Mother Pelican
A Journal of Solidarity and Sustainability

Vol. 13, No. 9, September 2017
Luis T. Gutiérrez, Editor
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From Linear Thinking to Complementary Thinking
and Relational Thinking

Carmine Gorga

September 2017

Quite apart from spectacular advances in the sciences and technology, why has Western thought—and Eastern thought as well—been stuck in a rut for the last 2,500 years: to cite one example, why the dichotomy between Being and not-Being? Why do we really not care that much about the environment? Why have inherently social human beings been reduced to isolated atoms in the lonely city?

These and many other outwardly disparate events of crucial importance in our daily life stem from a very same weakness in our modes of thought and expression: Our thought process is stuck in singularities and dualities, namely in linear thinking and at best in complementary thinking. Formally, these are expressions of Rationalism. We have to shift our thought processes toward triads, namely toward relationality and Relationalism.

God or, if you will, Nature or Evolution, gave human beings the unique facility to develop languages in accordance with their needs and their diverse abilities. Hence, the many different languages. Hence, various languages that reflect the cultural context in which they evolved. Hence, we are still in need to discover their limitations and potential.

Any language starts with the Word—which, for Christians, stands for the second person of the Trinity, through whom all things, described by words, are created. We will consider three words: singularity, complementarity, relationality.

We will discover three constants:

1. Singularities lead to domination of one entity over everything;

2. Complementarities lead to potential conflicts between two entities;

3. Relationalities lead to harmony among three entities.

These are words that soon blossom into three complex mental constructs, which surreptitiously help or hinder the clarity of our thinking processes: linear thinking, complementary thinking, relational thinking. We will observe these phenomena in seven settings: language, economy, ecology, sociology, politics, philosophy, and religion.

Singularity, complementarity, and relationality in language

Singularity covers anything composed of only one word or entity; complementarity covers anything composed of two entities; relationality covers anything composed of at least three entities. The specific content of these mental constructs, to compound the difficulties hidden in their existence, varies from discipline to discipline in which they independently operate, and they operate in an exclusive way: Where there is one, there is not the other.

Even though they pass under the radar of our conscious observation, and just because they generally are unobserved, these constructs are so powerful that they actually force us into three widely divergent modes of thought: linear thinking, complementary thinking, relational thinking.

Linear thinking

As a reflection of the culture at large, our language is linear; our predominant modes of thought and expression are linear. Any analysis starts with single words; inevitably with single words: singularities them all. No matter in how many guises one observes them, and indeed the custom is to observe singularities from here to eternity or to infinity, and back, no matter the complexity of the investigation, one observes always the same thing.

Hence, to persist in the analysis of single words and to end the analysis with single words is a major curse of humanity. Reductionism is the result. Scientists, especially, ought never to forget that, no matter its complexity, what they observe under the microscope, or in a photograph, or in a brain scan is a past, isolated, dead thing. Through reductionism, clarity is gained, but at the great cost of excluding much from our understanding of the world as a whole. We know the parts; we lose the whole. We know everything about the leaf; the vastness of the forest escapes us. Yet, we are hard pressed to abandon reductionism, because it has yielded outstanding results especially in physics and chemistry. Indeed, we should never abandon reductionist methods; we ought only to remain cognizant of their limitations. Reductionism, contrary to appearances, is not a major limitation of science alone. In analytic philosophy, especially, but not exclusively, one starts and ends the analysis with single words.

This is far too long a discussion to be exhaustively held here. It must be trumped with any such expression as this: Will future generations ever forgive us for the libraries of books devoted to the understanding of such individual words as Being, not-Being, Becoming, Beauty, Goodness, or Truth? Vain works them all.

Individual words are mesmerizing entities. They tend to dominate everything in sight; they tend even to destroy everything in their progress. Can the analysis of Being ever tolerate the analysis of Becoming? Why this inevitable structural weakness of linear thinking? Why this ability of words to devour anything they/we create? To repeat, any analysis starts with single words, a singularity par excellence, but to persist in the analysis of single words and to end the analysis in single words is a major curse of humanity, because nothing exists or can ever exist in isolation. Notwithstanding the great discoveries of science and the sublime apperceptions of philosophy, we must realize that there are no singularities. Singularities are the creation of our imagination. Singularities are the manufacture of our language.

