The Unified Logic: Fundamental Structure of Human Reason

John W. Nageley



A work in progress


"Human reason has this peculiar fate that in one species of its knowledge it is burdened by questions which, as prescribed by the very nature of reason itself, it is not able to ignore, but which, as transcending all its powers, it is also not able to answer."

Immanuel Kant
Critique of Pure Reason

 
"For all the promise of future revelations it is possible that certain terminal boundaries have already been reached in manís struggle to understand the manifold of nature in which he finds himself."

Lincoln Barnett
The Universe and Dr. Einstein


Overview

Newtonian principles provide a reduced model for studying and understanding physics. DNA provides a reduced model for studying and understanding biology. For studying and understanding human reason, The Unified Logic: Fundamental Structure of Human Reason provides such a model.

The Unified Logic presents the Unified Logic, a comprehensive, fully integrated logical characterization of the fundamental deep structure of human reason and principles, or laws, that govern its use. The Unified Logic comprises these three logics:

Principles distinguish between the surface features of reason and its deep structure.

Site Contents

For now, this site presents a description of The Unified Logic: Fundamental Structure of Human Reason. This site will be expanded as publication date approaches.

Presented are overviews of chapters included in The Unified Logic.

This site also includes Flight of the Pure Analogy, a paper for presentation at the Ninth International Kant Congress, Berlin, Germany, 2000. This paper is based on Chapter 5 of The Unified Logic, "Confronting the Terminal Boundary of Human Reason."

Contents

Summary
Introduction
Chapter 1 - Searching for Structures of Reason
Chapter 2 - Isolating the Fundamental Scheme of Objects
      Part 1 - The Unified Logic: Theoretical and Philosophical Foundation for Logic of Objects
      Part 2 - The Unified Logic: Logic of Objects
Chapter 3 - Accessing the "Beasties" (Sets)
      Part 1 - The Unified Logic: Theoretical and Philosophical Foundation for Logic of Sets
      Part 2 - The Unified Logic: Logic of Sets
Chapter 4 - Expanding the Universe of Hierarchies
      Part 1 - The Unified Logic: Theoretical and Philosophical Foundation for Logic of Hierarchies
      Part 2 - The Unified Logic: Logic of Hierarchies
Chapter 5 - Confronting the Terminal Boundary of Human Reason
      Part 1 - Theoretical and Philosophical Foundation, Terminal Boundary
      Part 2 - The Unified Logic in Terminal Region
          Section 1 - Terminal Analysis and Terminal Boundary
          Section 2 - Terminal Analysis Near Terminal Boundary, First-Order Terminal Objects
          Section 3 - Terminal Analysis At and Beyond Terminal Boundary, Second-Order Terminal Objects
          Section 4 - Terminal Synthesis Near and At Terminal Boundary, Terminal Synthetic Objects
          Section 5 - Terminal Analogy Near and At Terminal Boundary, Terminal Analogical Objects
Appendix A - Critique of Continuum Theory: Flaw in Real Number Line
Afterward
Presentations/Published Works
Flight of the Pure Analogy

Reduced Model

Philosophers have been discovering and introducing structures and principles of reason at least since the time of the ancient Greeks, as did Plato and Aristotle. Such great seventeenth and eighteenth century philosophers as Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume and Kant wrote extensively on epistemology. Some wrote from the perspective of rationalism and some from the perspective of empiricism. Kant tried to reconcile the disparity between the two positions.

Writings of classical philosophers on reason focused on, for example, "substance and attribute," "genera and species," and "classification." Kant explored and then, as no other philosopher has, characterized limits of human reason: "Human reason has this peculiar fate that in one species of its knowledge it is burdened by questions which, as prescribed by the very nature of reason itself, it is not able to ignore, but which, as transcending all its powers, it is also not able to answer." In this age, such questions often arise, for example, in particle physics at one extreme and cosmology at the other.

Thus, fundamental structures and principles of human reason then have been known for a long time, for centuries. To characterize them, many logics have been created. Aristotle created one. Many others have also been created since, culminating currently in modern symbolic logic and computer logic. Many grammars have also been created to characterize the fundamental structures and principles of reason. Chomskyís transformational grammar is well known. Undoubtedly, therefore, essential fundamental structures and principles of reason have been logically or grammatically characterized by one system or another.

