The Motherly Impulse

by David Fleischacker

Generally, one thinks of nurturing when one thinks of motherhood. Usually I find a guttural revolt against this idea today. Yet intrinsically “to nurture” is a good thing. I would like to give some thought as to what this means in relationship to motherhood. To nurture involves feeding and caring for basic needs but also the spiritual needs of another. In other words, it is a type of mutual self-mediation (To read about mediation, see Lonergan, “The Mediation of Christ in Prayer”). I would argue that starting in the organic matrices of the female body, there is a special attunement for nurturing. Even the neurons in her skin are far more numerous than in a man. Her senses are designed to be attracted spontaneously to details about the body and disposition of another. Girls can read faces more accurately at a much younger age than boys (boys do not catch up until age 40 on average — at least in some studies I have read). These needs stir sympathy in a woman much more rapidly than in a man. [Though for another blog, men more often than not are stirred to sympathy through their mothers and wives and sisters.]

Socially, this care impulse leads a woman to want to create a physical and personal space about her that is inviting. She takes care of it to reveal the dignity in another for which she naturally cherishes. She has a finality that is fulfilled when she can illuminate their worth and fulfill the conditions for their existence and their flourishing. This can get distorted of course, like all good things, but intrinsically it is supremely good, and its privatization is a supreme evil in a culture.

Psychologically, the motherly care stirs a warmth and love in the recipient of that care. It communicates to them their inherent dignity and worth. It lets them know that their lives matter, because the care is coming from a beautiful creature whom God has made and sent to care for them. A motherly smile and embrace can undo anger and bitterness. It can open up the heart to higher things. Through simple meals and gifts of nurture–another human being comes to thrive.

The mother is created to care about all the details of those who come into her realm. Naturally and most intimately it is “her man” and from this relationship, his children conceived in her body. I say “his” because I think it highlights the nature of begetting and bearing at the moral level and the depth of the bond between a man and a woman. It is hers too of course, and the man will stand in awe of “her” children. It is hers as received, even though the egg was formed in her. It is his seed, and his act to love her. She of course may have called him forth. And in the bonds of marriage, this bond has a sacred character, found in the promise in God’s witness of “until death do us part.”

The descriptive conjugates that develop regarding the motherly impulse must be multiple. There would be conjugates naming various desires of course, as well as her body and its contours, and how it changes during adolescence, and her apprehension of caring for a child, and for loving a man, as well as falling in love. There are conjugates of family too.

When I think of how we have degraded life in the home, it is incredible. We have made everything in it seem trivial, and thus a burden. How sad. We have really targeted the mother in the home–turning everything she does into a land of waste and a life not worth living. But in fact the opposite is true. Every act of love in the home provides conditions for the existence and thriving of her husband and her children, her neighbors and extended family, and even her own well-being. This includes the simple things that analogically extend from her own bodily schemes of care and protection of the unborn, whether that is the order of the kitchen and a place for everything in it, or the cleansing of the body, the routines of washing clothes, and the cooking that transforms and mold hearts.

Providing such conditions is a participation in the divine light that shines in the world upon each individual who comes into that house. They are called by name in that light. It shows each person that even their bodies and psyches are sacred and pondered within the mother’s heart.

It is built into the body of a woman to be radically attentive to others in this world, to see easily into their hearts and souls, to see even the littlest of physical, psychic, and spiritual needs.

As a note I do not want to say men can do none of these things, but at this point I just want to focus on the glory of a woman.

The woman’s body is structured to receive so as to give. Her body is the space in this universe in which life comes to be and then has its first days and months nurtured and protected. Her body feeds the new life so that it will grow and develop.

It is important to realize how the human heart works here. The essence of the human person is a capacity to self-transcend. A person lives through all of the conjugate forms, schemes of recurrence, integrators of lower levels and operators for higher, all within the unity that is the subject, the concrete person.  Below summarizes the higher and lower levels of proportionate being (see Insight for more). All of these come together in the human person.

  • Faith and love
  • Decision
  • Judgment
  • Understanding
  • Experience
  • cellular conjugates
  • chemical conjugates
  • subatomic conjugates

The woman’s body has a form made to receive the seeds of life, to call these forth from her man, to be awakened in her own intentionality and life by this calling and reception. Even chemically and organically this is true–from the hormonal responses to the illumination of regions and cortices of the brain and how neural demand functions transform conscious attentiveness. I wish these could be discussed in their beauty rather than in the light of the fall and the horizon of concupiscence, and modern gender theories.

When the seed has transformed the egg, and life comes into existence–yes existence!–Then the mother’s body becomes the child’s home. Warmth and sustenance are provided from the very schemes of the woman’s body, designed for this right down into the quarks and quark compounds emergent upon which are the atomic and atomic compounds, the cells and cell compounds of her body.

During gestation, her psyche undergoes a transformation rising from an increased rate of synaptic formation in various parts of her brain, prompting her, calling her from the depths of her being to rise into the glory of motherhood. Literally her brain increases its synaptic connections so that she will develop more rapidly in her concrete insights that will constitute some new schemes of motherhood, and so she will form a vibrant horizon of affectivity that will call forth acts of love and care. In other words, when all is allowed to come forth, this organic realization of her motherhood emerges into a neural set of schemes which bursts forth into a generous affectivity that re-orients motor-sensory operators toward a new attentiveness to the growing life inside of her.