Built on this weak base, linear thinking sooner or later leads us to infinity, but it does not grant us much certainty in its vagaries. Hence, the unending discussions about everything today, especially today. Hence, the constant disagreement.

To avoid these inherent limitations of linear thinking, whether aware of it or not humans have at their disposal a different form of thought and expression: complementary epistemology; complementary thinking.

Complementary thinking

Jewish language, the language of the Bible, the language of Jesus is complementary: One statement is not followed by a linear statement that perhaps arbitrarily adds a new—possibly contradictory—thought, but a complementary statement that completes and clarifies the first thought. The Beatitudes are perhaps the supreme example of this mode of thought and expression. Another extraordinary such statement is: “Who sees me.... sees my Father.”

Most interestingly, Lao-tzu, in the fifth century before Christ in China reached the same conclusions. The founder of Taoism, he described Tao with the following words: “These two things, the spiritual (Yin) and the material (Yang), though we call them by different names, are one and the same in their origin. The sameness is a mystery of the mysteries. It is the gate of all that is subtle and wonderful.”[1] My most favorite quote by Lao-tzu has always been: ” He who knows, does not speak. He who speaks, does not know.

This mode of thought, however, can lead to circular reasoning.[2] Thus, stasis, and perhaps even confusion. This, undoubtedly, is the reason why, especially in the West, we do not use it much.

The Discovery of Equivalence Relations: Toward Relational Thinking and Relationalism

We resolve confusion and potential conflicts between two words or two sentences with the help of a third word or a third concept, or, in general, with the help of relational thinking. Thus, we automatically pass from the observation of two words—or two entities—at a time, to the observation of three words or entities at once. The analysis is not finished unless all three elements are considered.

While there are no rules that control the use of linear and complementary language, the rules of equivalence are many and they are firm; they are well established, mirabile dictu, both in mathematics and philosophy; that is why equivalence relations are such extraordinary guides to clear thinking. Mathematicians do not advance one step, unless they are firmly guided by the rules of the principle of equivalence (Gorga 2010). Each term in an equivalence relation must be identical to itself throughout the discussion, it must respect the rules of the principle of identity and those of the principle of non-contradiction; each term of an equivalence must also be symmetric and transitive. One can exchange a term with the other and must obtain non-contradictory results. With the opening of a term to another, the analysis is not static but inherently dynamic.

More technically, safety from confusion is found, not in the destruction of the principle of non-contradiction as Hegel did, but in the full embrace of the principle of equivalence. With this mode of thought and expression, we search for a third term and we establish equivalence relations among them. If we cannot find such relations, we had better pause and ponder. These are some of the relations that we build: Day ↔ Night/Day (sunrise) ↔ Night; Night ↔ Day/Night (sunset) ↔ Night; matter ↔ energy ↔ spirit; Beauty ↔ Goodness ↔ Truth. And so many other such relations. This is Relationalism in its full glory.

We will be more open to Relationalism the more we become ready to abandon Rationalism. With Relationalism we do not abandon Reason. Far from it. We do not even abandon Reductionism; we shall only remind ourselves that, once we have finished the analysis of the “Part,” it is then that the work of understanding the Whole begins. We will only abandon unbridled Reason, and we will let reason be guided by ancient principles of logic: the principle of identity, the principle of non-contradiction, and the principle of equivalence.

Knowledge, as usual, is key. As pointed out by many sources, we are at the end of an age, the Age of Rationalism, the end of linear modes of thought and expression. We are at the end of this age especially because we have been tempted to apply rational, linear thinking to the social sciences. Economics, most notably, with its enormous influence on practical affairs has been seduced by the study of Homo Economicus, a non-existent entity.

Singularity, complementarity, and relationality in the economy

The field of study of economics concerns the operations of a singularity: Homo Economicus. Translated into English, Homo Economicus is The Producer, and in mainstream economics everything is subordinated to The Producer—conceived as a singularity, a dominant entity that is Everything. Thus, for instance, the consumer is reduced to a purely passive role.