Though Kant explored and then, as no other philosopher has, characterized limits of human reason, he did not provide a logical characterization of the fundamental structure of human reason. Lacking still in the study of human reason has been what Gregory Bateson (in paraphrasing John von Neuman) identifies as "a reduced model [of human reason] that would do for epistemology what Newtonian principles did for physics." At least it has until now. The Unified Logic provides one. That logic characterizes fundamental structures and principles of reason within a single comprehensive, fully integrated complex, using a single comprehensive system of symbols and logical structures. It also includes structures and principles needed for logically characterizing what happens to reason when it reaches the final frontiers of knowledge, its terminal boundary. Three Logics of the Unified Logic

The Unified Logic, the logic presented in The Unified Logic, comprises these three logics:

Each is distinct yet fully integrated into a unified logic. The Unified Logic devotes a chapter to each of the three logics. The first chapter presents the logic of objects, the most fundamental logic of the Unified Logic. Out of it arises the logic of sets. Out of the logic of objects and sets arises then the logic of hierarchies. Each of the three chapters includes laws, or principles or rules, governing probable and possible logical relationships between objects and structures characterized in it. Each integrates such fundamental functions as analysis and synthesis.

The Unified Logic concludes by exploring and logically characterizing what happens to human reason when it reaches final frontiers of knowledge as it does, for example, in particle physics at one extreme and in cosmology at another. The book refers to that limit as the terminal boundary of human reason.

Chapter 1 - Searching for Structures of Reason

Chapter 1 provides philosophical and theoretical background for the Unified Logic. It explores positions cognitive scientists have taken on whether it will ever be possible to describe the fundamental deep structure of human reason. This provokes the question of whether a common veridical view of the world is possible. For centuries philosophers have maintained steadfastly that it is impossible to describe the fundamental structure of human reason, and many have tried.

Contemporary cognitive scientists continue to maintain that it is impossible to do so, and yet many are still trying. This chapter concludes that underlying all human reason is a fundamental structure and that that structure can be described, or characterized. An objective of the Unified Logic then is to characterize that structure.

By analogy, the Unified Logic can be thought of as a logical characterization of the anatomy of human reason. In other words, just as there is a fundamental structure of the human body so also is there a fundamental structure of human reason. True, ways people apply their reason vary from person to person. But so also do ways people apply their anatomies. Yet all have the same fundamental anatomy.

Thus, on the level and in the manner characterized by the Unified Logic, variations in reasoning between people cannot be attributed completely to differences in the fundamental structure of their reason, any more than variations in their physical performance can be attributed only to differences in the fundamental structure of their physical anatomies. Anomalies in reason created by impairments of that fundamental structure are also addressed.

Therefore, as presented in The Unified Logic, all variations in reasoning are characterized as surface features, or manifestations, of the fundamental structure of reason. The book maintains throughout the distinction between the deep structure of human reason and its surface features, or manifestations.

Chapter 2 - Isolating the Fundamental Scheme of Objects

This chapter presents the Unified Logic, logic of objects in two parts. First it present the theoretical and philosophical foundation for the logic. Then it presents the logic itself.

Part 1 - The Unified Logic: Theoretical and Philosophical Foundation for Logic of Objects

The first part of this chapter provides theoretical and philosophical background for a logic of objects. It reviews history related to the search for an understanding of human perception and its relationship to human reason. This part touches on the contributions of such philosophers as Plato and Aristotle as well as those of several seventeenth- and eighteenth-century philosophers. Classical concepts of "substance and attributes" provide a foundation for structures introduced later in this chapter.

This chapter then focuses on and explores the perspectives of cognitive scientists on human reason, particularly as they relate to the psychology of perception. It lays thereby the foundation for the logical characterization of "mental representations," or "mental models," of perception presented in the second part of this chapter. In this process, it reveals the deepest, most fundamental structure of human reason--the one that underlies the relationship between an object and its attributes (variously referred to as its "properties," "characteristics," "qualities" and so on).

This part of Chapter 2 identifies that structure as the fundamental universal structure of human reason. Its structure is comparable in simplicity to the simplicity of this structure in algebra:  X + Y = Z. As demonstrated in Chapters 3 and 4, the fundamental universal structure also underlies the logic of sets and the logic of hierarchies.

Part 2 - The Unified Logic: Logic of Objects

The second part of this chapter presents the actual logic of the logic of objects. It logically characterizes fundamental relationships possible between an object and its attributes. Numerous diagrams illustrate fundamental relationships possible between them. Finally, this part logically characterizes the fundamental universal structure of human reason.

In this context, Chapter 2 presents a logical characterization of Immanuel Kantís das Ding an sich (or also die Sache an sich selbst), that is, the thing in itself.

Chapter 3 - Accessing the "Beasties" (Sets)

This chapter presents the Unified Logic, logic of sets in two parts. First it presents the theoretical and philosophical foundation for the logic. It then presents the logic itself.

