable theorem or might not be a probable theorem. Once again, I can’t give a fixed set of rules that says that one statement is likely to be a theorem and another is not. No mathematician wants to attempt a proof if it’s not likely that the statement is a theorem; that is, that it can be acceptably proved. Then even if it’s likely to be a theorem, you must convince yourself that you have the ability to construct an acceptable proof. Otherwise, the statement is simply a conjecture. The second fact is that there are within each mathematical discipline “tricks” or procedures that allow one to construct a proof by drawing little pictures, by taking finite examples, by examining the patterns presented by finite strings of symbols, by adding hypotheses, and all sorts of concrete visual processes, which are then translated into words and statements that don’t reveal the original “tricks.”
The last fact is related to exactly how much one writes in a proof. A proof is written in a special style and only in enough detail to be convincing to those individuals you expect to read the proof. I’ve been privileged to write a few thousand interesting and acceptable proofs for new statements. I’ve learned to write them in extreme detail. Then I cut down on the detail to a great degree, but always have available in my notes the detail removed just in case someone needs to be more deeply convinced. One can consider such pure mathematics as an art. But an art that has an extreme absolute character and that’s controlled by explicit logical rules. The research (abstract) mathematician finds the discovery of new mathematical truths (results) a very exciting and rewarding endeavor. [For a recent and interesting perspective on this area of mathematical activity, see Thurston, 1994.] Unfortunately, the Nobel committee is forbidden to award a prize for this type of pure non-applied abstract “science.” Later I’ll show that in one and only one case does this theorem proving aspect, this absolute mathematical truth aspect, have any relation to theological questions.
Mathematical ModelingFor areas that are not considered as sub-disciplines of Mathematics, mathematical modeling is the most important of all categories. This can be a difficult mathematical activity depending upon the material with which one works.
The phrase abstract modeling usually means that you model a particular mathematical theory in terms of general set-theory or some other mathematical theory (also called a structure). This is the least important to the non-mathematics community. In general, the most significant aspect is when a mathematical theory is used to model terms and descriptions that depict entities or processes described by a language taken from a scientific discipline distinct from pure mathematics.
This is often confusing since a particular mathematical theory might use some of the same terms that appear in the other discipline’s dictionary. Many textbooks and research papers do NOT distinguish between these different uses of the same term or expression.
In a physics paper, the reader is to simply “know” that the term “velocity vector” refers to an object that measures two quantities associated with the physical concept of motion and not the pure mathematical entity associated with the mathematical vector as used in the subject linear algebra. Do basic physics book ever mention the differences between these two concepts?There are two types of processes terms as mathematical modeling. The first type requires that there be at hand an actual mathematical theory and, at least, a discipline dictionary. Giving a general description for the modeling procedure is very easy but applying the vague operations of the procedure can be very difficult even in this first case. For informally presented mathematical theories, the most straightforward approach to this first type of mathematical modeling is to define a domainor universe of discourse for a mathematical theory. This is a collection of mathematical objects, usually from the informal mathematical theory, that comprise the entities from which mathematical functions and relations are built. After this is done, the next step is to construct a fixed correspondence between entities that are named in the discipline dictionary and elements in the domain of the mathematical theory. Then you take relations that are assumed to exist between the discipline entities and correspond them to mathematical relations between elements of the domain. The correspondence constructed is called aninterpretation. (These ideas can all be related to a weak set-theory structure and the objects take on names from formal logic such as predicates, variables, constants. A formal notion of interpretation is relative to such a weak set-theory.)
It is not the purpose of this article to discuss, in any detail, how an interpretation is constructed. It takes many, many years of technical training within various disciplines before the interpretation methods are understood intuitively. Even after such training, the actual construction process requires a certain amount of talent and creativity that cannot be described properly. One must intuitively know, somehow of other, that certain paths will not lead to such a construction while other paths are the most likely to pursue. And, one must also know the “acceptable” construction methods as well since each construction will be examined by others in order to see that it conforms to specific rules.
Within each physical science discipline certain basic statements are accepted.