Her growing affectivity also spontaneously turns to her beloved and calls forth from him and awakens in him this growing intention in her toward the new life. She awakens him with the goodness and glory of life. [And of course the reverse can happen as did happen with Eve and Adam.]

This new set of operators turns to the physical world to build a home that will welcome the new life. It will provide protection and nurturing of all the needs of the newborn. It will communicate to the world the love of this new one. It will bring about shifts of habits–all of which have been organically and psychically prepared in the body of the woman. As a note, I know there are similar neurological and psychic shifts in the man which I suspect is tied to his commitment and bond to her–but I have not seen studies on this.

Of course, culture can help these neurological and psychic reorientations to thrive or it can crush them.  And there are potential for deformations at every single point along the way.

The mother also then finds something glorious about which to converse with others, namely the new life growing within her. Those who care about life will wonder and be in awe about her. They will come to her, embrace her, confirm her, raise her up into the light of being. She has become one in whom life comes into existence. Her mother and father will come alive to this as well as should his parents.  But lurking around the corner is a massive if.  All of this mutual self-mediation of the glory of a new one and the glory of motherhood only rises to its rightful heights if the scale of values constitutive of their moral horizons are properly formed.

One can see this welcoming of life throughout Scripture. Think of the sadness women and men in the old testament experience at infertility. Sometimes the shame goes too far–and others would mock the barren woman, and she would even think of herself in this highly negative light. But sadness would be a proper response, as would the turning to God with wonder about her fate and hope for the gift of a child. Children were her glory and rightly so. And how blessed a man is with a good and faithful wife and the gift of children conceived in her and born of her (think of the Psalms and Proverbs).

When we turn to the New Testament, this glorious element of motherhood reaches heights unheard of before. Mary of course is that central icon, The Mother, the woman of the fiat, the one in whom God was conceived in her womb, the one from whom the source of all life is born. This dignity bestowed on a woman is a great and beautiful mystery. And look at what it calls forth from Joseph! Even before he has his revelation from Gabriel, he is moved by her and will not divorce her publicly. Already, her intrinsic virtue is calling him forth. Once he is illumined to what has been given to her, he immediately becomes her protector, her husband, whose heart is now filled with her and the fruit of her womb.

And think of how her own son brings life to her. His death will be a sword through her heart because his heart became sin for us, our sins, the sins of all humanity, and these sins form the blade. But he makes all things new. He is the eternal fruit, life, and salvation of all. And she is his mother.

God created Mary, the woman, who would be the sacred temple in which he would enter this world. He became us so that we might become like him. We too are the fruit of a womb, conceived into existence in the womb, born of a woman.

Even more so are consecrated virgins in the likeness of Mary as the mother of God. They bear the Son into the world. The firstborn of their consecrated wombs is the eternal Word of God. Eternity flows through them with motherly tenderness to each person who enters their midst, and their hearts sing of their man from whom they have received life into their souls and bodies. They are his betrothed in a personal and intimate way. They have entered the inner love and exchange of our triune God. The fruit of the womb is theirs, and their hearts will be pierced with the sins of the world, sin which rejects their beloved and the fruit of eternity which flows through them.

The motherly impulse in each and every woman springs from the finality of each and every quark of their being. It rises with greater and greater degrees of freedom until it reaches the apex of a state of being in love with the unrestricted one. In the biochemistry of their genetics this bursts forth into cellular differentiations and entire cellular systems that constitute their hormonal cycles, their biophysics, the bio-schemes of their womb, and the neural structures of their body. It bursts forth into sensitivity with attentiveness, trust, and affective love. It calls forth caresses and nurturing, which then rises into the intrinsic freedom of the spirit, which transcends the empirical residue in the world of common sense, artistry, drama, intelligence, culture, personal life, and faith. If fully integrated as in Mary, the finality becomes an operator of God’s maternal love in this world, and an integrator of the Trinification of the world, of each individual, of family life, of parishes, and of a civilization of love that is in a pilgrim state as a sacrifice within the cities of Cain and civilizations of death.

Oh how glorious is the motherly impulse of a woman.

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Note

I would like to say a little bit about how I am making most of these arguments in this blog.

  1. From the generic intelligibility of the levels of consciousness.
  2. From the generic intelligibility of metaphysics or generalized emergent probability.
  3. From specifying these generic intelligibilities to a) Higher and lower viewpoints, b) Higher and lower levels of schemes and things.

Though I have read studies that move from higher and lower levels into species with regard to female cycles, I am not discussing those here except in a generic fashion. I do have some other blogs that have looked at some of these–see for example 40 years since humanae vitae, female procreative schemes.

Perfection in Suffering on the Cross: A Transcendental-Metaphysical Analysis

[If you had seen this piece during the first hour it was posted, I apologize for the confusion. I dictate these blogs into a program, and then edit them.  What you saw was the unedited version.]

by David Fleischacker 

“….and he was made perfect through suffering”

Hebrews 2:10

Good Friday seems to be an obvious misnomer, at least when I was a kid I always thought so. Once one realizes that the crucifixion of our Lord is the greatest act of love, then one begins to glimpse why it the greatest and most perfect good. But why through suffering? Lonergan’s insight into sin as an act of inauthenticity can shed light on the magnitude of this goodness, and on why it must take place through suffering.  To see this, we must first understand the nature of inauthenticity, which means we must understand the nature of the transcendental notions and why these are so important.

