Using Axioms to Unblock Civilization's Progress

5 May 2015

Using Axioms to Unblock Civilization’s Progress Frank J. Lucatelli, Personal Intelligence, LLC Rhonda C. Messinger, Career Momentum, LLC a chapter in CRISIS AND RENEWAL OF CIVILIZATIONS The 21st Century Crisis of Ideas and Character

Marek Celinski, editor ABSTRACT Through this chapter, society’s indiscriminate, over-abundant and ongoing collection of data is identified as a block to civilization’s progress. It is argued that the data deluge is caused by insufficient methods for validating constructs - creative ideas and intuition. APriori Modal Analysis (APMA) is offered as a holistic axiomatic method of logic for establishing construct validity and overcoming the obstacles our data driven society faces. Through APMA, the progress of civilization from simple ideological forms of governance to purposeful forms of governance, in response to increasing global complexity, is charted. Governance and healthcare are used to demonstrate how systems can evolve in intricacy, how control can be dispersed more widely and how civilization can progress without burning out. APMA reveals how axioms and postulates can guide and limit data collection to only the critical information needed, and how statistical data can be meaningfully sorted and validated with increasing complexity. Keywords: Axiomatics, Construct, Validity, Axiom, Postulate, Logic, Statistics, Holism, Civilization, Complexity, Data, Intuition, APMA, Society, Governance, Healthcare Civilization’s Progress Is Blocked We are living in an age of data. We have so much data, in fact, that we don’t know what to do with it all. Computers help us by sorting and storing the data, and we can eliminate the use of chunks of data by picking and choosing which parts we want to use, but this is often random. Who hasn’t heard the line that we can prove anything by rearranging the statistics or that data only proves what we want it to prove? As Mark Twain said, “Facts are stubborn, but statistics are more pliable1.” According to Nate Silver2 an American statistician and writer who analyzes baseball and elections, “Information has gone from scarce to superabundant. Every day, three times per second, we produce the equivalent of the amount of data that the Library of Congress has in its entire print collection…” In 20103 The Economist reported that “…when the Sloan Digital Sky Survey started work in 2000, its telescope in New Mexico collected more data in its first few weeks than had been amassed in the entire history of astronomy. Now, a decade later, its archive contains a whopping 140 terabytes of information. A successor, the Large Synoptic Survey 1 Mark Twain. (n.d.) BrainyQuote.com. Retrieved December 20, 2014, from BrainyQuote.com Web site: www.brainyquote.com/quotes/quotes/m/marktwain163414.html 2 Nate Silver. (n.d.). BrainyQuote.com. Retrieved December 20, 2014, from BrainyQuote.com Web site: http://www.brainyquote.com/quotes/authors/n/nate_silver.html 3 The Economist, Special Report: Managing Information, Data, Data Everywhere, February 25, 2010

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Telescope, due to come on stream in Chile in 2016, will acquire that quantity of data every five days.” The Economist goes on to say, “Such astronomical amounts of information can be found closer to Earth too. Wal-Mart, a retail giant, handles more than 1m customer transactions every hour, feeding databases estimated at more than 2.5 petabytes—the equivalent of 167 times the books in America's Library of Congress.” It is likely that we have amassed more data at this point in time than we could possibly use in many life times. And, because we value numbers and measurement so highly, we are paralyzed as we wait for all this data to be sorted and used so we can move forward as a society. The volume of data we have collected and need to analyze is so large, at this point, that it is actually blocking our progress as a civilization, rather than helping us to evolve. The Axiomatic and Statistical Balance At one time, we recognized with greater significance the necessary interplay between data and ideas. Data and statistics provided the perspective of measurable evidence. Ideas, intuition and axiomatics held the perspective of structural logic. Statistics and axiomatics were used to balance each other when inventing, discovering and understanding what was real and what was not. Together they were able to dance us into the future, each providing the checks and balances for the other with neither holding the lead all of the time. Civilization grew as a result of this balance. When an idea, such as the world is flat, was formulated, it held power until data could be used to prove or disprove the perspective. Likewise, when data was gathered and interpreted as truth, such as a child could inherit the experiences of a parent4, that information held power until the idea could be examined as logical or not. Truths and knowledge grew from entertaining both perspectives; each perspective serving as a counter balance to check the other’s validity. As far back as Plato and Aristotle, both evidence in the real world and heart felt knowing balanced each other. Plato used emotional awareness and relied on an intuitive ability to know what was right and accurate to form his conclusions. He used mental exploration of why things occurred as his compass. Arthur Koestler shared this observation in The Sleepwalkers5: “... Plato maintained that true knowledge could only be obtained intuitively, by the eye of the soul, not of the body; ...”6 Aristotle, although submerged in the Platonic world, used tangible evidence whenever he could, and relied on what he saw in the world around him to examine the qualities that Plato proposed. From there he would form his accurate conclusions looking at how each situation played out. Aristotle’s approach led to creating and studying volumes of data and drawing conclusions on how the observed data was connected. 4 reference French naturalist Jean Baptiste Lamarck 5 Koestler, A. (1959) 6 Ibid, p. 160.

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“... Aristotle had stressed the importance of experience – empiria – as against intuitive aperia ...”7 As science developed, the Aristotelian approach was hampered by the technological inability to measure some things. For those things, Plato's intuition carried the day. Circles were accepted as perfect forms during those times. Plato assumed that planetary orbits, because of their presence in the unchangeable heavens, must be perfect and therefore must be circles. It wasn't until Tycho Brahe's pioneering astronomical measurements of the motion of the planets, many centuries later, that Johannes Kepler was able to prove that the orbits of planets were not circles, but ellipses. Because of the long-standing belief in circular orbits, Kepler himself believed that he had merely discovered a flaw in the universe rather than a universal truth. Newton's laws of gravitation, years later, were required to give context to Kepler's discovery. The legacy of Plato is that he demonstrated that the mind could perceive truths that exist beyond the messiness of material existence. He showed that there is a logic that transcends material reality and that logic is useful as at least a second opinion in any scientific activity. That Plato's assumptions were not always correct is obvious. His contribution to helping us move forward and not stay stagnant as a society as we waited for evidence is also obvious. The point and purpose of Plato's approach was to make a prediction about the nature of things and provide an opportunity to statistically prove or disprove an application of the prediction. This checks and balances of data and assumptions is what advanced science. Dmitri Mendeleev, the creator of Periodic Law, used intuition and axiomatic hypothesis formation as the start of his discovery. In a race to create gold from base metals, alchemists had developed a long list of materials and their properties. Mendeleev felt that many of these materials could be sorted or arranged in some way that would make sense. Overwhelmed by the data, he made an assumption that comparing weight with physical properties might be a way to sort materials. He developed a chart that crossed weight with the properties and then placed the alchemist’s materials on the grid. As he discovered open spaces in the chart he predicted that, if the chart were correct, a new material could be found to occupy each empty space. His work brought us the Periodic Table of Elements and continued to transform the craft of alchemy into the discipline of chemistry. Charles Darwin, the father of biological evolution, made his discovery using data and statistical examination. By observing and documenting what he was seeing, he was, over time, able to draw conclusions about what he had chronicled. Darwin noticed that birds of the same species living in a common geographical area had similar songs. Those of the same species living in a different area had a different type of song. He concluded that the birds living geographically close to one another were influenced both by their geography and each other, and therefore, there must be an interaction between organisms and their environment that influenced how the species evolved.