It is very important to realize that these statements include information about entities that can be observed by the human sense, or that cannot be so observed. The entities that cannot be so observed are often originally conceived of as “real” or “imaginary” during the development of a mathematical model. Then as the model develops assumed real entities are accepted as imaginary or assumed imaginary entities accepted as real.
Infinitesimal quantities and instantaneous velocities and accelerations were apparently considered by Newton as real measures that correspond to real physical behavior. The theory of indivisiblerequires one to believe that certain measures, dx, for real physical behavior are of such a “small” nature that they can not be further subdivided into yet “smaller” measures. The basic criticism of all this postulated behavior is that it does not correspond to observable real world objects. However, the extreme success achieved in modeling and predicting behavior of gross matter that can be observed using these postulated measures is what has produced almost all of our technical advances. New mathematical processes have shown that the behavior of accepted physical entities actually yields, through rational argument, the necessity that infinitesimal measures exist for other distinct “physical” entities. It is but a matter of choice as to whether or not these new entities are accepted as real objects in some form of objective reality that requires infinitesimal quantities in order to characterize their behavior.When photons (apparently not the original term used) were first postulated, Einstein stated that they were imaginary representations for energy. But since Einstein win a Nobel Prize based upon the postulate that these Planck defined energy elements are “instantaneously absorbed” by an electron and the electron acquires the total energy of the photon, almost all members of the scientific community insist that they are real physical objects. The same imaginary concepts were first assigned to other famous virtual particles and processes, and especially to “strings.” In all of these areas where there are new entities postulated that cannot be directly observed by human senses or machines, it is not their ability to predict, often but approximately, the behavior of gross matter that has led to their acceptance as objectively real objects. The reasons for any such acceptance are largely philosophical, political and economic rather than scientific. Notwithstanding this fact, the methods of mathematical modeling are not dependent upon such acceptance and apply whether or not the postulated objects, along with their postulated behavior, exist in object reality.
In order to construct a mathematical model, physical statements must be expressed in the same manner as the mathematical theory (usually the first-order predicate calculus) so that they can be interpreted. When these statements are interpreted, they must also be consistent with respect to the axioms that generate the mathematical theory. When this is the case, then the discipline statements can be adjoined to the mathematical theory as additional hypotheses. There are two general methods, with combinations of these methods, used to construct an interpretation relative to the physical sciences. To examine the behavior of a particular physical system, measuring devices are constructed from a strict set of rules. Although human senses are not considered as infallible devices, it is sometimes necessary to include such senses among the measuring devices. These devices are then placed in a strictly defined specific manner to record specific information. The next step is to correspond this information, in a strict and never varying manner, to entities within an existing mathematical theory. The measurements are used as a representation for physical behavioral characteristics. Often, when a physical law is stated, it’s stated in a manner where it is understood that the law states a relation between those measurements that characterize the behavior.
There is a second approach to this first type that is often necessary and that depends upon human comprehension of simple descriptive terms. In quantum physics, you have such a statement as “. . . all interactions between any two particles take place through the emission of a ‘field particle’ from the first particle and its subsequent absorption by the second particle.” [Duff, 1986, p. 26] The descriptive terms “emission” and “absorption” can be strictly related to mathematical “operators” that when interpreted yield the exact same description as appears in this quotation [Herrmann, 1983].
Then there is the concept of “symmetry” within quantum physics and how this concept is modeled by abstract mathematical group theory. There is no measuring device, as such, other than human mental imagery that yields a “picture” for symmetry or non-symmetry at the microscopic level of quantum physics. Group theoretic symmetry corresponds in a strict and absolute manner to a description for the behavior of fundamental particles. Duff states relative to quantum electrodynamics, for example, how the concept of symmetry correspondences to a description for behavior. When one uses this theory to describe the probabilistic behavior for a particle such as an electron, this description has associated with it a wave-function with an arbitrary “phase-factor.”