Setting the Stage of Suffering: The Transcendental Notions

The transcendental notions are at the root of why the transcendental precepts ring true to all of us. Just to recall those precepts–they form a hierarchy-[and I do mean a sacred order by this term, not its degradation in today’s culture to a negative bureaucratic term.]

The Transcendental Precepts

  • be loving (highest)
  • be responsible
  • be reasonable
  • be intelligent
  • be attentive (lowest)

 For those who have studied Lonergan, you know that these precepts articulate the way of perfection of the levels of consciousness. Each level is distinct because it is governed by a different fundamental operator and integrator of development (see chapter 14 of Insight for more info on operators and integrators). One gets a first glimpse into these operators when attending to the wonder at the root of our questions–hence as Lonergan distinguished in Insight-wonder sorts into questions for -understanding and questions for reflection–the first leads to insights, the second to judgments. Hence asking questions for understanding and getting insights is equal to being intelligent. And asking questions for reflection and making true judgments is equal to being reasonable. As those of us who have been around Lonergan for a while know, he later added questions for deliberation and decisions which when perfected are equal to being responsible. He also suggested a fifth level which is really tied to the entire capacity for self transcendence and its actuation in a state of being in love. This fifth level is a bit different than the others, because it not merely emerges from them, but both is a comprehensive perfection of one’s horizon and it is the starting point and orientation of every level of one’s horizon. It completely embraces every operator or integrator, and every operation and integration in the human subject from the lowest levels to the highest. It is a kind of alpha and omega of human existence. This is why it is a little bit more than a level. In any case it is what underpins the precept “be loving.”

 To expand one’s apprehension of these acts of wonder so as to see that these are not merely questions but operators and integrators, one can shift one’s attention to the interior fact that the same wonder is also the same that gazes upon an answer, and it is the same that moves beyond any finite answer. In Plato and Augustine, this is why wonder is more aptly called an interior spiritual light–the light of being. With Aristotle and St. Thomas, it is called the agent intellect. The symbol of light is a wonderful image because a light source is that by which we are able to look for something, and then, it is the same reality by which we gaze upon something found, and finally, it is the same reality by which we can look again for something new.

Realizing that questions really are operators and integrators is not easy.  We all start in the world of the senses and I, like many others, became trapped in that world. Plato describes this in his book the Republic.  All human beings are born in the cave.  They do not understand how they know, what they know, or the source by which they know.  Only with great difficulty does a brave soul break the chains of the cave, and then move from the cave into the bright light of the sun.  This is a wonderful description of what it is like to turn to and then discover the landscape of interiority. But once one does make this leap, the world of one’s own soul begins to open into something far more grand than anything seen with the eyes.

Included in this grand interior landscape is the lights that illuminate it. Those lights are the wonder that is both the operator and integrator of that landscape, and as Lonergan argues, it sorts into three types already mentioned–questions for understanding, questions for reflection, questions for deliberation. The union of these three into a whole is the real character of the essence and heart of the human subject, a character which is defined by Lonergan as the capacity for self transcendence (see chapter 1, Method in Theology). And if this capacity is not actualized by a love that is the true alpha and omega of one’s soul, one dwells in a dark abyss staring off into the unrestricted reaches of nothingness.

Lonergan more precisely called these fundamental operators and integrators the transcendental notions. They underpin, penetrate, and go beyond all operations and integrations in the subject. They start as questions, discover answers, and through these the subject transcends to new questions. In short, they are transcendental because they make it possible for a subject to self-transcend and even yearn for that which is absolutely Transcendent, our Lord, and more supernaturally, to receive the absolutely Transcendent as a gift called the beatific vision.

Linking the Transcendental Notions to the Finality of All that Exists

In Insight, Lonergan makes an amazing statement about the human mind that helps us to move an inch closer to understanding the kind of suffering Jesus experienced on the cross. Namely, Lonergan writes that in human beings the finality of the universe becomes conscious.  Linking this to the transcendental notions reveals a profound unity in the whole of the created world.

In order to move toward this link, let us first say something about the nature of finality. Finality is rooted in the potency of the universe, in each of its parts and as a whole.  That potency is one of emergent probability. In transcendental terms, it is an emergent probability of intelligibility, being, and goodness. Are you beginning to see the link to the human person? When the human being as part of that emergence is infused with the transcendental notions, then there emerges a creature that is a self-conscious intelligibility that is intelligent, a self-conscious being that is reasonable, and a self-conscious good that is responsible. And the unity of it all? A potentiality that is actuated in love.  The actuated orientation of finality in its highest reaches is a state of being in love. If you have been following these points, hopefully you can see what individuals like St. Thomas are thinking about when they say that all things have God as their end. 