7 Ibid

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By using axiomatics and statistics in harmony, as a holistic system, both Mendeleev and Darwin were able to refine their understanding of the world around them. They did not stand alone in combining their use of axiomatics and statistics. The Pythagoreans of early Greece believed that all numbers were integers, which they confirmed with their experiments with music and the whole number divisions of musical strings. The discovery that the diagonals of integer sized squares could not be expressed as whole numbers created a crisis within their mathematical conception of reality. Later mathematicians were forced to reframe their mathematical number theory to include irrational numbers. The square root of two, a heresy to the followers of Pythagoras, was simply another number to future mathematicians. In geology, the locations of earthquakes and volcanoes on a map of the earth presented a confusing panorama of seemingly arbitrary continental shapes and geological activity. The axiomatic sorting of continental drift and resulting colliding plate tectonics provided a perspective with which the accumulating data made sense, confirming a generalized movement within the Earth’s geology. In each of these examples the axiomatic and statistical methods support each other. The confusion of large amounts of data stimulated axiomatic hypothesis formation and hypothesis formation caused people to collect data. So why are we not using axiomatics now to address the large amounts of data that are paralyzing us? The answer is simple. The development of axiomatic systems that could be used as the balancing perspective to data collection was not developed at the same rate as data collection methods due to specific landmark mathematical events during the 1930’s. A Stunt in Axiomatic Development During the 1900’s David Hilbert and Kurt Gödel, leaders in the field of mathematics, were trying, once and for all to establish the basis of knowledge. At the International Congress of Mathematicians in Paris during 1900, Hilbert presented a list of twenty-three mathematical problems8 that he believed were the most important to be solved. From that list he selected two personally interesting items to work on that had the potential to demonstrate the value of axiomatics in mathematics.  Hilbert's Problem Number 2: The compatibility of the arithmetical axioms.9 Prove that the axioms of arithmetic were consistent. This is essential to justify the use of axiomatic systems as trustworthy vehicles for ascertaining truth.  Hilbert's Problem Number 6: The mathematical treatment of the axioms of physics. Identify a method to systematize the use of axioms in physics. If this can be done, then the method could be generalized to other disciplines.

8 Hilbert, D., (2014). Also see: . 9 If problem number 2 is shown to be true, then applying axiomatics to the sciences is justified, especially in “the theory of probabilities and mechanics.”

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Hilbert chose geometry10 as the topic to demonstrate his ideas. If he succeeded in developing a system for geometry, he believed that he would then be able to demonstrate that his program or system would lead us to certainty in all mathematics. His most famous quotation, inscribed on his tombstone, is: “We must know. We will know!” Adopting the popular belief that axioms and postulates were indistinguishable, Hilbert produced five sets of axioms for a total of 20 axioms11 in all, without once mentioning the term “postulate12.” His attempt was to make the axioms of geometry clearer, and consequently make geometry more solidly grounded in logic. If axioms were accurate and consistent, and a system for revealing the truth derived from axioms13 could be systematically developed, axiomatics could be established as a trustworthy source of knowledge. In 1931, seventeen years before the end of Hilbert's life, as he prepared to publish his work, Kurt Gödel, also a skilled mathematician, produced an extremely clever mathematical Incompleteness Proof14 that prematurely discounted Hilbert’s work and brought an end to creating a method for revealing “truth” from axiomatic systems. Both Hilbert’s and Gödel's failure to distinguish axioms from postulates set the stage for the consequential derailment of mathematics, and civilization as a whole, from a balanced interplay between axiomatics and statistics, to a predominantly statistics driven field. Not understanding that postulates were critically different from axioms and also, like axioms, not provable, mathematicians failed to see that Gödel's proof was at least a confirmation of the need for recognizing postulates as a unique category of logical statements, rather than a death sentence for axiomatic systems. The general misunderstanding of the implications of Gödel's proof began an eighty-four year focus on the statistical examination of what is important and where these important elements occurred, while holistically systematic methods for examining how and why things were happening were not established. Society began to wait for axiomatic hypothesis formation to provide information on how and why things were occurring as a result of being overwhelmed with data. We began to rely on conceptualization as a defensive mode, only using it when we needed to reduce data down into manageable categories, buckets, or nuggets. For example, prior to WWII there was a tradition that men worked outside of the home and women worked within the home. During the war, women volunteered to fill the labor slots that men had traditionally held. After the war most women returned back to the home due to the volume of men in need of jobs; nostalgia to return to the way things were; and a new 10 Hilbert, D., 1980 (1899). The Foundations of Geometry is regarded as David Hilbert's major work of axiomatics. 11 Hilbert, D. (German ed. 1899, Translated 1971, seventh printing 1994). p.3. 12 The “use of 'postulate' derived directly from classical Latin 'postulo' = to demand, to request; in Medieval Latin 'postulatum' (pp participle) took on the meaning of 'something assumed,' i.e. something that 'demands belief.' ” Quoted from Griffin, R. (2014). The “something assumed” are the axioms, which the postulates demand to be believed in order to appreciate the constructed meanings of the postulates derived from those axioms. -fjl 13 It is my opinion that David Hilbert made a common mistake made by mathematicians who enter the discipline because of their attraction to intellectual abstractions and the purity of thought that they represent. What this motive is prone to overlook is that axioms, which are the basis of any logical development, are not intellectual objects but rather emotional intuitions that ring true as accurate. -fjl 14 Gödel, K. (1931, 1962)

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American wealth allowing for single income homes. Traditionally, men worked outside of the home and women worked within the home. Data, driven by new wealth and the volume of men needing work, drove many women back into the home despite an expanding awareness that they had skills that were valued outside of the home. Were an axiomatic approach used to address the larger question of women’s role in society, the assumption of men and women needing separate places could have been challenged, and a new view could have been formed where they could live and work side by side. Other approaches to work might have been formed which would have continued to advance our civilization. By looking only at data, women took several steps backwards and it took years to get women re-engaged in the workforce and sharing their values in the corporate world. The motivation to holistically address the issue and ask how and why we could change the male/female dynamic was abandoned in lieu of data collection, because, in large part, the methodical development of axiomatics was cut short. As methods of data collection grew and were refined, supporters of axiomatic conceptualization went “out of fashion” and were marginalized in favor of reliance on hard facts. All un-measurable approaches to science were discredited. Activities like astrology, hand-writing analysis and alternative medicine were brushed aside as unreliable superstitions because of the difficulty of measurement associated with them. Even religion and its unscientific view of the world became a marginalized activity. The science of behavioral psychology, as practiced by B.F. Skinner, emphasizing evidential, statistically analyzed, observations took center stage. The work of Carl G. Jung and other psychoanalytical theorists with a holistic approach and with leanings toward less measurable notions of archetype15 and other allegorically based ideas were marginalized. The messages originating from conceptual sources were perceived as scientifically unreliable, since validation by data analysis was not possible. There was no method to rigorously examine and validate constructs. As time moved forward and statistical efforts increased, we used an intuitive ability to sort data into logically sounding groups until all of it was either accounted for or discarded, yet we lacked the capability to determine if we had all the data that was needed, and whether we had the appropriate number of groups. A structure was missing that would determine if we had everything we needed and whether it was sorted correctly. A Structure for Examining Axiomatic Assumptions Over time many great people struggled with creating a framework for logic. This was in part due to the complexity of the task, and also due to society’s renunciation of all things non-measurable. With mathematics stuck in the belief that axiomatics was incomplete and unreliable, thought leaders were discouraged from even trying. But some verification work persisted. Historically, Euclid's Elements of geometry form the basis for all work in logic. The role that Euclid most likely played has contributed to our misunderstanding of axioms and postulates. It is commonly believed that Euclid was primarily a compiler of 15 Jung, C. (1959, 1973). The concept of “archetype” was introduced by Carl G. Jung as an effort to acknowledge a broader scope of influences in the life of an individual. The archetypal construct or concept allowed Jung to bring ephemeral assumptions into a concrete forms that could be examined and appreciated as important driving forces in one's life.