Part 1 - The Unified Logic: Theoretical and Philosophical Foundation for Logic of Sets

The first part provides theoretical and philosophical background for the logic. It clearly establishes that the logic of sets presented in The Unified Logic is not "set theory": that is, it is not classical Cantorian set theory; it is not axiomatic set theory; it is not naive set theory. In short, it is not based on any existing theory of sets. This part of Chapter 3 presents a logic of sets based on a theory of the continuum, independent of Georg Cantorís fundamental theorem.

"There are sets; beautiful (at least to some), imperishable, multitudinous, intricately connected. They toil not, nor do they spin." This is according to Paul Benacerraf and Hilary Putnam (1985) in their introduction to Philosophy of mathematics: Selected readings. They are attempting to come to grip with the meaning of "set."

Benacerraf and Putnam write with elegance and feeling, almost poetically. They then add, "Nor, and this is the rub, do they [sets] interact with us in any way." As a result, they point to this problem: "So how are we supposed to have epistemological access to them?" This is very problem that anyone wanting to understand "sets" must face. How are we supposed to have epistemological access to "sets?" That question has been asked many times since Georg Cantor first introduced "set" as a mathematical concept over 100 years ago.

As regards gaining epistemological access to sets, Benacerraf and Putnam answer their own question in this way: "To answer, Ďby intuition,í is hardly satisfactory. We need some account of how we can have knowledge of these beasties, some account of our cognitive relationship to them." Examples indicative of the difficulties attendant on the concept of set are numerous. These difficulties are explored in this part of Chapter 3. Presented in Part 2 is a logic of sets to which epistemological access is possible--and useful.

Part 2 - The Unified Logic: Logic of Sets

The second part of this chapter presents the actual logic of the logic of sets. Numerous diagrams illustrate fundamental structures and logical relationships possible between objects and sets to which they can belong. This part then characterizes the fundamental universal logical principles governing relationships possible between objects and sets to which they can belong. The fundamental universal structure of human reason, abstracted and characterized in Chapter 2, provides the foundation for the logic of sets.

Chapter 4 - Expanding the Universe of Hierarchies

This chapter presents the Unified Logicís logic of hierarchies in two parts. First it presents the theoretical and philosophical foundation for the logic. It then presents the logic itself.

Part 1 - The Unified Logic: Theoretical and Philosophical Foundation for Logic of Hierarchies

The first part of this chapter provides theoretical and philosophical background on hierarchies. It includes historical background on categorization, or classification, from Aristotleís work on it to its place in modern science and cognitive science.

Part 2 - The Unified Logic: Logic of Hierarchies

The second part of this chapter presents the actual logic of the logic of hierarchies. It logically characterizes ways sets presented in Chapter 3 can be organized into hierarchies. Principles presented in Chapter 3 then govern membership of objects in sets organized within hierarchies. Numerous diagrams illustrate fundamental hierarchical substructures logically characterized by the Unified Logic and relationships and principles governing them.

The hierarchical substructure (and its governing principles) logically characterized in this chapter underlies all hierarchies, irrespective of the system of categorization, or classification, employed or kinds of objects to be categorized, or classified. In short, this part of the chapter logically characterizes the fundamental universal hierarchical structure.

With its introduction, the Unified Logic becomes a comprehensive, fully integrated logic, characterizing the fundamental structure of human reason. Just as it does for the logic of objects and logic of sets, the fundamental universal structure of human reason, abstracted and characterized in Chapter 2, provides the foundation for structures and relationships in the logic of hierarchies.

Chapter 5 - Confronting the Terminal Boundary of Human Reason

Human experience has clearly demonstrated that human reason has limits in its quest for understanding. That it does becomes clear when, for example, reason encounters questions it cannot answer and problems it cannot solve, as happens for example in particle physics at one extreme and astronomy and cosmology at another. In effect, unanswerable questions and unsolvable problems of this order and magnitude mark the terminal boundary of human reason. There reason reaches its limit.

Thus, part 1 of this chapter presents the theoretical and philosophical foundation for concluding that there is a terminal boundary to human reason.

Part 2 illustrates and logically characterizes what happens when human reason confronts its terminal boundary. The focus first is on what happens to reason near the boundary, then what happens to it when it is at the boundary. The focus then shifts to what happens to human reason when it attempts to transcend that boundary.

Part 1 - Theoretical and Philosophical Foundation, Terminal Boundary

The first part of this chapter then presents what happens to human reason when it reaches its terminal boundary. This part explores this phenomenon as manifested in, for example, science and theology.