Transcendental Notions: Created Participations in the Divine Light

There is a profound mystery in the transcendental notions. Nothing in this world can cause them. They emerge almost as if from nothing.  There are just there.  And they are a great power in us.  What begins to dawn on the humble soul is that these lights must come from something great.  And as one discovers that they have no intrinsic bounds, and they give us that yearning to understand all that could be understood, and know all that has being, and enjoy all that is good, we begin to wonder about the “hither” of all things and of these yearnings.  It is the question of God that arises when we move to this state.  And when we begin to discover the good without bounds, the true without conditions, the intelligible that is total, we discover as well the real source of these interior lights, the transcendental notions.  They are created participations in the Good, the True, the Intelligible, that is responsible without bounds, reasonable without conditions, intelligent that is total, and loving with mercy that embraces death on a Cross.  This discovery sheds a more magnificent light upon the finality that has become conscious in us.

Oh what a great good each child, each woman, each man is in this world! We are in the image and likeness of our Creator.

Sin and Evil as a Violation of Finality

Human beings as illumined by the transcendental notions have a radical freedom that introduces the potentiality for the sinister. That radicality is the possibility of violating finality. Human beings can turn against the transcendental notions, they can betray the interior lights. In short, they can fail to be intelligent, to be reasonable, to be responsible, and to be loving. This betrayal reverberates into the fabric of the entire universe of being.  Even the quarks that are sublated in our atoms and cells, in our neurons, and elevated into the realm of consciousness are privated by sin of a vertical finality that they were made to enjoy.  This is the nature of evil. And this is just one person. We could go on to discuss how these acts of inauthenticity are then mutually self-mediated to others and the fabric of our universe.

Original Sin and the Deformation of the Finality of the Universe

The doctrine of Original Sin tells us that at the beginning of the human race, sin entered the world through one man and one woman that led them from the harmony and joy of the garden into the cities of Cain. Speculating on how that sin is transmitted from one generation to the next, St. Augustine narrowed the culprit to a disorder that exists between men and women in the procreative act–concupiscence. The heaven and earth were doomed to be corrupted until the end of time.  This reality led the inspired prophets to hope for a new heaven and a new earth.

The Revelatory discovery of the fallen state is complemented by a profound inverse insight that helped the Greeks to discover evil as a privatization of being. With Lonergan we can transpose this privation into a violation of the finality of intelligibility, being, and goodness. A surd is introduced into history and the cosmos. Things really are not as they should be. In our own souls, we directly recognize such violations as a turning away from the light, a light that is a voice which we commonly know as the voice of conscience. And because those lights are a created participation in the divine light, conscience is also recognized as the voice of God in our own souls.

To summarize, inauthenticity is a betrayal of the finality of one’s own soul, and since one’s soul is a conscious realization of the finality of the universe, it is a betrayal of the finality of the universe. Every sin is such a betrayal.

Authentic Suffering as a Response to Evil and Sin

Now we can turn to the character of suffering. I remember the first time that I began to realize that suffering was not intrinsically evil. I was brought up in life to think so and of course, I was a good son of our world.  Suffering was the great evil to be avoided, and  in mercy, we must help others to avoid it too.

There is a truth in this. Suffering is not where we should be. However, St. Augustine taught me something different about suffering. He helped me to understand the church’s disposition to suffering that had always puzzled me. Why embrace the cross? Why lift our sufferings up to the Lord? Why did Jesus suffer unto his death?

St. Augustine taught me that suffering is not intrinsically evil but good. It is that which is good responding to evil. Suffering is the seeking and yearning for healing. This is why it is good. Even our physical bodies operate this way. Pain sensors tell us something is wrong. They prevent further harm. They get us to change behavior to avoid damage and/or to allow for healing. If my hand did not hurt when I put it on a burning stove, I would destroy it.

Suffering the Evil in Others

Human beings can suffer the evils of others. This should be a kind of surprise until you realize that the interior lights, the transcendental notions, mean that we are the kind of creature in which potentially the entire universe that exists could dwell. And that which is more beautiful and profound will dwell in us in a way more beautiful and profound manner. This is why our friends and family are so much a part of us. And what hurts them hurts us. If a good friend dies, part of us dies. When we encounter violations of the finality in others or in creation, this violation is ours because of the nature of who we are. It becomes and is my privation, even if it was your sin. Your sin becomes my sorrow [just as is the opposite true — your good becomes my good (if I do not have envy).]

 So suffering evil is the yearning that evil be overcome and that goodness be restored. This is why authentic suffering is actually good. We should be sorrowful over any and all acts of inauthenticity, and any and all violations of the finality of creation.

Jesus’ Suffering on the Cross

Now let us return to the suffering of Jesus on the cross. That suffering was a response to the sin of the world that dwelt in the soul of Jesus. Jesus is present to all acts of inauthenticity throughout history, and as present, these dwell in him through his subjectivity. This begins to explain the depth of what Jesus meant on the Cross when he proclaimed “My God my God why have you forsaken me?” HBy becoming one of us, all of our privations became his.  He yearned for the healing of the privation of all sin, of all inauthenticity, of the totality of deformed and fallen creation. This was the depth of his love and begins to shed light upon what he meant when on the Cross he turned to his Father, “Father, forgive them for they know not what they are doing.”

Sin is a radical darkness that betrays the light. It turns one against the One who is the source of the transcendental notions.  It is not a mere turning either, but active turning against.  Inauthenticity has a dramatic and murderous plot to it. This begins to explain why we are the cause the crucifixion of our Lord in every act of sin. 