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the work of many other mathematicians who preceded him, and this might explain his lack of clear distinction between common notions (axioms) and postulates. Some centuries later Ramon Llull16, pioneered a conceptual logic system that strove to unite meaning with structural form. Llull wrote extensively on the subject but in the end, his work died with him. Llull joined meaning with structure in an intuitively logical system. He struggled however, with sorting conceptual information in a logically consistent way. During the late nineteenth century and early twentieth century people like George Boole17, Charles Sanders Peirce18, Abdul'l-Baha19, Gottlob Frege20, David Hilbert21, Ludwig Wittgenstein22, Bertrand Russell23, and Kurt Gödel24 all contributed to questions concerning the fundamentals of logic. They individually and collectively raised issues that our culture, still today, has not fully resolved. Ludwig Wittgenstein played an especially important role in developing Peirce's concept of logical connectives, which are the differentiated elements of a powerset25. He wrote about the powerset of three in his book Tractatus Philosophicus,26 and in other writings explored the powerset of four. Shea Zellweger27, a contemporary researcher, developed elaborate models that demonstrated the tightly integrated relationships among Wittgenstein's sixteen logical connectives. It is in response to the work of these individuals that the A-Priori Modal Analysis (APMA) system, presented below, was developed by co-author Frank Lucatelli. A-Priori Modal Analysis (APMA) Frank’s impetus for developing APMA, like most innovations, was to solve his own personal difficulty in appreciating the importance of having a common scale of detail for understanding terms and concepts, and a difficulty in sorting factual detail. In his experiences, metaphors were more highly valued in conversation than analogies, puns were seen as the lowest form of humor, and satire was considered clever. Both metaphors and analogies, are just another way of bringing 16 Llull, R. (2014) (b. 1232-d. 1315) 17 George Boole (b.1815-d.1864) with his ground-breaking book, Laws of Thought, introduced the concept that thought itself obeyed consistent principles. 18 Charles Sanders Peirce (b.1839-d.1914) began original work in the area of logical connectives. See: 19 Abdu'l-Baha (b.1844-d.1921) 20 Frege, G. (1893) (b.1848-d.1925), building upon the work of George Boole, applied the logical method to search for the fundamentals of mathematics in his application to arithmetic. 21 David Hilbert (b.1862-d.1943) was one of, if not the, leading mathematician from the late 1800's until the publication of Gödel's incompleteness theorem. 22 Ludwig Wittgenstein (b.1889-d.1951), following Peirce, was a pioneer in the exploration of the use of truth tables. 23 Bertrand Russell (b.1872-d.1970), co-author of Principia Mathematica with Alfred North Whitehead, is also known for his discovery of the Russell-paradox. 24 Kurt Gödel (b.1906-d.1978) is best known for his publication of his “incompleteness” theorem, which many believed that it showed that David Hilbert's program was not achievable. 25 The work of Charles S. Peirce in the development of truth-tables, and further developed by Ludwig Wittgenstein and Bertrand Russell. 26 Wittgenstein, L. (1922, first English edition; 1961) 27 Shea Zellweger has pioneered a logic alphabet that translates the logical connectives of axiom levelfour into a typography that is used to graphically show logical transformations. and

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a term to life, but the impact each of these forms of speech had on people puzzled him. Hierarchies of terms seemed to exist in other disciplines as well, and Frank’s investigation of several led him to Euclid’s confusing presentation of common notions and postulates in The Elements. The tragic story about Hilbert's dream and Godel's proof finally made sense, and was confirmed for Frank by a statement by Abdu'l-Baha: “By nature is meant those inherent properties and necessary relations derived from the realities of things28. Culture had misinterpreted Godel's proof. Upon additional examination of Euclid’s lists, with the benefit of hindsight and history, Frank created these working definitions of axioms and postulates. 

Axioms are un-provable points of truth. They represent fundamental knowing, ring true without factual evidence and do not require an explanation.



Postulates result from the necessary relationships that exist among axioms. They are created by the convergence of two or more axioms, they require thought, they are synthetic, and they also are not provable.

It can be noted that, as part of his work with axiomatics, Frank has refined the term postulate by identifying that postulates are not provable. For close to a century postulates have been alternatively confused with axioms as if they were synonymous with them, and with hypotheses, where multiple ideas are brought together to form a new composite idea comprised of axioms, theorems and/or postulates. A hypothesis, therefore, can be proven true with data and other forms of evidence. APMA distinguishes postulates from hypotheses, and from the previously held notion of postulates being synonymous with axioms. The postulates discussed in this chapter are simply the combination of a level's given axioms, where the number of axioms identifies the logical level. Although postulates cannot be proven or disproven, they must ring true and not contradict evidential data. APMA uses axioms and postulates, as defined in this chapter, to create a grid in which key concerns (axioms) can be separated from their subsets (postulates) when examining a system, process or organization. By separating the key concerns from their subsets, systems can be more carefully examined and discussed. Unlike other grids, APMA provides confidence that each key concern is of equal weight and value to the next, and that all the important and appropriate subsets are covered. It is a holistic approach to examining concepts. Axiomatic thinking using the APMA model is a progression of logical levels, distinguished by the number of axioms within them, and becoming more complex and sophisticated as the number of axioms increases. The model includes the application of axioms and postulates as distinct types of assumptions.

28 (Abdu'l-Baha, Baha'i World Faith - Abdu'l-Baha Section, p. 340)

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In the following diagrams the number of axioms increases incrementally, one additional axiom at a time, and the corresponding number of postulates increases exponentially. This is the primary reason that it is so important to distinguish between axioms and postulates. When treating both axioms and postulates as the same logical objects, we can quickly become overwhelmed by the complexity of the systems. Separating them helps to focus on the essential axiomatic qualities of each logical system while postulates provide the necessary complexity as systems evolve. A One-Axiom System In a one-axiom system there are only two items. One-Axiom System No. of Items 1 1 0

Category Null Axiom Postulates

Name Not A1 A1

Diagram 1 - A one-axiom system

Axioms are symbolized by letters. The subscript next to the letter identifies the axiomatic system number. The axiom in a one-axiom system therefore, has the subscript 1. In a twoaxiom system each of the axioms carry the subscript 2. Postulates are automatically generated by the necessary relationships among the axioms. Single letters represent axioms and letter combinations represent postulates. Postulate subscripts also match the system from which the postulate was generated. Using the subscript helps to illustrate that an AB combination from a two-axiom system is different from the AB of a three-axiom system. The total number of items or objects within a system will always be 2n, where “n” is equal to the number of axioms in each system. The 2 represents the presence of “nothing” and the presence of “all”. Every system has a “nothing” and “all”. Ludwig Wittgenstein expressed the relationship between the known and unknown in this way, “What we cannot speak about we must pass over in silence.”29 A philosophical example of the one-axiom system is existentialism. In this philosophy, as in the one-axiom system, an object either exists or it doesn't.

A Two-Axiom System In a two-axiom system there are four items.