In concluding, though, this part emphasizes the need to continue trying to answer what are for now unanswerable questions and solve what are now unsolvable problems. It also emphasizes the need to search for new tools of reason, ones that may someday allow us to break through or transcend that that terminal boundary.

Part 2 - The Unified Logic in Terminal Region

This chapter provides an overview of the challenge of the terminal boundary to human reason. It provides a rudimentary logical characterization of reasonís encounters with questions and problems that it cannot answer at the terminal boundary. Thus it provides a primer on ways reason attempts to extrapolate logically from the known to the unknown using the logic of objects and the logic of sets and logic of hierarchies.

This part of Chapter 5 comprises five sections.

The first section presents fundamental logical constructs and principles that can be employed at the terminal boundary.

The second section explores and logically characterizes the fate of human reason near the terminal boundary. Human reason creates there first-order terminal objects, based on the fundamental structure of human reason.

The third section logically characterizes the fate of human reason at and beyond the terminal boundary, based also on the fundamental structure of human reason. It focuses on logical characterizations of attempts to move beyond that boundary.

It then focuses on what happens to human reason as a result. Contexts explored include astronomy and cosmology and theology. This section presents products of speculating at and beyond the terminal boundary as the second-order terminal objects.

The fourth section explores and logically characterizes the use of synthesis in attempts to understand and then move beyond the terminal boundary of human reason. Fundamental examples from mathematics and mysticism are explored and logically characterized. This process creates second-order terminal objects.

The fifth and concluding section explores and logically characterizes the use of analogy in the quest for understanding near, at and beyond the terminal boundary to human reason. The process required also creates second-order terminal objects.

"Flight of the Pure Analogy"    Flight of the Pure Analogy   presented below is based on section 5, Chapter 5 of The Unified Logic. It was written for presentation at the 9th International Kant Congress, Berlin, Germany, March 2000. The paper explores reasonís use of analogy at the terminal boundary of human reason.

Appendix A

In Appendix A, critiqued is the prevailing theory of the continuum, based as it is on Cantorís fundamental theorem. Questioned then is the validity of the concept of the real number line, based as it is on Cantorís fundamental theorem. At the conclusion, provided is a new definition of the continuum. That definition provides fundamental ideas underlying the logic of objects, the logic of sets, and the logic of hierarchies presented in The Unified Logic. Twenty five principles, or laws, are derived from presentations provided in this appendix.

Afterword

Only the fundamentals of the Unified Logic are included in The Unified Logic. Thus, areas for future and further study exist. Each of the three fundamental logics, for example, can be expanded and refined. Also, logical relationships possible between the fundamental structure of human reason and surface features of human reason need to be explored, identified and logically characterized. Fields for such study include, in general, cognitive science and, in particular, psychology and linguistics. The Unified Logic can function as a reduced model for studies in those areas.

Fruitful also should be further research in the use of this logic in science and engineering, particularly in designing systems and processes. For example, based on the Unified Logic, a team comprising engineers, a computer scientist and programmer at Battelle, Pacific Northwest Laboratories, designed and developed a functional prototype of software to characterize and assess the operating efficiency of non-nuclear power plants, called Computer-Assisted Site Assessment (CASA) system. For its design and development, over the life of the project, the Laboratory allocated $80,000 in exploratory research funding. In creating computer programs, the Unified Logic could supplement or even replace neural networks, for it comprises simpler, more intuitive structures without loss of descriptive power.

As one focus of academic study, the Unified Logic could supplement if not replace traditional and some symbolic logic. The intuitively obvious character of the Unified Logic makes it more amenable to student learning. (Generally, only students who do well in mathematics also do well in traditional courses in logic.)

Current Studies in Human Reason

As introduced, humans have been conducting studies of human reason, or human cognition, within the Western analytic tradition at least since the time of the ancient Greeks. As reviewed in The Unified Logic, those studies continue as cognitive science. Cognitive Science includes studies in psychology, philosophy, computer science, artificial intelligence, neuroscience and linguistics. Extensive work in this area is now being conducted, for example, in the Stanford University Metaphysics Research Lab, which can be accessed over the Internet.


Conference Presentations

Conference presentations on The Unified Logic include these:


Kantís Transcendental Dialectic:

Flight of the Pure Analogy

by

John W. Nageley

Section 15: Kantís Logic

Cosmologists and astronomers and particle physicists, among others, roam the frontiers of knowledge and fly through Kantís "empty space of pure understanding," searching for breakthroughs in understanding. Some look out from Earth into the vastness of the universe while other look deep within the structure of matter. All wonder at what they find and what it all means. As Kant (1965) points out, however, "Human reason has this peculiar fate that in one species of its knowledge it is burdened by questions which, as prescribed by the very nature of reason itself, it is not able to ignore, but which, as transcending all its powers, it is also not able to answer." Scientists and others as well, though, continue to seek answers to such questions. They create ideas and theories to explain what they think is there and what it means. In doing so, though, they often attempt to transcend all their powers, to go beyond the limits of experience. Whereas Kant maintains that it is natural for them to do so, that metaphysics is entirely natural, he also maintains that the results are entirely without a basis and cannot be a science.