A physical alteration in the behavior of the electron is often modeled by a “mathematical transformation” of this wave-function or various aspects of this function such as the phase factor. In this case, “Successive transformations, each of which changes the phase, have an end result which does not depend upon the order of the sequence in which they are applied.” [Duff, 1986, p. 58] This is further related to the behavior of emission and absorption of virtual photons. Such transformations are modeled by the general notions associated with the mathematical theory of Abelian groups. Hence, such physical concepts correspond not to the numerical measurements recorded on a device, but correspond to behavior patterns as they can be comprehended by the human mind – patterns that are interpreted as being the same patterns exhibited by an abstract mathematical structure.
The previous example for physical modeling is related to a strict correspondence between specific terms taken from a physics dictionary where entities are described that appear to yield the described behavior. There is a very important example where we only model actual observed behavior by means of a strict correspondence and are not concerned with hidden entities that yield such behavior.
Humans apply various logical processes to collections of words, sentences and other written forms. All of these forms can be considered as a collection of finitely long strings of symbols. In the discipline of linguistics, the entire collection of symbol strings formed from a finite alphabet forms an intuitive set. Denote this set by the symbol W where W is the name for the set within the mathematical theory called standard ZF set-theory.
The logical processes used are applied to members of W. This is interpreted by saying that we apply a logical process to “subsets” of W or, in the standard set theory notation, the logical processes are applied to any A subset of W. Thus a logical process can be interpreted by a special function that exists within standard set theory and is often denoted by C. When this logical process is applied to subsets of W, subsets of the same set W are obtained. This is interpreted by the informal statement “for each A subset of W, one has C(A) subset of W.” One of the basic processes that differentiates certain types of logical deduction from other games we play with strings of symbols is that, when we apply such deduction to a set A, we at the very least get the set A back again. Although it’s hoped we get more than this, this minimal property is modeled by saying “If A subset of W, then A subset of C(A).” If I list one or two more very obvious and basic properties about such logical processes and if they are modeled within standard set theory, then what is obtained is the basic axiom system for things called finite consequence operators.
Now using properties of standard ZF set theory and these consequence operator axioms one can, by mathematical reasoning, obtain new properties about finite consequence operators. These properties can be easily verified in the “laboratory” for they predict perfectly what occurs when the human being takes a collection of statements, logically combines them together and writes down a set of conclusions. Finite consequence operator theory is probably the most empirically verified mathematical theory that has ever existed. At no time is one concerned with any aspect of the physical brain activity that may yield the written statements. Note that finite consequence operator properties do follow from the rules of classical logic and they can represent a collection of physical processes.
All aspects of physical mathematical modeling assumes one underlying condition. The condition is the one condition assumed by science. It states that
***the physical-world behavior we perceive, comprehend or even perform, at the very least, is associated with the logical processes used to produce the mimicking mathematical theory, a theory that is strictly associated with the terms taken from a discipline dictionary and that predicts such behavior.
The second type of mathematical modeling occurs when one first observes or imagines within a nonmathematical discipline certain recurring and simple patterns that can be described in written form. Further, logical deduction from these patterns seems to follow the deductive processes used within mathematical reasoning. In a few cases, an individual may not know of a particular mathematical theory that would serve as a bases for an appropriate interpretation. When this happens, one may be forced to abstract the properties and create a new mathematical theory prior to the interpretation process. Apparently, this is what Newton did when he created infinitesimal calculus as it applies to mechanics. Newton considered perceived knowledge of mechanical behavior as the bases for his geometry. His geometry used concepts for observed physical motion, concepts that Newton claimed involved infinitesimally small measures. However, once the mathematical theory is created, the exact same requirements for a strict interpretation must be maintained.