29 Wittgenstein, L. (1922, first English edition; 1961). p. 74.

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Two-Axiom System No. of items 1 2 1

Category Null Axioms Postulate

Name Not AB2 A2 and B2 AB2

Diagram 2. A two-axiom system

The first and only non-null postulate in a two-axiom system is the combination AB2. It does not require thought to create the postulate at this level. It is the only possible option. Examples of this logic are the Taoist concept of the Yin-Yang's passive and active interactions. Watsonian and Skinnerian behaviorism, based upon stimulus-response, the passive and active elements of behavior, provides an additional example. In each case the single postulate is the expression of the whole system. In Taoism it is the overarching unity of nature and its silent and harmonious ebb and flow, it is the reality within which the influences of yin and yang co-exist. A Three-Axiom System In a three-axiom system there are eight items. Three-Axiom System No. of Items 1 3 4

Category Null Axiom Postulate

Name Not ABC3 A3, B3 and C3 AB3, AC3, BC3, ABC3

Diagram 3. A three-axiom system

The three-axiom system has served as the model for most of Western culture's organization of ideas. For example, Christianity is embedded in the Father, Son and Holy Spirit, science speaks of solid, liquid and gas, music uses rhythm, tone and melody, and psychology speaks about the ego, id and super-ego. Governance, as a three-axiom system, is likely one of the most familiar in Western culture. It includes executive, legislative and judicial distinctions and is the model upon which the Roman Empire was built. We will use governance in the pages that follow to demomstrate APMA. As you read through the diagrams, it is important to note that the descriptions used for governance can apply to all levels of collective activity including work, school, community action and volunteerism. The goal of any three-axiom system is to describe and establish a state of stability30. Even in architecture, it is commonly accepted that a three-legged stool or triangular truss with three points of contact, is stable. It is also important to note that every axiomatic system is addressing a whole concept. Diagram 4 is an example of what a three-axiom system in science might look like.

30 See Lucatelli, F. (2014a) for a formal example of the application of APMA levels two and three used to correct the identification of Euclid's Common Notions and Postulates as valid axioms and postulates.

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A Three-Axiom System in Science No. of Items

Category

1

Null

3

Axioms

4

Name

Description

03 - Not ABC3 - Null3

Everything that does not belong in this construct.

A3 - Solid

That which holds its own macroscopic shape.

B3 - Liquid

That which is constrained by local gravity and conforms to the shape of its container.

C3 - Gas

That which is not constrained by local gravity and fills whatever space it is in.

Postulates AB3 - Slurry

A material that straddles solid and liquid states, for example, melting ice and flowing lava.

AC3 - Sublimate

A material that transitions directly between the solid and gaseous state, such as snow in Colorado and dry ice.

BC3 - Vapor

A material that straddles the liquid and gaseous state such as water's steam.

ABC3 - Physical Chemistry

The name of the discipline that includes all solid, liquid and gaseous states of materials.

Diagram 4. The Axioms and Postulates of a System in Science A Three-Axiom System in Governance No. of Items

Category

1

Null

3

Axioms

4

Postulates

Name

Description

03 - Not ABC3 - Null3

Everything that does not belong in this construct.

A3 - Executive

The power to put plans, actions or laws into effect.

B3 - Legislative

The power to make laws.

C3 - Judicial

The clarification of ambiguous laws and the resolution of conflict.

AB3 - Public Life

Making everyday decisions in the context of an accepted system.

AC3 - Enforcement

Correcting erroneous behavior that is not allowed according to clarified policy.

BC3 - Policy

Ratified and unchallenged agreements.

ABC3 - Governance

The name of the discipline that concerns itself with executive, legislative and judicial issues.

Diagram 5. The Axioms and Postulates of a System in Governance

The three axiom system of governance in Diagram 5 focuses on the executive, legislative and judicial axioms. Most people recognize and accept that these are the three branches of governance. Additionally, to identify the postulates for governance in this example we combined axioms using the formulas in the grid. The formula for the first postulate in this three-axiom system is AB3. In this example, the blend of The power to put plans into effect and The power to make laws is described as making everyday decisions in using an accepted system. We might then call that function Public Life. These are just terms and descriptions we've chosen as an example. The work that APMA requires from users looking to validate or examine a system is an in-depth scrutiny and conversation about 11

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the key concerns (axioms) and the subsets (postulates) of the whole system. More traditionally, conversations dive directly into problem identification and solving, or issue raising and idea generating. When looked at from a statistical approach, often only parts of the system are addressed. The use of APMA requires that more time will need to be spent, upfront, on clarifying key concerns of the entire system and what the concerns specifically encompass. Focus will need to be applied to terminology and specific meanings. In this case there must be discussion about whether the combination of axioms A3 and B3 is successfully making everyday decisions, and if the label, Public Life, is suitable for that description. More often than not, groups will find that there is an array of understandings about common terms. Even slight variations may need to be clearly established in service to isolating the axioms and postulates of the system. This upfront, initial work may take quite a bit of time, but it will be worth all of the work when it becomes evident that the whole system is being examined versus just some of the parts. More people will be able to understand the system and share their insights and support simply because the key concerns and subsets will be more focused and defined. This renewed focus and definition will open doors for more individuals to recognize their roles and their impact within the system, offering opportunities for additional involvement and advancement. Comparing Three and Four Axiom Systems If we use pie charts to represent the whole of any given system, with the pie wedges of equal size representing the axioms, we can compare a three-axiom system to a four-axiom system.

Diagram 6. A comparison of the scope of axioms in evolving systems

In Diagram 6 we see that axioms collectively represent whole concepts. The axioms are each independent and non-overlapping. In order to transition between a three and fouraxiom system, each of the four axioms must cover less territory within the circle than they do in a three-axiom system. This means that each axiom becomes more focused in meaning as the number of axioms increases. It also means that the “A3” in a three-axiom system cannot be identical to “A4” in a four-axiom system. Unlike a static three-axiom system, which expresses stability of any given topic, a fouraxiom system is a transformative process, which introduces dynamic opportunities. For comparison, a four sided shape made of Popsicle sticks and joined by a single pin at each corner will not hold its shape like a triangle of sticks joined by a single pin at each corner.

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Like a three axiom system, the three sticks form a fixed relationship that keeps the structure stable. With four corners, as in a four-axiom system, we can say that the connections, points or specific relationships are stable, but the entire structure has flexibility to move. Try it with your hands or Popsicle sticks and you will see how easily the overall structure moves in a four-axiom system and not in a three-axiom system. The flexibility illustrated in the four-corner system is an important capability for making process adjustments possible. It is not a weakness of the system's integrity, rather an indication of the strength of the four-axiom system. In our daily lives the three and four axiom systems are reflected in the difference between ideologies and processes. Ideologies are a three-axiom system because the intent is to maintain a status quo. The work being done is toward the application and acceptance of the ideology. Think back to previous examples of governance, science and religion. Three axiom systems are global in nature because we expect them to apply to all different situations and applications. Processes are four-axiom systems. They are contextually sensitive. Processes, like four-axiom systems, are inherently about movement and change from one position or state to the next. In order to initiate a process you need to have a goal or purpose to reach. Applications of four-axiom systems can be found in manufacturing, recursive algorithms in software, and in any activity where continual improvement is sought. Diagram 7 provides an example of governance using three and four axiom systems. The four-axiom system is a redefinition of the three axiom system. As mentioned earlier however, all new terms must be identified because A3 is not equal to A4, B3 is not the same as B4 and so on. You can't just add a new axiom when moving from 3 to 4 and call it good. There are three steps for this transition. 1. Determine two ways to make each axiom or postulate in the three axiom system happen. In the governance example, Diagram 7, the two ways that the executive axiom, A3, can be achieved is by Clarification of purpose to ensure that there are policies in place and Maintain the system to make sure policies are being used correctly. 2. Choose which of the two ways identified belongs in which place based on the formula. Looking at the formulas, A3 gets divided into A4 and BCD4 in the 4axiom system. That means a decision needs to be made whether Maintain the system or Clarification of purpose becomes a new axiom and which becomes a postulate. A rule of thumb is to select the placement based on how complex the tasks are in the system. Axioms will be the simplest and postulates will be more complex. In the example in Diagram 7, we selected Clarification of purpose as the axiom and Maintain the system as the postulate. 3. Put the axioms and postulates into the grid and define/clarify the terms. Descriptions of axioms and postulates helps to understand them. Discussing and identifying terms that are acceptable to everyone within the system allows for greater involvement and acceptance.