Analogies are often used in attempts to transcend the limits of experience. The kind used for such attempts is what I call the pure analogy. The objective of this paper is to characterize the logical structure underlying that kind of analogy. By way of illustration, the logical structure of two fundamental orders of pure analogies will be characterized. With the first kind, a first-order pure analogy, a noumenon can be compared with a phenomenon. With the second kind, a second-order pure analogy, one noumena can be compared with another noumena. Kantís (1965) definition of noumena as "mere objects of understanding" will be employed as will his definition of phenomena: "Appearances, so far as they are thought as objects according to the unity of the categories, are called phaenomena." To be accounted for is Kantís position that "...analogies have significance and validity only as principles of the empirical, not of the transcendental, employment of understanding."

Generally, Kant notes that logic deals with the forms or structures of thought, not with its content. Therein lies the power and value of logic. As James Newman (1956) writes in a

commentary on George Booleís work, "Symmetries, relations and resemblances which in ordinary language are hidden or veiled are made to stand out boldly in a symbolic idiom." Furthermore, he adds, "An efficient symbolism not only exposes errors previously unnoticed, but suggests new implications and conclusions, new and fruitful lines of thought." Thus, the objective of this paper can be stated as making perceived analogical relations between phenomena and noumena "stand out boldly in a symbolic idiom." In the "symbolic idiom," when logical structures are formulated independent of experience, such as pure analogies, though, the only measure of their validity is the internal consistency and integrity of their structures. Therefore, the emphasis will be on characterizing their fundamental logical structure.

To begin characterizing the fundamental structure of pure analogies, the fundamental structure underlying a common analogy, which is one that compares two phenomena, will be examined. For example, a simple comparison between two phenomena is contained in the simile "My Love is like a red rose":

(R) Rose

---

(L) My Love

"My Love" is a phenomenon and is being compared with a "Rose" also a phenomenon. (All phenomenon will be enclosed in parentheses to distinguish them from noumena, which will be enclosed in brackets.) The objective is to reveal otherwise unknown or unexpressed characteristics of "My Love." A one-to-one comparison between the two cannot, though, be made directly, that is, not directly between the two things themselves. Instead, it must be based on a comparison of their attributes.

For example, attributes of the unknown "My Love" in the relation can be logically characterized in this way, independent of content:

(l1, l2, l3,...) Î (L) (My Love)

That is, there is no known limit to the number of attributes that can logically belong to (L). For the known "Rose" in the relation, its attributes can be logically characterized in this way, also independent of content:

(r1, r2, r3,...) Î (R) (Rose)

where, again, there is no known limit to the number of attributes that can logically belong to (R). With this much established, it is now possible to construct a common analogical relationship between "My Love" and "Rose" as shown here:

 

(r1, r2, r3,...) Î (R) (Rose)

× × × ---

(l1, l2, l3,...) Î (L) (My Love)

Thus, the unknown (L) is compared with the known (R), one phenomenon with another, based solely on their attributes. Logically, then, the more attributes that can be mapped one-to-one between the two, the greater will be the descriptive power of this analogy.

As stated, with a first-order pure analogy, a noumenon can be compared with a phenomenon. Stanford University's Sidney Coleman (Hoffman 1990), a leading theoretical physicist, provides a contemporary example of such an analogy. In his theory of multiple universes, the universe contains other universes, one within another. Kantís identifies the "universe," or "cosmos," as a product of pure reason: "Pure reason thus furnishes the idea...for a transcendental science of the world (cosmologia rationalis)...." The idea underlying Colemanís theory then is a noumenon, represented here by [U]:

[U] [Universe of Universes]

(Brackets are used to identify noumena.) So that others can understand what he means, Coleman presents the universe containing other universes as analogically related to a balloon that contains other balloons, one within another. It is a phenomenon, represented here by (B):

(B) (Balloon of Balloons)

Based on Colemanís illustration, a first-order pure analogy shown here can then be created:

[U] [Universe of Universes]

---

(B) (Balloon of Balloons)

As such, Coleman is comparing a noumenon with a phenomenon. Again, because things themselves cannot be compared, to fully characterize their analogical relationship, a one-to-one correspondence must be created between their attributes. For the phenomenon "balloon of balloons," (B), attributes that can logically belong to it can be characterized in this way, again independent of content:

(b1, b2, b3,...) Î (B)

To complete this first-order pure analogy, the noumenon "universe of universes" must be included. By definition, though, a noumenon has no known attributes. Though noumena are "mere objects of understanding," still it is possible to logically characterize attributes that can theoretically belong to one, based simply on the fundamental structure of human reason. Attributes that can in this example logically belong to "universe of universes," [U], can be characterized in this way:

[u1, u2, u3,...] Î [U]

Furthermore, as is the case with a phenomenon, even though [U] is a noumenon, there is no known limit to the number of attributes that can logically belong to it.

Therefore, between the phenomenon (B) and the noumenon [U], a first-order pure analogy can now be constructed:

[u1, u2, u3,...] Î [U]

× × × ---

(b1, b2, b3,...) Î (B)


That is, it is logically possible to construct a one-to-one correspondence between attributes of the phenomenon and those of the noumenon. Ideally, by creating this pure analogy, we should be able to better understand Colemanís idea of the universe that contains other universes by understanding its relationship to the idea of a balloon that contains other balloons. Theoretically, also the more attributes that can be mapped between the two, the greater will be the descriptive power of this analogy.

A variation in the use of first-order pure analogies can be found in Hermetic philosophy, specifically, in its Doctrine of Analogy, contained in the Emerald Table, which is attributed to Hermes Trismegistus (Underhill 1967). That doctrine, the alchemistís "Key of Creation," reads, Quod superius sicut quod inferius. For alchemists, it had this meaning: "What is below is like that which is above, and what is above is like that which is below, to accomplish the miracles of one thing." In that Doctrine, according to Underhill, is "an implicit correspondence between appearance and reality, the microcosm of man and the macrocosm of the universe, the seen and the unseen worlds." Created by that doctrine then is this analogical relation:

Superius

¯ 1 ­ 2

Inferius

Here, first the inferius, or phenomenon, is shown as a projection of the superius, a noumenon; then the superius is shown as a projection of the inferius. Again, because things themselves cannot be compared, to fully characterize the analogical relationship between the inferius and superius, a one-to-one correspondence must be created between their attributes. For the inferius (I), attributes that can logically belong to it can be characterized in this way:

(i1, i2, i3,...) Î (I) (Inferius)

For the superius (S), attributes that can logically belong to it can be characterized in this way:

[s1, s2, s3,...] Î [S] [Superius]

Because logically there is no known limit to the number of attributes that both can have belonging to them, this first-order pure analogy can now be constructed:

[s1, s2, s3,...] Î [S] [Superius]

× × × ---

(i1, i2, i3,...) Î (I) (Inferius)

With second-order pure analogy, on the other hand, as introduced, two noumena, can be compared. Even though noumena are beyond experience, the fundamental structure of human reason allows this to occur. For example, logically, Coleman's idea of the universe of universes, [U], which is a noumenon, can be compared with the idea of the set of all sets, [S], used in set theory, which is also a noumenon:

[U] [Universe of Universes]

---

[S] [Set of All Sets]

Again, because things themselves--even noumena--cannot be compared directly, to fully characterize an analogical relationship between two noumena, a one-to-one correspondence must be created between their attributes. To this end, attributes that can logically belong to "set of all sets," [S], can be characterized in this way:

[s1, s2, s3,...] Î [S] [Set of All Sets]

Thus, based on attributes that can logically belong to the noumena "universe of universes" and the "set of all sets," this analogical relationship can now be constructed:

[s1, s2, s3,...] Î [S] [Set of All Sets]

× × × ---

[u1, u2, u3,...] Î [U] [Universe of Universes]

Thus attributes of the two noumena can be logically mapped one-on-one. That is, the logical structure of one noumenon is here related to the logical structure of another noumenon. The structural similarity of both is thereby revealed. Theoretically, again, there is no known limit to the number of attributes that can belong to each, and the more attributes that can be mapped one-to-one between the two, the greater will be the descriptive power of this analogy, even though both are noumena and thus devoid of content.