Mathematical Truth and Its Correspondence to Discipline TruthThe classical two-valued mathematical “truth” relative to “proving things” transfers to the discipline being modeled in one and only one way. Certain statements using the discipline dictionary, when expressed in a very simple manner (an informal first-order language) will under the strict interpretation correspond to statements within a mathematical theory. Under the above condition ***, if certain statements from the discipline correspond to the non-logical axioms or hypotheses of a mathematical theory, then any other interpreted discipline statement E is said to be a rational statement, where “rational” means that statement E can be obtained through application of the exact same logical processes that produce the mathematical theory. This type of mathematical “truth” relates only to rational comprehension with respect to specific logical processes and nothing more. If the discipline language claims to describe a physical event, then mathematical “truth” (proofs) under this basic transfer is not related to whether or not an event will or will not occur. It is not related to the concept of “truth” as a measure of “fact.”
The concepts of “what is a physical fact,” “what is not physical fact,” “what is an observable” or “what is not an observable” are not aspects internal to mathematics but are exterior notions. The assumptions that must be made cannot be established with mathematical certainty. Only the concept of rationality is certain with respect to mathematical modeling. [Note: This does not mean consistency is certain, since the consistency of much mathematics is only empirical in character and, hence, not certain.] I re-state a previous contention.
If the discipline statements are “accepted” as behavioral “facts” about physical events, then under the basic assumption that what we can comprehend or observe about nature satisfies the same logical patterns as those exhibited by a mathematical proof, then the conclusions of the mathematical theorems of the theory, when strictly interpreted, will yield “facts” about how nature behaves.
Thus, under these conditions, if a set of discipline statements are listed in a category called “accepted fact,” then any statement deduced from the mathematical model would also need to be listed as “accepted fact.” [There are certain aspects of quantum theory that have been postulated to possibly follow other non-classical patters. But the logic used to demonstrate this is classical logic.]A significant prototype for the concept of “accepted fact” within physical science is the notion of “occurrence.” Under the above conditions, if H contains statements that describe the occurrence of physical events and the deduced conclusions contain statements K that describe other distinct occurrences, then when these H events occur, the K events should occur. All of this, however, requires that each such discipline statement used be constructed in the special form of classical logic and statements are combined together in the same special way as they are within the mathematical theory. This means you use a simple intuitive first-order predicate language or other specific types for those portions of the discipline language that have been mathematically modeled. In summary, you have replaced mathematical “truth” with a new interpretation that follows the same pattern as mathematical truth. Because of other concerns, the statements so modeled may be “assumed to be fact.” The concept called “fact” that you are modeling by, say, classical two-valued mathematical modeling must follow such properties like the statement D_1 is fact, the statement D_2 is fact, then the statement “D_1 and D_2” is fact.
However, the most misunderstood aspect of such scientific discourse is that what has been accepted as a set H of “factual hypotheses” relative to statements that cannot be directly verified by scientific means need not be the correct set of hypotheses that yields the verified behavior of a physical system. There are many different sets of hypotheses that postulate different entities or different processes and, that after strict mathematical modeling, yield the exact same verified statements that are deduced from such an H set of hypotheses [Herrmann, 1983, 1993b, 1994b].
Thus, it is impossible to know which set of hypotheses is the correct set if only indirect verification is possible. This is one of the strongest arguments that, in general, the acceptance of certain sets of hypotheses is not based upon science but rather is often based upon philosophical, political or economic concerns.Yet, another important aspect of these modeling procedures is also often misunderstood or, even purposely, omitted. All of the above depends upon the interpretation. If the interpretation is altered, even in the slightest, then usually the above correspondence to discipline “fact” or “rationality” no longer exists. Although physical system behavior can still be observed and described, it is no longer being modeled by the specific mathematical theory simply because the interpretation has been altered or, indeed, for some mathematical theories, no known physical interpretation exists. The relation between physical system behavior and whether or not biological creatures can comprehend such behavior completely, as such comprehension is based upon the use of strict logical patterns, leads to some interesting philosophic concerns. I point out that Marx advocated that science change its reliance upon certain logical patterns associated with mathematics and rely, instead, upon his dialect. Of course, this has not happened.