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Two Levels of Governance Three-Axiom System Category Null Axioms

Name

Four-Axiom System

Descriptions

Name

Descriptions

03 - Not ABC3 - Everything that does not belong in this construct. Null3

04 - Not ABCD4

Everything that does not belong in this construct.

A3 -

The power to put plans, actions or laws into effect.

A4 – Clarification of Purpose

Knowing what is to be achieved.

The power to resolve conflict.

B4 – Insuring Safety

Protect from danger, risk or injury.

The clarification of ambiguous policy.

C4 – Shape Attitudes

Mentor and provide advice.

D4 – Cooperative Partnerships

Provide sustainable, relevant and quality programs.

AB4 – Encourage Leadership

Help people to recognize their innate abilities.

Executive B3 -

Legislative C3 -

Judicial

Postulates AB3 -

Public Life AC3 -

Enforcement BC3 -

Policy ABC3 Ideological

Governance

Living in the context of an accepted system or policy.

Correcting erroneous AC4 – Gain Support behavior that is not allowed according to clarified policy.

Identify methods to create stakeholder buy-in.

Ratified and unchallenged agreements.

AD4 – Identify Guidelines

Clarify the parameters for development.

The collective concerns of executive, legislative and judicial issues.

BC4 – Encourage involvement

Use methods for including stakeholders.

BD4 – Evaluate Agreements

Determine whether policies fit needs.

CD4 - Eliminate barriers

Remove anything that may deter progress.

ABC4 – Gaining new knowledge

Expand general personal awareness.

ABD4 – Learning the system

Learn and practice using the system.

ACD4 - Innovating

Welcome new ideas and methods.

BCD4 - Maintain the System

Monitor and replenish the system.

ABCD4 – Functional Governance

Managing the operations of the entire system.

Diagram 7. Governance as level-three and level-four systems.

The relationships between these levels is shown in the Formulas for Transition column between each level in the diagram. Stepping from one level to the next is a process of breaking down each axiom and postulate of the current system into two new terms that will be used to fill in the content of the next higher level. The formulas tell us the relative complexity of the terms to be used in the next higher level. Single axioms, expressed by single letters, are the simplest terms, and terms with multiple letters are correspondingly more complex depending upon the number of terms in the letter symbol. The all-postulate is always the most complex term of any system and is the name of the system. As the structural level advances by the number of axioms, the focus of each axiom narrows and additional detail must be handled. New axiomatic systems do not replace old systems; they encompass them, adding new functionality with each new concept, and

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retrospectively, increasing the contextual subtlety of every former concept. Although any level may be operated in isolation, the levels are also tightly integrated like Babushka dolls maintaining a holistic view while adjusting the amount of detail desired.

A Five-Axiom System Jumping ahead to a five-axiom system, the postulates could quickly overwhelm one's ability to keep up with the advancing complexity. In a five-axiom system there are thirtytwo objects. Diagram 8 is a rough draft of a five-axiom decision-making system. It shows how much more detail is possible as the levels advance. To be complete this system would require additional examination, discussion and statistical testing to insure that the content is valid. While this may seem unnecessarily complex, artificial intelligence will ultimately require this level of detail, and more, if it ever hopes to mimic human abilities. While four-axiom systems enable us to create dynamic systems for the development of functional solutions to a wide variety of problems, five-axiom systems create a logic for deciding which options provided in four-axiom systems are timely and worth pursuing. Making the distinction between axioms and postulates allows one to focus on and understand the relatively few axioms at each level rather than being encumbered by the rapidly expanding set of both axioms and postulates combined. The postulates then, are a secondary expansion of the axioms. Postulate expansion is dependent on axioms. The purpose of this transformation is to expand the categories for a given subject so that a finer analysis of the topic may be done. Marshall McLuhan has pointed out in Understanding Media that, as the resolution of detail increases, the system complexity must change to match it as well. As we learn to operate more complex systems we can then develop comprehensive solutions to a wide variety of problems, knowing that the focus of our research aligns with our reality. Content validity remains the domain of statistical methods; construct validity is in the domain of theoretical axiomatics. This evolution in complexity of axioms from one to two, three, four and five is, in essence, an evolution of our civilization. The focus of a one-axiom system is the existence of the topic. A one-axiom system asks the question, “Is this a reality or something imagined to be real?” The system gauges if something is happening or not. It can be compared to the use of a light switch. This is a question that is ultimately decided by evidential investigation. The focus of a two-axiom system is to discover the active and passive interaction among two things. For example, introverted and extraverted is a two axiom system, when in a crowd, introverted tends to be passive and extraverted tends to be active. As the number of axioms increase, so does the complexity of purpose. A threeaxiom system is designed to keep things stable. It is designed to be difficult to change. That might explain why institutions like religion and governance adopted three-axiom ideals as their first organizational model. Because this is the way we do things here might be a good way to characterize three-axiom systems.

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A four-axiom system focuses on flexibility and adaptability while keeping operations running smoothly. A four-axiom system doesn't replace a three axioms system, it provides a new level of categorical complexity to address the accumulating questions that have arisen in the previous level. A four-axiom system examines functional processes. Gaining involvement from a broader population while maintaining and coordinating the process for products and services that are being developed would be the goal of groups operating in a four-axiom system. Organized involvement might be the moniker of a four axiomsystem. Evolving Civilization Category Null Axioms

Governed as a Three-Axiom System 03 - Not ABC3 A3 - Executive B3 - Legislative C3 - Judicial

Formula for Transition 03 ↔ 04 + ABCD4 A3 ↔ A4 + BCD4 B3 ↔ B4 + ACD4 C3 ↔ C4 + ABD4

Governed as a Four-Axiom System 04 - Not ABCD4 A4 – Clarify purpose B4 – Insure safety C4 – Shape attitudes D4 – Form cooperative partnerships

Postulates

AB3 - Public Life AC3 - Enforcement BC3 - Policy ABC3 - Ideological governance

AB3 ↔ D4 + ABC4 AC3 ↔ AB4 + CD4 BC3 ↔ AC4 + BD4 ABC3 ↔ AD4 + BC4

Formula for Transition 04 ↔ 05 + ABCDE5 A4 ↔ A5 + BCDE5 B4 ↔ B5 + ACDE5 C4 ↔ C5 + ABDE5 D4 ↔ D5 + ABCE5

Governed as a Five-Axiom System 05 - Not ABCDE5 A5 - Clarify facts B5 - Identify principles C5- Discuss relationship of facts and principles D5 – Agree on relationship and implications E5 – Take appropriate action