We can see how these consequences flow from the limitations of linear thinking: No matter the complexity of the analysis, one observes always and everywhere the operations of such singularities as Homo Economicus or The Producer or The Market.

Even though it inveigles the discourse in circularity of reasoning, and gives natural rise to a tug of war between the two entities, complementary thinking leads to the inclusion of The Consumer in economic analysis. This expansion is essential on many accounts, but especially for The Producer to continue in his or her operations and prosper. When economic theory fixes its gaze on the complementary world of The Consumer, vast new/old horizons open up to view. The world of exploitation of the Consumer by the Producer is left behind, and a new world awaits to be discovered. First, this is the world of Economic Interdependence; interdependence is the norm in economics: Selfish competition is a voracious machinery that destroys the competitor and destroys the fabric of society. The world of Morality and Economic Justice, comes soon after into view. Economists will find that the need for respect of natural resources stands in between.

All this, and more, is operational in Concordian economics, a reconstruction of economics based on sound epistemological and anthropological foundations. This is a paradigm in which the world of producers and consumers is rounded up by inclusion in it of the world of Distribution of ownership rights—ownership rights over money and real wealth. Thus, we enter the world of relational thinking.

The inclusion of the economic value of ownership rights is not dictated by arbitrium or whim, but by the logical necessity of completing the complementarity of Production to Consumption into an equivalence, the equivalence of Production to Distribution and to Consumption. These relationships are best caught in the following geometric representation of the economic process:

Figure 1 – The Economic Process

Even in the purchase of a chocolate bar we find these three elements: real wealth, money, and ownership rights as evidenced by the purchase slip.

Among other implications, the economic discourse is no longer tied to a linear mode of thought and expression; the linear world of economics thus goes beyond complementarity and is immersed into the spherical world of equivalence relations—in other words, as applied to economics, this is the gradual transformation of Rationalism into Relationalism, the world in which everything is systematically related to everything else. As Pope Francis stresses: “Everything is connected.”[3]

This is the world we are gradually, no matter how briefly, going to explore in a few other fields.

Singularity, complementarity, and relationality in ecology

The relationships between singularity, complementarity, and relationality can be made more directly manifest, and—due to the ignorance of the writer—more succinctly expressed, in the context of ecology. When ecology is reduced to such a single term as Natural Resources, the economic drive to exploitation is irresistible.

We begin to have some respect for the existence of Natural Resources once we put them in relation to human beings, as in the contemplation of Natural Resources and Human Beings. In this context, however, a continuous tag of war persists. It takes this form: Who should prevail, Human Beings or Natural Resources? Trees or jobs? Clearly, such a conception is better than that of trees alone, jobs alone.

The aspirations of true ecology will be realized only in the context of this set of equivalence relations: Natural Resources ↔ Mother Earth ↔ Human Beings. The chances of success for humanity will increase in the same proportion as we see Mother Earth as a living, spiritual entity.[4]

Singularity, complementarity, and relationality in sociology

Singularity is Man; singularity is Woman. These are non-existing entities that lead to the mentality of dominion as expressed in the practice of patriarchy and matriarchy. There is nothing to say about the subjugation of women by men—or men by women—except to express pity for the degradation of humankind. How low can we get! We are destined for greater things, we have room for greater worlds than pitiful exaltations of superiority.

Superior/inferior relationships between men and women are sustained by ideological unreality. Women are not cardboard figures; they have blood and minds and feelings. The first observation must be that blood and minds and feelings are exactly the same as those of men. The second is that, above all, we all have history; or, as my wife says, we all come with “baggage.” Hence, we all have to tolerate the shortcomings of the other. Two useful statements to deflate the heat of clash of wills. My mother-in-law, Marie Mohr: “That’s why there is chocolate and vanilla.” My fairly recent discovery: “Disagreements are the spice of life.”