Georg Cantorís series of transfinite numbers (Cantor 1955) can be also be viewed as a product of a second-order pure analogy. Based on his "diagonal proof," he posited the possibility that there are transfinite cardinals beyond the cardinality of the real number line--even an infinite number of such transfinite cardinals. To those cardinals, he assigned the symbol À , aleph. This then became his sequence of transfinite numbers:

1, 2, 3,..., À 1, À 2, À 3,..., À n

Cantor is credited thereby, by some, as having established in mathematics the existence of "actual" infinity (Fraenkel 1966). Though Cantor did not create his sequence of transfinite numbers by way of analogy, nevertheless, when reduced to its essential components, it becomes evident that that sequence contains two analogically related noumenal ideas. First is the sequence of rational numbers:

1, 2, 3,..., n

Second is the sequence of transfinite numbers:

À 1, À 2, À 3,..., À n

Thus, underlying Cantorís complete sequence is this analogical relationship, one between rational numbers and transfinite numbers, both noumena:

[À 1, À 2, À 3, ,..., À n] Î [T]

× × × × ---

[1, 2, 3,..., n] Î [R]

Summary

In summary, a crucial point in this: underlying all pure analogical constructs is the fundamental structure between a phenomenon and its attributes:

Common Analogy

(x1, x2, x3,...) Î (X) (Phenomenon)

× × × ---

(y1, y2, y3,...) Î (Y) (Phenomenon)

These then are two kinds of pure analogical constructs, each based on the fundamental structure underlying the common analogy:

 

First-Order Pure Analogy

[x1, x2, x3,...] Î [X] [Noumenon]

× × × ---

(y1, y2, y3,...) Î (Y) (Phenomenon)

 

Second-Order Pure Analogy

[y1, y2, y3,...] Î [Y] [Noumenon]

× × × ---

[x1, x2, x3,...] Î [X] [Noumenon]

Third order pure analogies and beyond also can be constructed, but they are even further removed from phenomena than are the first two.

As demonstrated, when pure analogies are made to stand out in the "symbolic idiom" noted by Newman, it becomes obvious that the fundamental logical structure on which pure analogies can be constructed is the same fundamental structure on which common analogies are constructed. The major difference between a common and pure analogy is simply in the kinds of things that can be compared.

Commentary

In resorting to pure analogies in their sciences, Coleman and Cantor are, according to Kant, resorting "to principles which overstep all possible empirical employment," thus what they do cannot be science. For example, compared with the bulk of the pronouncements of cosmologists about the nature of the universe, ones that they can prove through the "empirical employment" of reason, are relatively few. There is little scientifically valid evidence to support, for example, the existence of dark matter at the edge of the universe.

In creating his theory of transfinite numbers, Georg Cantor can also be seen as overstepping "all possible empirical employment." As Alexander Calder (1979) points out, for example, "Cantor's proof that the set of transcendental numbers was infinite failed rather spectacularly ...in that it did not give rise to one example of a transcendental number, let alone an infinity of them." A.A. Fraenkel (1966), for his part, is forced to this conclusion about Cantorís work: "Unfortunately,...some of the results of set theory, in particular Cantor's theorem and its proof, bring us close to a gulf which is difficult to bridge. The matter is serious enough to have induced important mathematicians to speak of Cantor's work as a Ďpathological entanglementí which later generations would look upon with bewilderment." That entanglement is attributable to the paradoxes his theory creates when carried to its mathematical and logical conclusions (Barker 1964).

Thus, both Coleman and Cantor, and alchemists as well, can be seen as flying on the wings of the pure analogy in "the empty space of pure understanding," up there with Plato: the world of the senses is just too limiting. Even though no resistance is encountered in that space, Kant holds that no advancement in understanding can be achieved. He (1950) warns, "All knowledge of things merely from pure understanding or pure reason is nothing but sheer illusion, and only in experience is there truth." Though Coleman, for example, ventures out into "the empty space of pure understanding" on occasions, he is firmly grounded in the fundamental principles of applied science. So he does not rely on pure understanding or reason for his science. Cantor (1955), though, does rely on it. Cantor believes that "conceptions in pure mathematics are free, and not subject to any metaphysical control." It was in refusing to acknowledge that he encountered resistance during his flight that he was able to conclude that he has discovered actual infinity.

Whereas, Kantís warning about the use of analogies as principles of the transcendental employment of understanding must be heeded, within the larger context of his epistemology, though, pure analogies play an important role in exploring the limits of human reason. For example, such ideas as black holes, wormholes in the fabric of space, and dark matter at the very edge of the universe, and other exciting ideas in cosmology and astronomy, are predicated on pure analogies and have been productive (Hawking 1988). Similarly exciting ideas created by pure analogies are found in particle physics, such as superstrings (Davies and Brown 1992). Without pure analogies, many sciences would be pretty dull stuff, really.

A standard of excellence in the conduct of science is Einsteinís. Lincoln Barnett (1968) in The Universe and Dr. Einstein, in speaking of the "new and extraordinary truths about the universe" that have been discovered by applying Einsteinís theories, points out, "These truths can be described in very concrete terms. For once he had evolved the philosophical and mathematical bases of Relativity, Einstein had to bring them into the laboratory, where abstractions like time and space are harnessed by means of clocks and measuring rods."