But Why is This the Way of Perfection?

Now we have to introduce one other element to explain why Jesus was perfected in his suffering, and why we are too. 

Self-transcendence that is authentic and unfolds with the finality of emergent probability is a great good. However this is natural. It is expected. It flows from what should be. But there is an even greater set of goods which are authentic responses to evil. In other words these are acts of suffering, which really yearn for the reconciliation of the sinner and the healing of evil. They are a gift, a surprise, and intelligibility that goes beyond natural finality. It is to be intelligent in the darkness of the unintelligent, to be reasonable in the lies of the unreasonable, to be responsible in the wickedness of the irresponsible, to be loving in the hatred of life exuding from those privated of love. Virtues such as courage and temperance would not be if it were not for evil. The authentic response to evil bears forth a good that is beautiful and magnificent, beyond what nature provides. The greatest of course is to die for one’s enemy as Christ did for us. This is a perfection that transcends the natural finality of this universe and is highly intelligible, true, and good.  It is the magnitude of God’s love for us.

Finality, Love, Marriage: Tensions and Contradictions in Love, Part 9

by David Fleischacker, Ph.D.

I have decided to return to an earlier sequence of blogs that are commentaries on Lonergan’s 1943 essay, “Finality, Love, and Marriage” — an essay which can be found in the 4th volume of his collected works.  My objective is twofold; 1) to understand Lonergan’s intentions in the essay, 2) to examine what will happen to the essay in light of Lonergan’s later writings on Insight and Method in Theology.

This blog will examine a single paragraph in the essay — section 2.2 — titled “Tension and Contradictions.”  I will start with a quote,

“But besides this multiplicity of aspects, to be verified in any instance of love, there also is a multiplicity of appetites and of loves generating within a single subject tensions and even contradictions.”

Section two of this essay as a whole is focused on the “concept of love.”  In 2.1, Lonergan articulated how love is “the basic form” of all appetite, and how appetite is a principle that unites subjects both in seeking and enjoying a common end.  In 2.2, the section we are considering in this blog, Lonergan brings up the simple fact that appetites are never single in the human subject.  These multiple appetites have multiple ends that are loved and that unite subjects in both seeking and enjoying.  But because of this multiplicity there are “tensions and even contradictions.”

If love is the basic form that unites subjects, why then does it cause tensions and contradictions.  Let us begin the answer by starting with a more precise understanding of the multiplicity.  Notice Lonergan’s three examples of appetite.

  1. Hunger
  2. maternal instinct
  3. rational appetite

The proper object of the first is “my goal” — more specifically nourishment of my body. The proper object of the second is the good of the child.  The proper object of the third is the “reasonable good.”  The way to these objects is specific as well, hence food for the first, care of the child for the second, and the discovery of the reasonable good for the third.

One can imagine that the appetite for my good, the appetite for the good of another, and then an appetite for what is absolute will find themselves in conflict some day if not every day. However Lonergan notes that the third appetite  is the doorway to a liberation from this interior war.  The third “moves on an absolute level to descend in favor of self or others as reason dictates.” (as a note, Newman says this in a number of his writings).  Somehow, reason will discover and project a harmony in the multiplicity.

What happens in the later writings of Lonergan help to expand and fill out the way that reason transcends “on an absolute level” the tensions and contradictions, and can “descend” to set a harmonious order of human action in the multiplicity of appetites.

Insight and Appetite

In Insight, Lonergan seems to have the same notion of appetite operative even though he is heading into a transposition of faculty psychology into intentionality analysis.  One sees this transposition in

  1. Lonergan’s formulation of image and affects via Freud and depth psychology.
  2. His linking of image and affect to neural demand functions.
  3. The linking of image, affect, and neural demand functions into patterns of experience.
  4. The three levels of human development that are interlocking higher and lower levels of integration and operation (organic development, psychic development, and intellectual development) — (chapter 15)

And this list is not exhaustive. These developments provide a glimpse into Lonergan’s deepening understanding of the nature of the levels and the relationships of the levels of being within the framework of the development of the human person.  Lonergan does not yet use the language of horizon, but but he is using “view” and “viewpoint” which reminds one of Newman’s use of the terms (and a number of other figures over the previous century).

In Insight, the language about the mind itself is transposed into intentionality.  Data of sense is combined with data of consciousness to provide the starting point for self-understanding and self-knowledge.  With the clarification of the notion of being (chapter 12), which transposes the agent intellect–and the manifestation of that notion in questions for understanding and questions for reflection, concepts, and judgments–the nature of the mind itself is more directly reflected in the terms and relations.  This in turn provides a basis to answer the most fundamental problems of modernity and post-modernity in their challenge to epistemology and metaphysics.

This shift to interiority analysis greatly expands upon Lonergan’s articulation of “reason.”  Insight and Judgment constitute the acts which when rightly exercised are isomorphic with being.  His explorations of insight in math, science, common sense, and philosophy provide a deeper understanding both of the harmony and unity of intelligence and reason.  When he shifts to an explanatory account of epistemology and the general character of metaphysics in the second half of the book, he completes the circuit which allows one to discover not only the harmony of the mind, but the harmony of being within a framework of generalized emergent probability.   This significantly expands what can be said about unity and plurality of appetites, and the contraditions they might generate, and why reason can objectively discover a unified order in the multiplicity of appetites.