At closer look, if a distinction needs to be made between men and women one should not be misled by those superficial, tendentious, and ultimately wrong observations that women are somehow “weak” and “voluble” and “irrational” human beings. Both theology and science make these observations unsustainable. Can a person who goes through one or more childbirth experiences be considered weak? Voluble? I have changed my mind umpteen times while writing this piece, let alone about other issues, in the last few months, the last few months alone. Irrational? How many men do irrational things? Many times over. Is war rational?

Clearly, Men and Women were created for each other. The two are interlocked into a relation of complementarity. One cannot do anything living and vital without the other. In fact, great strides forward have been made during the last 100 years or so, especially since Elizabeth Cady Stanton’s Declaration of Rights and Sentiments (1848); great strides have been made with the discovery that men’s and women’s functions are complementary.

Man cannot regenerate humanity without Woman; nor can Woman regenerate humanity without Man. This is an activity that must be intended as a physical as well as a spiritual operation. To grow self-assuredly, kids need the direction of a father as well as the direction of a mother. Certainly, women must be free to do everything that men do. But there are things that they cannot do. They cannot be fathers. Equally, there are things that men cannot do. They cannot be mothers. Attempts to pervert such relations in The Literature or The Laboratory might liberate both men and women from the respective obligations imposed upon them by Nature (God? Evolution?). More likely than not, such efforts create Monsters. Goya, who lived through monstrous times, got it right: "Dreams of reason create monsters."

But even the characterization of complementarity between the sexes has to be declared fundamentally flawed. True, complementarity is the unity of two, not the control of one over the other. But this is not enough. This relation, which is our prevalent cultural expression, leads to a continuous tug of war between the sexes. If by the fruit you shall judge the tree, it is evident that the conception of complementarity of men and women leads mostly to strife—hence, it is not healthy, it is not sustainable.

Safety will be reached with the help of the declaration and the practice of a fundamental human equivalence between the sexes, a conception that is automatically included in the definition of a person as the unity of body, mind, and soul. Both men and women are persons.

Perhaps, the next most important stage in human development can be characterized in this fashion: Men and Women are not just complementary but equivalent to each other. They have the same value. Relational thinking propels us to ask, what is the third term that links men and women indissolubly together? What is the third term that resolves all potential conflicts and misunderstandings between the two persons, a Man and a Woman? Spirit definitely is one such term that fundamentally applies. Thus, we create the Spirit of Humanity, the Haman Race. Formally: Man ↔ Humanity ↔ Woman.

Humanity is too general, too abstract a term to link men and women together. The traditional third term into which both Man and Woman disappear is Family. Thus, the traditional equivalence between men and women is this: Man ↔ Family ↔ Woman. And family must be intended, not as it has been brutally described in some quarters as a “baby factory,” but as described in Pope John Paul II’s Theology of the Body, family as Communion of persons.[5] Thus, the relation between the sexes can be described as Man ↔ Communion ↔ Woman.

This natural relationship between women and men will not be established—or re-established—without a cultural evolution away from secular/religious patriarchy driven by the tripod money → power → honors. Patriarchy is a cultural disorder, just as matriarchy is a cultural disorder. Patriarchy is an abomination, if it is not balanced by matriarchy. (My father used to say: "I am the owner; but who gives the orders is my wife.") In my humble opinion, the horrors of patriarchy have to be faced head-on. And they have to be resolved in each one of the spheres in which such horrors manifest themselves.

Most important of all, this re-evaluation of the family will not occur without a deep re-evaluation of the role of love in human relations, so that we will be truly satisfied when we reach the stage of writing down this equivalence: Man ↔ Love ↔ Woman. A great stride toward this goal has been made by Bruce Feiler (2017) in his thorough historical, cultural, and theological meditation on the relationship between Adam and Eve as a love story, a true love story; the first, yet ongoing love story.

In any case, in my ongoing work on Relationalism I have discovered many other equivalence relations. I will not repeat them here. One of the most important and conclusive, unobjectionable, immediate realization that Relationalism yields is the affirmation that women are not irrational. Women are relational. Indeed, I do not know any person more analytical—more rational—than my wife, Joan. She surely is so often more rational than I am. She is a mathematician, especially when she shops. She has decidedly made a contribution to economics that deserves a Nobel prize.