In any case, as Kant notes, the human inclination to go beyond the limits of experience is natural; metaphysics in other words is natural. He writes that, as such, humans are no more likely to give up metaphysical speculations than they are to give up breathing. So, when researchers roam the frontiers of knowledge or fly through the empty space of pure reason in seeking breakthroughs in understanding, ideas found there often seem quite natural to researchers, particularly, those arising from pure analogies. For example, metaphysical speculations were the foundation for the science of the alchemist, therefore most natural. Even though their science seems primitive now, they did lay the foundation for modern day chemistry. Sir Isaac Newton was deeply influenced by their work (Schlagel 1996). Coleman (Hoffman 1990), for his part, as to the naturalness of his thinking, claims, "Multiple universes is one of the most natural ideas in the world." Indeed, what could be more natural? To Cantor (1955), his theory of transfinite numbers also is natural. He states that his purpose in creating those numbers was "to generalize or to extend the series of real integers beyond infinity." He then adds, "Daring as this might appear, I express not only the hope but also the firm conviction that in due course this generalization will be acknowledged as a quite simple, appropriate, and natural step." About Cantorís "natural step," the highly respected turn-of-the-century mathematician David Hilbert (1925) writes, "No one shall drive us out of the paradise which Cantor has created for us."

Pure analogies then can be considered "natural" extensions of human reason. They can be at least inspirational and therefore often productive, even though, or even perhaps because, they do not have an empirical foundation. About extreme ideas found in cosmological, Coleman (Hoffman1990) states, "These ideas have the great property of making our head feel funny."

In other words, admitting that there are limits to the abilities or capabilities of human reason does not require that we accept those limits, that we cannot or should not challenge and possibly overcome them. Rather, the imperative is to recognize when questions are unaswerable, and then seek new tools and methods to answer them. That there are now questions that burden us that we cannot ignore, but still cannot answer, does not mean that we will never be able answer them.

Because metaphors are fundamentally analogies, Max Black (1962) provides this useful insight: "Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra." The paradigm he presents is not, though, complete. Instead, it should be stated, perhaps attempts to push the limits of knowledge can begin with pure analogies. They inspire and guide researchers in undertaking research that they might not otherwise undertake. To be valid, though, Kant would hold that pure analogies cannot end with algebra. Instead, science should begin with pure analogies and then move to algebra. To be valid, though, science must end with experience. In that way, ideas that have no grounding in experience today may have grounding in it tomorrow.

References

 

Barker, S.F. 1964. Philosophy of Mathematics. Prentice-Hall, Inc., New Jersey.

 

Barnett, L. 1968. The Universe and Dr. Einstein. Bantam, New York.

 

Black, M. 1962. Models and Metaphors. Ithaca, New York.

 

Calder, A. 1979. "Constructive Mathematics." Scientific American.

 

Cantor, G. 1955. Georg Cantor: Contributions to the Founding of the Theory of Transfinite Numbers. Jourdain, P.E.B. (Ed.). Dover Publications, Inc., New York.

 

Davies, P.C.W., and J. Brown (Eds.). 1992. Superstrings: A Theory of Everything? Cambridge University Press, New York.

 

Fraenkel, A.A. 1966. Set Theory and Logic. Addison-Wesley Publishing, Reading Massachusetts.

 

Hawking, S. W. 1988. A Brief History of Time: From the Big Bang to Black Holes. Bantam Books, New York.

 

Hilbert, D. 1925. "On the infinite." Philosophy of Mathematics, 1985. Paul Benacerraf and Hilary Putnam (Eds). Cambridge University Press, Cambridge.

 

Hoffman, P. 1990. "Physic Fugitive." Discover, July 1990.

 

Kant, I. 1965. Critique of Pure Reason. Norman Kemp Smith (Trans.), St. Martin's Press, New York.

 

Kant, I. 1950. Prolegomena to Any Future Metaphysics. Bobbs-Merrill Company, Inc., New York.

 

Newman, J.R., 1956. "Commentary on Symbolic Logic, GEORGE BOOLE and a Horrible Dream." The World of Mathematics, v3, J.R. Newman (Ed). Simon and Schuster, New York. ed.

 

Schlagel, R.H. 1996. From Myth to Modern Mind: A Study of the Origins and Growth of Scientific Thought. Peter Lang, New York.

 

Underhill, E. 1967. Mysticism. The World Publishing Company. Cleveland, Ohio.


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