But, what if classical logic is not the actual logic used by a science-community to predict physical behavior. Is there mathematical modeling procedure for this? Yes. It has been shown that the notion of the finite consequence operator can be used to represent all forms of known deduction. This deduction need not correspond in any manner to the “truth-table” model. It corresponds to the “occurrence” notion only. That is, if the (consistent) hypotheses occur and the physical processes satisfy the logic-system used by the science-community, then the “proved” conclusions will occur. Since, we really don’t know what future scientific endeavors will require including, as some suggest, changes in the logical processes used, finite consequence operators and their equivalent the “general logic-systems” would be the proper general method to employ when modeling physical behavior (Herrmann, 2006).
Errors in Mathematical ModelingWhen Newton began to mathematically model physical behavior, he needed general physical concepts that were absolute in that no physical process could alter the concept. He picked Galileo’s idea that time could be considered as such an independent variable. He had other absolutes as well such as measures of length. He allowed time and length to take on real number and infinitesimal values. From the time of Newton until 1824, the language of infinitesimals was the major numerical approach to the mathematical modeling of physical system behavior. In 1824, Able discovered that the then known language of infinitesimals was mathematically inconsistent [Able, 1824]. This led to an alteration in mathematical theories to exclude the terms and previously understood properties of infinitesimals. BUT the scientific community has continued, until this very day, to model physical behavior by use of the language of infinitesimals, hoping that they do not use the inconsistent part. This is one area where error is possible in modeling, but the major error occurred in 1905 and still presses.
A. Einstein, who had poor mathematical understanding at the time, developed what he claimed was a mathematical model for the strange behavior of light. He claimed that he was able to predict laboratory experiences from a more fundamental character of physical reality. His and almost all modern approaches to “relativity” questions make the same fundamental logical error [Herrmann, 1993b].
Einstein used a special physical language and an operational definition for how time is to be measured. This definition is then modeled mathematically via measures and the measures correspond to real numbers. He then makes certain “obvious” statements about these measures retaining the language of light propagation. He continues to construct what is claimed to be a fixed correspondence into a mathematical theory. However, throughout his assumed mathematical arguments, he often makes the error called the model-theoretic error of generalization.
This error means that he changes the interpretation and claims, without the possibility of proof, that what is established with respect to the language of light propagation and the first interpretation of measured time also holds true for the concept of time, in general, and, hence, any measurement of time. He and most modern scientists generalize these results, without possible proof, to another domain. Indeed, they generalize from a light propagation language for time measurement, and what they claim is not absolute time, to absolute time as measured by infinitesimals. These procedures can produce logical contradictions. [Reconsideration of the actual foundations for “relativity” and the correct theory of infinitesimals has led to a new theory that not only predicts as accurately as the Einstein theory but shows explicitly why certain measurements for “time” are, indeed, altered [Herrmann, 1993b].]
Suppose that you use a digital clock to measure time. The measures of time you use can be corresponded to the nonnegative integers. You use the theory of nonnegative integers as your mathematical theory for model building. One of the theorems of this theory states that if you take any nonempty set of nonnegative integers every member of which is less than 100, then there exists in this set a number m such that every member of this set is less than or equal to m. Under this correspondence, your measures of digital time also have this property. Suppose, on the other hand, you used as a measure Einstein’s defined of the time concept. Einstein “time,” prior to my alteration in the concept, must correspond to the real numbers since the calculus is used to derive relativistic conclusions. But the above theorem about nonnegative integers is not and cannot be a theorem about the ordering of the real numbers. A worse error, in the use of the calculus for physical modeling, is the continued use of infinitesimal arguments without applying the correct theory or considering the realism question.
The theory of infinitesimals was corrected by Abraham Robinson in 1961. It turns out that infinitesimals do not behave like the real numbers. We must not use them as a direct basic model for certain types of real world behavior. [They can have an indirect affect upon physical system behavior.] For example, any nonempty set of real numbers between 0 and 1 has an associated real number called a “least upper bound.” However, the set of infinitesimals between 0 and 1 does not have a least upper bound. This makes the infinitesimals difficult to conceive of geometrically. Further, we have the important question of realism. It’s claimed that we can indirectly observe the effects of the “top” quarks as predicted by a mathematical model and, therefore, top quarks must exist in objective reality although we cannot perceive them directly. Under the same reasoning, it must be the case that entities with infinitesimal charge exist, or that there can be a collection of infinitesimally many neutrons and infinitesimal time exists. Why? Since we use an infinitesimal mathematical model to model charge on surfaces and to predict the critical mass of plutonium.