AB4 – Encourage leadership AC4 – Gain support AD4 – Identify guidelines BC4 – Encourage involvement

AB4 ↔ E5 + ABCD5 AC4 ↔ AB5 + CDE5 AD4 ↔ AC5 + BDE5 BC4 ↔ AD5 + BCE5

AB5 – Establish premise AC5 – Document reality AD5 – Plan approach AE5 – Review Scope

BD4 – Evaluate agreements

BD4 ↔ AE5 + BCD5

BC5 – Identify obstacles

CD4 ↔ BC5 + ADE5

BD5 – Engage in process BE5 – Adopt the system CD5- Offer enhancement CE5 – Monitor and adjust DE5 – Appoint authority ABC5 – Establish range of impact

CD4 - Eliminate barriers ABC4 – Gain new knowledge ABD4 – Learn the system ACD4 - Innovate BCD4 - Maintain the system ABCD4 – Functional governance

ABC4 ↔ BD5 + ACE5 ABD4 ↔ BE5 + ACD5 ACD4 ↔ CD5 + ABE5 BCD4 ↔ CE5 + ABD5 ABCD4 ↔ DE5 + ABC5

ABD5 – Confirm performance ABE5 – Test enhancement ACD5 – Demonstrate competency ACE5 – Challenge assumptions ADE5 – Activate solutions BCD5 –Verify responsibilities BCE5 – Implementing process BDE5 – State expectations CDE5 – Engage participant ABCD5 – Identify leadership ABCE5 – Collaborate ABDE5 – Hear responses ACDE5 – Design safeguards BCDE5 – Sense direction ABCDE5 – Purposeful governance

Diagram 8. The Governance of Civilization evolving from level-three to level-five

In a five-axiom system the focus is on becoming conscious of the big picture. This is evident in the number of axioms and postulates that must be created for this system to work. When looking at the big picture using a five-axiom system, evaluations can be made as to which choices will serve the system best. This is because more issues are being examined by more people in service to the same big picture. Community 16

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commitment might come to mind when thinking about the five-axiom system. Our current three-axiom system of governance provides a stable construct or design that can be validated. APMA and construct validity focus on the structure, scale and clear distinction between axioms and postulates. They do not validate the content that resides within the grid, rather, they validate the appropriate relationship to other content within the grid. The content itself originates from subject matter experts, workers in the field and others. The terms must be confirmed by content validity. The terms used to fill-in the lettered categories are always going to be a best guess with the information available. Best guesses, however, prove to be evolving assessments of the nature of each category, and ones that are better examined by rigorously statistical methods. Many of the terms in each of the axiomatic systems of governance are recognizable. It is easy to say ”We do that already.” What is unique to using the axiomatic approach is that the number of variables and solutions are fixed, scaled appropriately to each other, and we know that the categories are valid and sufficiently defined for further statistical investigation. It is easy to see how using APMA to address organizational and community topics could be valuable. Using APMA in our governance, education, work, social action and more could improve our civilization as a whole. We would all benefit from recognizing our individual impact and role in the governing system. We could all benefit from finding ways we can be involved or at the very least buy-into our governance. In understanding the levels of APMA people might then be compelled to zoom to the highest axiomatic level they can in structuring a group or examining an organization or topic. It is important, however, to take a step back and first identify which system best represents the current environment. Charting current practices is a precursor to movement. Knowing where you are, and where you came from, helps to define where you want to be. Before moving to a new level you need to know where you are. Then, it is important to move only one axiomatic level at a time. Adding more than one axiom into a system at a time before re-stabilizing the group, organization or idea will make it difficult for the population to adapt, adopt and take action to realign its efforts. This also introduces a non-judgmental view of past systems, which always seems over-bearing and suffocating at the moment of transition to a higher level system. Taking one step at a time will allow the evolution of a topic to happen in a valid, stable, methodical and conscientious way.

Using APMA When using APMA, you develop the ability to recognize whether a particular level of logic is properly fitted to the content requiring your attention. Because axioms are always assumptions, knowing the subject and having clarity about the number of assumptions included in the subject allows you to select the appropriate axiomatic system required. Conversely, if you know how detailed you want to be, but do not know the assumptions, you can pick the appropriate level of complexity and search for the specified number of assumptions. This greatly simplifies the search for the necessary assumptions because you always know how many to find. Additionally, because the ranking of all of the logical objects of any level is from simple to complex, the order in which axioms and

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postulates occur is predetermined. You can sort the terms of a system in ascending order, according to the number of axioms in each term (for example AB contains two axioms A and B) and then sort each subgroup with equal counts in alphabetical order. Each axiomatic system has a characteristic quality that becomes familiar and recognizable as one masters the APMA system. An example of this is the debate between behaviorism and cognitive psychology. The debate never seems to reach a point of resolution because the two disciplines reside at different structural levels. Behaviorism is a level-two discipline with only active and passive axioms to use: stimulus and response; while cognitive psychology must be discussed no lower than level-three. B. F. Skinner31 is correct when he claims that behaviorism has no use for thinking because he is describing an evolutionary precursor to thinking at a level with only two axioms. Items at different levels have evolutionary relationships with each other and different topics at the same level have analogous relationships with each other because they share the identical logical structure. For example, one could say that doing is like solid’s behavior in physical chemistry, feeling is like liquid’s behavior, and thinking is like gas’s behavior. Such analogies are extremely helpful in finding appropriate names for any new topic at any given level. These analogies make the development of new topics at the same level much easier to produce and also aid in transforming from one level to another. It cannot be over-emphasized that within axiomatic systems there must be a unity of meaning and quantity. The phrase “true to form” is an axiomatic necessity. Guided selection and the unity of meaning and quantity is precisely what allows axiomatics to organize and reduce data. Starting with axiomatics, we know which data to collect and analyze because we have created valid qualitative associations, or types among a fixed number of assumptions. We have created the construct. Within most academic books that address the issue of determining validity of assessments, the instructions fail in providing an operational definition for construct validity. APA's Standards for Educational and Psychological Testing32 specifically states that: “Rigorous distinctions between categories are not possible.” Although this is true for causal logic, it is the nature of relational logic, as used with the APMA method in this chapter that rules for establishing rigorous distinctions between categories is essential to the unitary nature of constructs. APMA is a method for establishing construct validity.

Conclusion: Re-balancing Statistics and Axiomatics We, as society, have been working backwards since the 1930’s, expecting collected evidence to tell us what to do. As a result, we are becoming paralyzed by the volumes of data we have collected. The costs for data collection are becoming unsustainable, human privacy is being compromised and the data, itself, is producing less than satisfying results. Axiomatics can provide a logical system, framework and proportional understanding of our data explosion. When applied judiciously, it offers the relief of 31 Skinner, B. (1953) and Watson, J. (1924, 1925). 32 (AERA), (APA), (NCME), (1999, 2014), p. 9.