And men, of course are nothing, no they are bullies, they are brutes, if they are not relational.

Singularity, complementarity, and equivalence in politics

As Alasdair McIntyre has pointed out, the last five hundred years or so have seen the creation of a new entity, the Individual. Prior to that, one conceived of human beings as fully integrated in community. This is an old story. Exploitative elites have always found that it is so much easier to control isolated human beings to do their bidding than human beings organized in communities.

Individualism is the political ideology supporting this disintegration of society; similarly works the ideology of self-reliance in sociological affairs. The most destructive development of them all has been the isolation of human beings: the lonely crowd, the lonely city is the result.

These are phenomena that in the end have created the battle of the sexes: men against women; patriarchy versus matriarchy. This is the foundation of social unrest. To abate terrorism, we have to develop a system of “companions” ready to offer continuous fellowship to many such lonely persons. Not so long ago, before the megalopolis, the public square used to be the place to encounter natural fellowship.

Who does not believe and support individualism? And yet, this is an intellectual construction that springs from an abstraction: The Individual. The Individual does not exist. To balance this extreme conception of society, intellectuals during the 19th Century built Socialism, a political construction based on a greater abstraction: the primacy of Society over the Individual. The battle is still raging. Peace, it is fervently to be hoped, will eventually be found in a new construction that this writer likes to call Somism, the integration of Men and Women in society, namely, the creation of the social man, the social person. This then is the proposed equivalence:

Individualism ↔ Somism ↔ Collectivism.

Singularity, complementarity, and equivalence in philosophy

The conception of Being is the great singularity on which Western philosophy is built. In our syncretic terms, we can say that Western philosophy is all built on the conception of One Being. During the last five hundred years or so, in the Age of Exploration, the West, with its insatiable curiosity, has discovered many other civilizations—and tried to conquer them all. When unable to conquer them intellectually, most often the West has destroyed the civilizations it has encountered.

For me the most notorious expression of the inherent destructiveness of linear, rational thinking is this: We had to destroy the village in order to save it.

During the last century, a peculiar phenomenon has taken place. Western philosophy has discovered that at just about the same time it was born, about 2,500 years ago [Socrates (469-399 BC), Plato (428-347 BC), Aristotle (384-322 BC)], a coincidence to be marveled at, a competing philosophy was also born: Eastern philosophy [Lao-tzu (604-531 BC), Buddha (563-483 BC), Confucius (551-479 BC)].

The conception of not-Being is the great singularity on which Eastern philosophy is built. In our syncretic terms, we can say that Eastern philosophy is all built on the conception of not-One, that is Zero, the empty set, the empty space, not-Being.

The world is gradually discovering that we need them both; we need both Eastern and Western philosophy, because the one completes the other; the one enriches the other.

Yet, the ongoing discussion is in the terms of complementarity. As we know, we need to go beyond complementarity. We need to go toward the world of equivalence. The hope of the future, and the invitation to the literati is to discover that Relationalism has the potential to resolve all the inner conflicts that exist between Eastern and Western philosophy. This is the proposed equivalence:

Western Philosophy ↔ Relationalism ↔ Eastern Philosophy.

Singularity, complementarity, and equivalence in religion

As we have seen, any analysis starts with single words, a singularity par excellence, but to persist in the analysis of single words and to end the analysis in single words is a major curse of humanity because, with the exception—for believers—of the birth, life, death, and resurrection of Jesus, there are no singularities in nature. Nothing exists or can exist by itself.