Let me suggest the heuristic solution to discovering this harmony.  In Insight,  occurrences and events fall within the framework of systematic and non-systematic process, which is developed more precisely into schemes of recurrence, conditioned series of schemes, and at the height of generic intelligibility, emergent probability itself.

If one shifts from a mere multiplicity of occurrence (hence a statistical apprehension defined by classical correlates) to emergent process, then one begins to grasp a horizon-scape (land, water, sky) in which these appetites are naturally ordered.  Hunger for example is within the nutritional cycles of the cells and cellular systems of the body.  Eating is part of these systems in its sublation into the cultural mores of a people.  The same is true of all appetites.  They have their natural rhythms and cycles. These really do form a harmony within the context of generalized emergent probability, which incorporates even dead ends and catastrophes (see chapter 4 of Insight).  But even dead ends and catastrophes do not involve a real conflict of appetites, even if a conflict of appetites can cause catastrophes. The gravest catastrophes result in death, and no one subject to death has an appetite for such an end.

Yet, to the casual observer, there does seem to be real conflicts. Are these objective? If so, how?  The answer is yes.  Ultimately for such a conflict of appetites to occur, there needs to be a free subject who can violate the intelligible order of emergent probability.  Intentionality analysis reveals the culprit.  It is the fallen spirit.  The one who rejects the light of intelligibility, being, and the good.  In other words, the subject who fails the dictates of seeking and finding understanding, truth, and value.  Such failures of spirit (spirit = that which is intrinsically independent of the empirical residue) will result in a failure to descend to unite the multiplicity of appetites into their order within the emergent world.  In turn, this failure will descend into the fabric of the polis and even the cosmos.    That is a larger scope of tension and conflict however than we find in section 2.2.  Yet, this larger expanse of being and the world that can be known by reason sheds some light on the source of the tensions and conflicts.  Motor-sensory and intersubjective appetites will conflict when a wise order is privated. That natural and wise order is one of emergent probability.

Method in Theology and Appetites

In terms of Method in Theology, the tensions and contradictions of the appetites can be easily transposed into intentionality analysis. Every cognitional and moral operator and operation is both conscious and intentional. As conscious, these intellectual, rational, and volitional levels of operators and operations intrinsically allow the subject to be self-present, but not necessarily understood and known.  As intentional, one can transpose the relationship of a specific appetite to a specific object.  Hunger can be transposed into the intention of food, motherly care into the intention of the well being of the child, rational appetite into the intention of intelligibility and being.  No conceptual revolution here.

Some areas of appetite are expanded in Method.  Borrowing from Dietrich von Hildebrand, Lonergan constructs a simple generic map of affective appetites — non-intentional states and trends are organized under one column, intentional feelings under another (and these sort into those at the level of sensate experience–pleasure–and those at the level of decision–value).   This generic pattern is further expanded when Lonergan focuses upon the intentional feelings that respond to value. In chapter two, he formulates the scale of values — vital, social, cultural, personal, and religious/transcendental.  One can see the beginnings of this scale in 2.2, with the examples mentioned above.  Hunger belongs to vital values, motherhood to intersubjective and social values (one could argue personal emerges as well), and rational appetite constitutes a type of cultural value (descriptive as well as explanatory).  Both in Insight, and more so in Method, the scale is expanded in scope.

Arguably the most significant transpositions and expansions found in Insight and Method are the Transcendental Notions (only that of Being is explicitly formulated in Insight, and in Insight it is identified as a notion, which falls within the framework of heuristic notions which are components of heuristic structures).  These notions are integrators of systems and operators of development.  This shift to the transcendental notions allows Lonergan to recast a more developed understanding of the dialectic of human development.  In Insight, that dialectic sprung from a tension of levels (eg. sensate and intellectual).  In Method, is becomes more precisely articulated as a dialectic of authenticity and inauthenticity. To return to 2.2, the tension considered is on the same level, that of motor-sensory appetite and conflicts between these appetites.  However, as the essay proceeds, Lonergan will move to a dialectic closer to that found in Insight, but not quite as precise.

The transition from this essay to Insight and Method will not in the end contradict or undermine the general arguments of the essay, but rather will strengthen the arguments as will be seen when we move to further sections.  Lonergan’s later writings provide a more penetrating heuristic for understanding the general character and specific features of the nature and life of marriage.

Next commentary on this essay will be upon friendship — 2.3.

 

Reality or Being — Are these the same?

by Dr. David Fleischacker

This one is more for those who have studied Lonergan a bit.  Sorry to those who have not.

Though most today might think of being and reality as the same, what is meant by both today is not the same as that of Plato, Aristotle, St. Augustine or St. Thomas Aquinas. For them, Being is not only “that which is,” but a that which is that is necessarily intelligible. Being is intelligible and actual intelligibility is being. And just because you can name something does not mean it is intelligible, hence just because you can name something does not mean it exists.