Most particle physicists ignore the fact that the theories used in subatomic physics to predict indirectly evidence for the existence of the top quark, also imply the existence of subparticles as the “true” constituents of matter and fields. These facts indicate clearly that scientists choose from a mathematical model certain entities and call them real while not applying the same procedure to other possible constituents for what can be shown are philosophic reasons.
For physical system behavior, infinitesimal models predict very accurately. They are the basic model for almost all macroscopic and large scalar physical system behavior. The fact that they are, at the least, accurate in various domains requires, in certain cases, frequent updating of information.
Theology and Theorem ProvingNow that the necessary mathematical ideas have been discussed, it’s possible to relate these concepts to theological notations. Indeed, I am the first to do this for significant theological notions. Any possible hidden relation between the Scriptures and numerology will not be discussed where numerology refers to patterns of numbers associated with the Greek or Hebrew alphabets.
Except in one respect, it’s conceded that the pure mathematical exercise of theorem proving is neutral theologically. If one assumes, that there is no God, that humankind evolved by chance, and that humankind is only related to physical processes, then the exercise of “proving things” is just that and can have no relation to the concept of a Divine being.
Suppose that you only hypothesize that there is a God who created the universe and, in His own image and some how or other, created humankind. In this case, the “proving things” has a very specific character. [Since portions of this article may be read separately, I repeat a statement made previously.] For humankind to apply mathematical models and obtain accurate predictions for the behavior of perceived physical systems, certainly means that, as far as perception is concerned, our simplest two-valued “absolute in character” reasoning processes, the processes used in mathematical reasoning, mirror similar processes in the physical world. There may be many other processes that cause a physical system to develop over time and of which we can have no comprehension, but an exceptional amount of evidence shows that those processes we can comprehend have the same logic-like properties as those we use to “prove things.” There is considerable testimony besides physical evidence, which indirectly verifies this basic belief.
As Nobelest deBroglie stated it: “[T]he structure of the material universe has something in common with the laws that govern the workings of the human mind.” [March 1963] But, from the perspective of Divine creation, our mental processes where created by God and, if Paul is correct [Romans 1:19-20], we can “clearly” understand what is to be know about God’s attributes. As C. S. Lewis states it: “What appears to be my thinking is only God’s thinking through me.” [Lewis, 1978, p. 29] “[E]vents in the remotest parts of space appear to obey the laws of rational thought . . . . There is in our human minds something that bears a faint resemblance to it.” [Lewis, 1978, p. 32.] “According to it what is behind the universe is more like a mind than it is any thing else we know.” [Lewis, 1960, p. 32.] “He is the source from which all your reasoning power comes: . . . . ” and “He lends us a little of His reasoning powers and that is how we think: . . . .” [Lewis, 1960, p. 52, 60].
Thus, with respect to the Divine attributes described within the Bible such as what He has created, the mathematical process of theorem proving gives evidence for of an absolute Divine mental process as well as evidence for the hypothesized statements. No matter what one may say about any non-two-valued logic such as the dialectic arguments used by liberal theologians and philosophers, two-valued classical logic is the most absolute in character.
Even thought some may doubt the existence of a supernatural God that satisfies the hypotheses, mathematical modeling and theology can be technically related in a specific way that expands upon the theorem proving aspect. This relation is discussed in the next and last section of this article. This relation actually enhances the evidence for the above hypotheses.
Theology and Modeling[The methods used to model mathematically theological notions are the same as those used to model physical behavior via a mathematical structure and interpretations. However, the great difference is that the structure used is called a nonstandard model, and the objects being consistently interpreted were not originally considered to have theological significance. A nonstandard model appears necessary for a significant modeling method that describes God’s attributes is the method of “comparison.” God’s Biblically stated attributes are often compared with those of His created.]