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simplification for making valid decisions both globally and locally; universally and individually. When used to lead us, axiomatics can help to structure our approach and critically challenge us to examine the global complexity of our decisions. We can reduce the over-abundance of paralyzing data by rebalancing our approach to axiomatics and statistics. Rather than relying on data collection as the only starting point for our new directions, ideas triggered by intuition and other gut driven approaches can be more broadly accepted and used if we embrace methods for validating our thoughts. The use of axiomatic systems to organize and structure the relationships among data can also be more fully incorporated into society. Further refinement and use of APMA will allow for a validation of axiomatic approaches that were stifled in the past. Once balanced, we can find new outlets for our work, the advancement of our civilization, and ourselves. In governance we can benefit from using axiomatics in collaboration with data to breathe flexibility into the system. The process of moving into a more complex axiomatic system, four at minimum and five ideally, will elevate governance to a higher hierarchical level and purpose. Our new found level of differentiation among concepts, tasks and truths will allow us to keep pace with corporations that currently operate at higher axiomatic levels with greater access to data. Governance will then be poised to limit corporate sway over issues that are better served from a societal perspective. Environmental and social issues are being usurped by corporations, when they should rightfully be managed by governance. Not all human activity is trade related, which is an erroneous assumption of the corporate model. Using four or five-axiom systems of governance, we can reconfigure how decision-making occurs at all societal levels, and the effort can be more widely distributed to smaller localities so that individuals are able to make choices based upon their own unique personal preferences, insights and special local conditions. This would truly transition us back to democratic decision-making. Health care is another prime example of a discipline in dire need of a thorough axiomatic analysis. At a 1998 health congress in Houston, TX33, Frank Lucatelli offered to lead a session as part of the conference-wide discussion on the definition of health care. His focus was on defining some of the field’s basic assumptions. Dr. John L. Gedye, one of the organizers of the congress, recorded his observations of that session in the forward to a book written by the congress' co-organizer, Dr. John C. Lowe34: “One of the consequences of the new paradigm is that we are required to be explicit about things which previously we handled implicitly. This, it seems to me, is literally a change in human consciousness. Frank Lucatelli (one of our group) has pioneered the development of a formal general methodology for making explicit what is implicit in our language of common and technical discourse. It is commonly assumed, for example, 33 The “Congress for Defining a new Paradigm for the Healing Arts” was held at the University of Texas Health Science Center, in Houston, Texas on May 28-31, 1998. 34 Dr. John C. Lowe was a co-organizer of the Houston Congress with Dr. John L. Gedye.

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that there is just one kind of scientific method; but Lucatelli's formal general methodology allows at least four types of science – operational, axiomatic, experimental, and exploratory – to be distinguished. At the present time, clinical science seems to be largely an operational science, and this suggests that there may be much to be gained from broadening its scientific base to include the other three types... But when, at the Houston Congress, we attempted to start to apply Lucatelli's methodology to the language of the healing arts in general, we were almost overwhelmed by the enormity of the task, and many of us were tempted to try to set it aside. The enormity of the task, however, is a pointer to its importance and to the profound consequences of accomplishing it successfully.35” As the above-mentioned session proceeded, it became clear to the participants that there was little agreement about the definitions of many of the common terms shared by the health care professionals present. Determining the number of axioms of the system and what they are was inhibited by a common struggle to speak the same language. Current healthcare incentives, based upon corporate ideals, are driving results that are counterproductive on a societal scale. An axiomatic approach to health care would reduce confusion, articulating a clear set of fundamental terms and meanings from which a new understanding of health care could then emerge. At this conference with health professionals from many diverse disciplines in the health field, the enormity of the problem described by Dr. Gedye was not resolved. Many factors conspired against the conference participants. Frank did not have a written description of how the process worked; the establishment of health axioms is an enormous undertaking; and there was a lack of commonality in the aspect of healthcare to focus upon. Personality theory is a discipline that could also benefit greatly from the application of a rigorous axiomatic system. The basis of current theory is driven by factor analysis; a technique that gathers responses from surveys and then statistically analyzes the results to find clusters of data points. The assumption being made is that clusters of data reveal valid unique personality perspectives. The attempt to find valid structures through the exclusive empirical study of detailed content cannot produce consistently valid results, because it is an inappropriate tool for that purpose. As a result, almost every personality theory is flawed by not having construct validity, with the exception of some that were developed using other methods, such as the Enneagram, and aspects of Carl Jung's typology. Most personality assessment models are out of scale with each other or are over simplified because of the inability of statistical methods to successfully handle more than five variables. The “big five”36 personality types which dominate much of personality theory today, open, conscientious, extraverted, agreeable and neurotic, originated from this limitation of statistics. In many disciplines there is confusion concerning what to do about current issues that 35 Lowe, J. (2000). pp. 39-40. 36

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affect values central to a humane existence. The environment, healthcare, worker’s rights, human rights and governance issues consume our conversations, yet these issues might not be getting the focus they deserve for lack of both a commonly agreed upon set of axioms and a clear understanding of their natural consequences. As a result, business models and power fights are being used for the allocation of energy, transportation, nutrition and education. We skirt issues such as understanding and appreciation of personal differences, though seeking a more thorough comprehension of the issues and choices that benefit most people would yield greater results. The news is currently reporting that multinational corporations are preparing to sue governments over laws that have impacted corporations’ ability to maximize profit. This is a clear sign that the profit motive vs. the values motive, has exceeded its bounds. Karl Marx once commented, “The last capitalist we hang will be the one who sold us the rope.”37 It's time to curtail this excess; the earth is now hanging in the balance. The world is evolving in its ability to handle increased complexity. We are emerging from a world in which static ideologies have sufficed into a dynamic world in which stasis is most often viewed as a form of death. As Lewis Carroll has prophetically said in Through the Looking-Glass: "Well, in our country," said Alice, still panting a little, "you'd generally get to somewhere else—if you run very fast for a long time, as we've been doing." "A slow sort of country!" said the Queen. "Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!"38 We are experiencing a speed-up of action, in which those standing still will be left behind. Plugging data into axiomatic systems, especially based upon four-axiom systems, is a first step toward providing sufficient detail to create and allow for dynamic movement and flexibility. The functionality of a four-axiom system can be seen in the difference between command and control management systems versus the functional management of productive activity in multiple venues, advocated primarily by W. Edwards Deming and which appeared upon the world stage by Toyota's adoption of it. The movement back to more locally controlled decision-making to replace central control is another example of this trend. However, becoming functional is still not going far enough. For too long we have let our systems determine the outcomes that we get. It is time to advance our logical grasp to where we are implementing those functions that we have decided beforehand are in everyone's best interest. Five-axiom APMA systems will help us to keep issues distinct and the solutions holistic. We will be able to reverse the trend from centralized control, characterized by level-three systems, so that the fine details and local variances may be expressed and optimized, which all contribute to the difference between successful decision-making and failure. 37 38 Carroll, L. (2010).