In contrast to intelligible being is that which is not.  Statements do not get any easier to make which are true. Darkness is an easy example because its descriptive correlate has a relatively easy explanatory basis. It is the absence of any visible light waves.  We can name it but it has no intelligible being (at least in the visible range of light — there may be being beyond the visible light spectrum as a note).  More difficult are those absences that seem so real there must be something intelligible.  So, for example, inertia seems like it must have some kind of intelligibility.  After all great minds searched for the answer to the  cause of inertia for centuries upon centuries. But in the end it belongs to the empirical residue (see chapter 1 of Insight) and one likely will need an inverse insight to get that it lacks intelligible being (also see chapter 1).  More difficult is something like evil. But it too lacks intelligibility.  In fact it not only lacks intelligible being, but it is a privation of being and so introduces the absurd. In either case, these lack intelligibility and thus cannot have being.

Here is another way to get at the same point.  Let’s make a distinction between being and reality.  Let’s say that reality includes experiential absences, partial constitutive components of being (eg. the empirical residue), privations of intelligible being, and concrete being that is intelligible.  This makes it a larger category than being because it includes named nothingnesses and absences and privations.  Concrete and real being that is intelligible however is only “part” of this world, a larger world that really is not.

The import of grasping what the ancients meant by being and us moderns do not has a number of ramifications.  Without realizing the ancient meaning of being, disciplines like metaphysics will be misunderstood.  Evil will make no sense.  Why?  Because the ancient statements about being cannot be applied to nothingness, absences, and privations without being unintelligible.  And so us proud moderns will tend to think that these ancients were simply careless and unintelligent.  But it is the reality of moderns that is lacking.

Judgment and the Recovery of Being

by Dr. David Fleischacker

Lonergan’s explanatory formulation of the interior structure of judgment dismantles one of the great culprits of the modern world that has left vast reaches of the Western world in a dark age. It is dark because it thwarts self-transcendence precisely in one of the great powers of the human mind.  Judgement makes possible a real presence of a person to that which is.  It mediates a true encounter with intelligible being. In other words, authentic judgment allows being to dwell within one.  This darkness is the real forgetfulness of being.  Heidegger was only partially right. He did recognize something that was true about the fallen state of us.  But he still left one with out the ability to enjoy and rejoice in the goodness of even the littlest beings in the world.  Those little, finite beings–trees, rocks, the human body, stars, planets–were merely ontic things.  For him Being– the Ontological–was all that mattered, and even that notion lacks in Heidegger the liberty that Lonergan comes to discover. It is after all a transcendental notion.

When one proclaims that all is mere perspective, or one announces that one can never be sure of what truly is, or one thinks of reality as out there but not in here (in my head), then one is proclaiming that being is fundamentally unknown.  It is as Kant said, in the noumena.  This is the darkness in which today we are chained and enslaved.  It is a self-inflicted cave of own’s own mind, and if one is completely honest, then Derrida is right, even that cave is a mere trace. It too resides in the darkness.  Even my own thoughts flow in the differance of lost presence.

For most, I think the world of entertainment and work keeps them from facing this haunting darkness which they have absorbed since their day of birth.  Many do escape into a world of common sense and do not bother with these questions.  But if pushed in a direction they do not like, then as an instinctual mechanism of self-defense, they pull out the darkness of the no-nothings.

I remember one day saying to a friend, “don’t you know that you can’t find happiness in hockey — he loved hockey to the neglect of nearly everything. He was able to deconstruct my simple quest with one stutter of his vocal cords and a brush of air sent my way in the wave of a hand.  I knew what he meant.  He meant you can’t really know the answer to what you are asking.  Don’t bother me with it.

Lonergan does not answer this deconstructive shallowness with the same brush of air and grunt.  No such simplicity can be found with his response.  Yet, amazingly, in one book he sends to the grave this particular darkness for any who want freedom from these chains that have been growing and entangling the Western world for 500 or more years.  I suppose one could argue that it has been longer and started with the nominalists, but the other day, someone I know — Dr. Chris Blum — pointed out rightly that without the founders of modernity (Descartes, Hume, Kant, etc.), these nominalists would have been forgotten.

Lonergan in one book opens the doors to the cave. That book is Insight. He let’s in some light. We can discover that the shadows and traces of being are not our genie lamp. With the great skill of a gifted surgeon, Lonergan, at the beginning of the book, asks the reader to examine in themselves the act of understanding. It begins a journey into a massive world of interiority and self-appropriation.  The attentive and careful reader who takes this journey is not asked to trust the writer in the end, though one must trust along the way.  He leads the reader from insight in math and science to that of common sense and things, all before he turns to the excavating work of exploring judgment.

It is a brilliant plan as anyone knows who has seriously read the text.  His first eight chapters remove the rocks that block the path to light and freedom, and then finally he removes the hinges of the locked doors of the cave.

Starting in chapter 9, he then begins to open the door.  In chapter 11, the reader gets asked to walk out of the cave unless he or she is too afraid to do so and simply refuses to see the beauty and the landscape of being.

In the next couple of chapters, through the notion of being and then of objectivity, Lonergan provides an explanatory account of why we can be present to being, and why being can dwell within us.  It gives the subject who has dwelt in the cave of the modern world a new wineskin and a new garment.  More technically, it is a new heuristic foundation to taste the beauty and glory of the real universe of being.