I mention again that there are certain logical processes called “dialectic” processes that are not two-valued in the above sense. Liberal theologians and philosophers seem always to use these procedures to argue for their more controversial notions. There’s a good reason for this. By application of clever selections of a set of theses, a set of antitheses and a specific synthesizing process one can give a dialectic argument for the acceptance of almost any pre-selected statement. Further, there are many dialectic arguments that have no possible dialectic consequences. These facts can be established by using the absolute nature of two-valued logic for most of the basic dialectic concepts can be model mathematically. [Gagnon, 1980. Herrmann, 1992.]
Of considerable significance is the fact that in 1981 at the Annual Meeting of the American Scientific Affiliation held at St. David PA, a paper was given entitled “Mathematical Philosophy – Status Report I” that, by means of pure logical analysis, specifically predicted with respect to Marxism and other such dialectic controlled philosophies certain consequences. “[C]ertain political and philosophical systems that have infiltrated numerous modern human cultures are based upon logically inconsistent and, thus, contradictory foundations. . . . [T]he paramount inconsistency which pervades these systems is closely related to human rights considerations and, in particular, how these irrational systems view human behavior, wants, and aspirations. It is the close proximity of these demonstrable contradictions to these highly emotional human factors which will tend to generate certain significant consequences. . . . Unless these fundamental errors are eradicated from these systems, then these cultures appear to be doomed to collapse from inherent logical inconsistencies. . . More importantly, it is a highly significant fact that the internal logical collapse of these inconsistent systems can take any diverse form. This collapse could easily involve an irrational action which would envelop all of mankind in an unprecedented holocaust.”
Dialectic controlled systems contradict a two-valued mathematical model for the behavior of the Divine mind [Herrmann, 1994a]. Two-valued logic is the basic logic utilized by humankind since the beginning of language. It’s especially common throughout cultures that believe that such concepts as freedom, life, and even forms of slavery, among others, are absolutes. The significance of the above quoted statements is that the prediction made in 1981, in many respects, has come true. This is a clear and exact example of the dangers in using the dialectic arguments for any purpose when two-valued logic is available. For this reason, I must reject the dialectic arguments put forth by any theologian or philosopher when a two-valued absolute logic will suffice.
Mathematical modeling of a nonmathematical discipline often depend upon absolutes. Absolutes are the simplest possible terms, descriptions or concepts that are fixed and do not vary in meaning. If they are rules or instructions, then their simplicity is such that the vast majority of individuals would arrive at the exact results after individually following the instructions. The use of discipline language absolutes is not necessary in order to construct a specific model. However, it is self-evident, that if absolutes are not modeled, then as discipline terms or concepts alter their meanings the model’s interpretation may fail or, indeed, there may no longer exist an interpretation. Science is mostly interested in fixed immutable physical laws and processes and, hence, the strict modeling of absolutes.
For a Christian, there are three biblical proof-texts that are so simple in character that their meanings could hardly be distorted by an alteration in their literalism. Malachi 3:6 should be considered as an absolute statement. Certainly, whatever God states or promises, and His attributes should be considered as absolute in character. Descriptions can be conditionally absolute, that is they are absolute except under a specific condition. Some of God’s statements and promises are conditional relative to time, but descriptions for Divine attributes can also be absolute “in time” from a viewpoint of the language used and the logic employed. Paul implies in Romans 1:19 that what can be known about God by all of humankind is “plain” [NIV, Today’s English, Jerusalem, RSV, New English], “apparent” [Greek Literal], “manifest” [KJ], “known instinctively” [Living], for God has made it “plain” [NIV, Today’s English, Jerusalem], “shown it” [KJ, RSV], “disclosed it” [New English], “manifests it” [Greek Literal]. For if this is not so, then human beings would have an “excuse.”