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Things won't stay the same anymore, they will improve or disappear. In this process, however, it is not advisable to skip levels. Unless we understand a thing as an isolated object, we won't understand how it may change39; and if we don't understand the dynamics of change, our decisions will be nonsensical. Ultimately, if we can master both axiomatic and evidential tools as two complimentary and necessary approaches for progress, we may easily resolve and continually improve critical issues in civilization. Each of the two approaches to science will need to make assumptions; axiomatics will assume axioms and evidential approaches will assume constructs. They each assume what the other works to establish by its method. The strength of one approach is the Achilles heel for the other approach. Axiomatic logic may be shown to be lacking or erroneous by demonstrating that one of its axiomatic assumptions does not conform to evidential knowledge. Evidential conclusions may similarly be shown to be lacking or erroneous by demonstrating that there is a flaw in the axiomatic logic of the construct used to explain the data. While this process of mutual correction may be guarded against, apprehensively, it is the resulting elimination of flaws in each other's work that continually improves our scientific, and hence, cultural knowledge. There are some things in life, like nit-picking monkeys, that we are unable to do for ourselves, and so it is with these two opposite approaches to understanding. Rebalancing statistics and axiomatics could, as suggested by C. P. Snow40 in The Two Cultures, connect literary and scientific minds. Technical minds could check for precision, while the literary minds express information clearly and succinctly. Like the periodic table that created spaces to place yet undiscovered elements, a re-balancing could point to new concepts that are required to understand increasingly complex ideas. As the periodic table gradually evolved and filled in, sub-nuclear particles became more evident at a smaller scale and the nature of molecules became clearer at a larger scale. This is a common progression, because the maturation of any framework automatically provides analogies for parallel structures and forces us into the next framework as the meanings at a given level are comprehended. This is a model for an infinite process of learning and there is no danger of running out of new ideas to explore. Hermann Hesse (1943) in The Glass Bead Game,41 foresaw a society in which a popular social activity would engage humanity. Hesse envisioned a serious game that when played would add to humanity's knowledge. It would be an activity that could be entered at any time and in any place, and could reap satisfaction for each individual. As a net result of wide participation, civilization would advance. Use of axiomatic systems with evidential support can help us realize Hermann Hesse's dream of unifying society and advancing civilization. About the Authors 39 Deming, W. (1982, 1986) and Deming, W. (1994). As the industrial revolution has emerged and is now approaching maturation, the fourth axiomatic level is being perfected within global corporations, particularly in terms of the work of Dr. W. Edwards Deming. 40 Snow, C. (1959, 1998). 41 This book about the quest for knowledge was published just five years before David Hilbert's death.

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Frank J. Lucatelli I recently heard an anecdote about the mathematician, Carl Friedrich Gauss, who was interrupted in his work to be told that his wife was dying. His response struck a chord with me, as I'm sure that it does with all those who are fascinated with mathematics. It is reported that he said, “Tell her to wait a moment, I'm almost done.” For all those patient moments, I am eternally grateful to my wife, Virginia “Dede” Lucatelli. Little did I realize it at the time but my high school geometry teacher, Florence Johnson Lytle, first encouraged my questioning the nature of the relationship between axioms and postulates – and it still fascinates me to this day. I am also indebted to my longtime friends and associates Thomas L. Kopinski, Gina Thomas, and Rhonda C. Messinger for being willing listeners and collaborators, and who have been invaluable for me to think through many interesting ideas in general and particularly the concepts presented in this paper. Rhonda deserves further appreciation from me for agreeing to bring her exceptional skills in organizing information for optimal learning to bear as a co-author for this chapter. Frank J. Lucatelli is Principle Consultant of Personal Intelligence™, LLC, Director of HEAR, Inc. (Human Educational Alternatives Research, Inc.,) and an architect, in Kalamazoo, MI. He currently works as an executive coach and thought leader for organizations, helping to rethink the way employees function within the organization and assisting executives in becoming more effective by using their inherent creative potential. His interest in creativity, and the question of whether it could be taught, has driven his independent research work since he was an architecture student at the University of Detroit Mercy. His research has led him along multiple paths, all merging around a new theory of knowledge, and the resulting recognition that personality is a major determinant of the brand of one's creativity. This general research into creativity has inspired further scholarship in the fields of psychology, personality theory, mathematics, nuclear and particle physics, the nature of the connection between the science and religion, as well as the close connection between the points of the Deming Management System and personality-focused creativity. Frank is the originator of the Personality Insight™ assessment instrument. Frank may be contacted at Linkedin: Rhonda C. Messinger Whenever I start a project my mind gets transfixed on the topic and the rest of my life assembles within the walls of what is being created. What I do, see and hear becomes information to bounce off of the new concept or idea on which I am focusing. Thank you to my family, especially my husband Alan, for bearing with me while I crawled into my hole of creativity and unleashed my new learning at random times. I’ve done it so often, I’m not quite sure you knew I was crafting a chapter over the past few months until you so graciously offered to read it to make sure it all made sense. A special thank you to Frank who gave me the opportunity to learn, explore and write about Axiomatic Systems with him. The journey of learning has been amazing.

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Rhonda C. Messinger is a professional educator and a board certified business, leadership and career coach. She has spent much of her career analyzing performance problems and developing instructional material in fields ranging from education to strategic marketing, medicine and manufacturing. She is the owner of Career Momentum, LLC, a career consulting firm, and author of various career development and self-help articles. Rhonda blends training, coaching and mentoring to help individuals and teams achieve their professional goals. Her work incorporates all phases of career development including job search, work transition, career discovery and performance enhancement. Rhonda holds a BA in K-12 Education, an MA in Educational Leadership, the Global Career Development Facilitator (GCDF) certification, the GCDF Instructor accreditation and is a Board Certified Coach (BCC) with designations in career, business, leadership, and life coaching. Rhonda may be contacted at Linkedin: ~~~~ References Abdu'l-Baha. (1943, Sixth printing 1976). Baha'i world faith - Abdu'l-Baha section, p. 340. Wilmette, IL, Baha'i Publishing Trust. American Educational Research Association (AERA), American Psychological Association (APA), National Council on Measurement in Education (NCME), (1999, 2014). The standards for educational and psychological testing. Washington, D.C., AERA Publications. Anellis, Irving (2004). “The genesis of the truth-table device.” n.s. 24 (Summer 2004) pp. 55-70. Russell: The Journal of Bertrand Russell studies. The Bertrand Russell Research Centre, McMaster U. See: http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1119&context=russelljournal. Accessed 12 January 2014. Bahá'u'lláh, (1945, 1952, 1973, 1975, 1978, Third edition reprinted 1986) The seven valleys and the four valleys. Translated by Marzieh Gail in consultation with Ali Quli Khan. Wilmette, IL, Bahá'í Publishing Trust. ISBN 0-87743-113-2, ISBN 0-87743-114-0. Boole, G. (1854, date unmarked in first American printing) Laws of thought. New York, NY, Dover Publications, Inc. Carroll, L. (2010). Through the Looking-Glass. Seattle, WA, PacPS: Pacific Publishing Studio. Casti, J.; DePaul, W. (2003). The one true Platonic heaven: A scientific fiction on the limits of knowledge. Washington, DC. Joseph Henry Press. Casti, J. (2000). Gödel: A life of logic. Cambridge, MA, Basic Books of the Perseus Book Group. Cronbach, L., Meehl, P. Construct validity in psychological tests. First published in Psychological Bulletin, 52, 281-302. Toronto, Canada, York University website: Classics in the History of Psychology. Deming, W. (1982, 1986). Out of the crisis. Cambridge, MA, MIT Center for Advanced Engineering Study. Deming, W. (1994). The new economics for industry, government, education. 2nd. ed. Cambridge, MA, MIT Center for Advanced Educational Services. Euclid. (Circa 250 BC, 1956). The thirteen books of the elements. Volumes 1-3. Translated by Thomas L. Heath, New York, NY, Dover Publications, Inc. Frege, G. http://plato.stanford.edu/entries/frege-theorem/. Accessed 28 October 2013. Frege, G. (1893) Grundgesetze der arithmetik. English translation published (1964) as The Basic Laws of Arithmetic. Berkeley and Los Angeles, CA, University of California Press. Gingerich, O. (2004). The book nobody read. New York, NY, Penguin Books. Gödel, K. (1931, 1962). On formally undecidable propositions of Principia Mathematica and related systems. New York, NY, Basic Books, Inc. Griffin, R. (2014). Private correspondence with the author, Emeritus Professor of Latin. Kalamazoo, MI, Western Michigan University.

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