Indeed, while a singularity in relation to human beings, in truth Jesus is an integral part of the Trinity. Thus, not even Jesus is a singularity. Indeed, being a Triune being, not even God is a singularity. An interesting exchange. Editor: “The Trinity, as one God, is a singularity, isn’t it?“ Writer: “The Trinity is one God, as the economic process is one process or why there is only one person even though we are composed of body, mind, and soul. For us humans to understand any complex entity, we need to break it down into its constituent parts, otherwise the entity disappears into total abstraction. One other consideration about the one God: Given that God is Spirit and us humans are composed of matter, energy, and spirit, whereby we can be united with God in this life and the thereafter, if God is one, he will crash our individuality in the afterlife. My understanding is that he will tolerate some humans as ‘gnats’ around him after our death. And I guess the most evident proof of the munificence of God is the conception that He does not call us gnats at all, but defines us as entities made in his ‘image and similitude.’ And just now perhaps I might have gotten it: Singularities are the reality of mystical unions: union with God, first; union with other human beings thereafter; or union with Mother Nature, for that matter.”

Indeed, for believers, the Queen—to recall Saint Thomas Aquinas—of all equivalence relations is this:

The Father ↔ The Holy Spirit ↔ The Son.


Ever since 1965, when I changed one equation in Keynes’ model of the economic system and found myself in a completely different new/old intellectual world, I have been riding a tiger by the tail. It has been a wonderful ride. Still riding it, I can utter only one word:


To solve the “issue” of the sexes, and many other social issues as well, we need a civilizational change; we need a shift in our culture. Certainly, we need to start with a deep meditation on love. But, we need much more than that. After we deeply understand love, we should practice “the art” of love—not individually, as “individual” isolated human beings, but as society as a whole. And that is not going to be easy. Why? The “issue” is systemic; it is complex. From a money based civilization, we have to pass to a virtue based civilization. We have reduced love to sex; we have to remember that love is the highest of theological virtues.


1. Quoted in Little, J. p. 3.

2. Science is inherently unable to study singularities: science needs at least two points or two events in order to function, in order to extrapolate at least a third term, which alone determines the sense of direction the observed reality goes. Otherwise, the reality to cover is a 360º field, which gives us no sense of direction at all. That is indeed characterized as "circular" reasoning.

3. Laudato Si’, #91, #117.

4. Conceivably, the equivalence of Matter ↔ Energy ↔ Spirit shall take some of the cobwebs out of our minds to reveal the “real” reality of Mother Nature and make the sacredness of the earth more readily acceptable (Gorga 2007).

5. Theology of the Body, especially chapters 9, 10, 29, 30, 31, 32, 41, 43, 45, 68, 69, 104, 107, 123, 126, 127.


Feiler, B. 2017. The First Love Story: Adam, Eve, and Us. New York, NY: Penguin Press.

Francis. 2015. Laudato Si’: On Care for Our Common Home. (LS)

Gorga, C, 2007. “On the Equivalence of Matter to Energy and to Spirit,” Transactions on Advanced Research, 3 (2): 40-45. Available at Relationism.

Gorga, C. 2010. “On the Transformation of Mathematics from a Linear to a Relational Discipline - Toward the Reunification of the Physical and the Social Sciences,” International Journal of Mathematics, Game Theory and Algebra, 19 (4): 235-244. Available at Relationism.

John Paul II. 1979-1984. Theology of the Body. Available at EWTN.

Little, J. (Ed). 1999. Bruce Lee, Artist of Life Boston, MA: Tuttle Publishing.


Carmine Gorga is president of The Somist Institute. The mission of the institute is to foster sensible moral leadership. He is a former Fulbright scholar and the recipient of a Council of Europe Scholarship for his dissertation on “The Political Thought of Louis D. Brandeis.” By inserting Hoarding into Keynes’ model of the economic system and using age-old principles of logic and epistemology, in a book and a series of papers Dr. Gorga has transformed the linear world of economic theory into a relational discipline in which everything is related to everything else—internally as well as externally. He was assisted in this endeavor by many people, notably for 27 years by Professor Franco Modigliani, a Nobel laureate in economics at MIT. The resulting work, The Economic Process: An Instantaneous Non-Newtonian Picture, was published in 2002 and has been reissued in a third edition in 2016. For reviews, click here. During the last few years, Dr. Gorga has concentrated his attention on the requirements for the unification of economic theory, policy, and practice calling this unity Concordian economics. He is also integrating this work into political science, which he calls Somism, and culture in general, which he calls Relationalism.

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