The only reasonable way that the absolute statements made by God can be “plain” to all humankind and absolute in their meanings (i.e. they do not change) is that they retain the same meaning from the moment they were written throughout historical time. The concept of “plain” to all humankind would require that the meanings not be hidden from common human comprehension and be “plain” only to a selected few philosophers or scientists. Further, 1 John 2:27 tells exactly how the Bible scribes obtained the word-forms. Whether it be under Old Testament or New Testament anointing, the word-forms are not those as might be selected by the scribes, but appear to be word-forms selected by the Holy Spirit of God. They are word-forms that, at the time written, have absolute meaning, whether literal, figurative or another special linguistic construction and that are understandable in their original meanings.
It is “plain” to me, that any other interpretation of the method used by the scribes to obtain the absolutes of the Bible, any other interpretation for the word-forms but the common meaning at the time the word-forms were presented and any other method for comprehending these absolutes except for a two-valued logic, would be contrary to instructions given specifically within the Scriptures. Special processes that influenced the selections of words that appear in the Bible have now been mathematically modeled [see influences]. Of course, the basic fixed contextual meanings of such words leads to conceptswhere concepts may be modeled by the describing set technique. This technique allows for the illumination or refining of concepts to improve comprehension.
To further justify these previous conclusions, consider the G-model [Herrmann, 1982] and the D-world mathematical model [Herrmann, 1994a] that, using a fix interpretation, model explicitly the Bible’s absolute description for God’s general behavioral attributes [see attributes] and how God’s mind functions when compared with human mental processes, respectively. [Note: The terms used in these older articles have been altered. See changes This model is now termed the GD-model.] Then there is the General Grand Unification model (GGU-model) that besides describing a cosmogony and solving the general grand unification problem [Herrmann 1988, 1994b] contains, using a strict re-interpretation, a mathematical description for various Divine creation scenarios, one of which is the MA-model. The MA-model establishes that a Biblically literal interpretation in terms of the explicit linguistic statements given in Genesis 1 leads to a scientifically rational creation scenario. (For examples, see the last sections of my book “Science Declares Our Universe IS Intelligently Designed, Xulon Press, 2002, and my personal belief statements starting with beliefdvd.)
For specific examples of absolutes, consider Job 33:12 “. . . I will answer thee, that God is greater than man.” Then seek out the attributes that when compared with man, one can say that God is “greater than man.” Psalm 95:3 states “. . . how profound your thoughts.” Using the Scriptures, one should assume that God’s thoughts are more profound than those of any biological life-form, that God is also more righteous than any biological life-form, that God is more intelligent than any biological life-form, etc. Using the first part of Herrmann, [1993a] and adjective reasoning one obtains a mathematical model for such God-like attributes. Consider statements such as Isaiah 55:8, 9. “For my thoughts are not your thoughts, . . . . As the heavens are higher than the earth, so are . . . . my thoughts than your thoughts.” These absolute concepts relative to thought patterns, and many more Scriptural absolutes, are modeled by the behavior of nonstandard mathematical objects.
It’s self-evident from this discussion, that the most significant theological aspect of (two-valued) mathematics is in the activity of mathematical modeling. Correct mathematical modeling of theological concepts demonstrates explicitly that the modeled Divine attributes and there relation to His created entities follow the most common form of human logical discourse.
Applying the exact same philosophy of science for the acceptance of entities by means of indirect evidence, entities that cannot be perceived by means of humankind’s basic five senses, the correctness of the predictions made when such theologically interpreted models are applied to science and to human behavior demonstrates beyond any doubt that such a Scripturally described entity, God, must exist.
However, if one should reject the notion of indirect evidence as a physically sound notion and since, with the exception of “proving theorems,” mathematics itself can never determine what is or is not fact, unless certain statements are actual fact and behavior follows certain explicit logical patterns, then mathematical modeling of theological notions does have a minimal effect.
It has been established, beyond all doubt, that with respect to a specific interpretation for Scriptural terms, that the basic statements made in the Scriptures relative to Divine attributes and behavior are rational statementsthat follow the most common and explicit form of logical discourse – modern scientific logic. (RH)
*** Will Myers
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Will Myers