I could repeat Lonergan’s answer with regard to the conditions required for true judgments and the principle notion of objectivity, and why these happen in us all the time.  But for the full meaning of these explanatory formulations to burst forth and make sense, one really does need to travel down all of those earlier chapters of Insight first.

Hence, this blog you are reading is merely an invitation to those who have some inkling that perspectivalism and relativism are unhappy conclusions, and that traces of others are not so joyful as their real presence in filial and agapic bonds of love.

By the way, for those who are not able for various reasons to move into the explanatory account of the freedom and light of true judgment, do not worry.  Lonergan’s account reveals that good sound judgment gives you that liberty even when you are unable to explain why.  You really can love–in a mutual indwelling presence–your friend, your spouse, your child….and God, even if the how remains a mystery.

 

Isomorphic Existentialism

Existential Isomorphism

By Dr. David Fleischacker

I would like to make a simple statement. The finality of the human person is one of existential isomorphism.

Why Existential?

I am sure some will think that I have committed an error in tying the word existential to isomorphism.  Some would be disturbed if they knew what I meant.  Some of the dead might twitch a bit. Nietzsche I am sure would turn in his grave. Most of the 20th century existentialists might will themselves to rise from the dead and burn me at the stake and insist that God is still dead. They might call upon their leader — Friedrich, Friedrich, where art though — so that he could lead them in their inquisition with his sharpened words and golden pen.   So, let me be clear as to my fears of the power of these willful mongers.  Will to power and its maturation in the 20th century notion of self-realization are not what I mean by linking the two terms. Yet, there is a truth in the 20th century existentialists that I would like to return to the world of being and goodness and beauty.  As St. Augustine said about heresies, there is always a great truth in them which is why they can arrest people and capture their imaginations.  The same is true I would argue with Existentialists such as Sartre.  That nugget of truth is that human beings do have something to do with their coming to be in this world (or in their self-destruction).

In other words, I want to recover the rightful place of human freedom or decisions.  I want to place it back into a normative framework of a naturally ordered universe that has its nature in a finality that is oriented as Lonergan argues in Insight toward increasing intelligibility and being and goodness. These transcendentals are the norm of the normativity of all existence, especially when they become conscious and active in the human soul as an actuation of the capacity for self-transcendence.  It takes wisdom to figure this out.

So, what about isomorphism?

In Insight Lonergan argues that the structure of cognition is isomorphic with that of being.  Hence, intellectually patterned experience, insights into conjugate and central forms, and judgments affirming those insights as true are isomorphic to conjugate and central potency, form, and act of beings.

J (judgement)   –>    Conjugate and Central Act

U (understanding)–>    Conjugate and Central Form

E (experience)–>  Conjugate and Central Potency

It is not just any E, U, and J that matters to this isomorphism.  The relevant conscious and intentional operations are those that have moved into explanatory accounts of this world–hence insights that emerge in intellectually patterned experience, and then are verified in judgments about the truth of those explanatory insights.

What this means is that in true explanatory knowledge, the human soul has come to be a mirror (as St. Thomas notes) of that which it knows, and it knows that which it knows by becoming a mirror to that which it knows.

Adding the term “existential” goes beyond what Lonergan does in Insight. And as mentioned, I want to expel it of the licentious willfulness that one finds in 20th century existentialist philosophers. I want to recover an older meaning of existence found in St. Thomas and Aristotle, one that links together being and becoming into a harmonious unity.  The act of will is only an act of will when it is based on an intelligibility, and thus it is an authentic volitional act when rooted on form, not on nothingness (which actually is impossible because we cannot create from nothing).  It really combines some of Lonergan’s later developments in Insight with those of his later life, namely the link of metaphysics and its isomorphism with intellectually patterned consciousness to the moral order and the level of decision.  In short, when decisions are based upon the fullness of the cognitive isomorphism with being, then one’s decisions shift one to an explicit participant in the unfolding potency of being [as a note, even one who operates in the world of common sense is a participant in the unfolding potency of being, but only implicitly.  Common non-sense however is evil because it is a failure to participate in this finality of the universe.], and thus participate in a moral isomorphism with the emergent universe and its finality.

I would like to add one other piece that identifies a more complete existential isomorphism, namely when the entire neural and motor-sensory operations, along with their landscape of emotions and passions join in on the isomorphism. For this to take place, the neural and motor-sensory levels need to reach an integrity in which they are intelligibly ordered in the higher levels of the moral and cognitive isomorphism (see what Lonergan does in his last chapter in Insight “Special Transcendent Knowledge”).  In other words, all levels of development when united in a sublating or subsuming fashion into the highest reaches of conscious intentionality form an authentic existential isomorphism of the soul with an emergent universe.

Interestingly, the university when setup right has as its specific end this existential isomorphism in which the totality of the person (organic and neural, motor-sensory, intellectual, rational, volitional, religious) is mediated toward this unity with the finality of the universe.

Just a thought that has tremendous ramifications.

From David Fleischacker

Just a quick note.  I will be publishing a reflection every Friday at 3 pm. Most of these will be short pointers and thoughts about the writings of Bernard Lonergan.

Kindly,

David Fleischacker

Trinitarian Reflections: The Transcentdental Notions and God, blog 1

by David Fleischacker

About two years ago, I started a new notebook on linking together the University and its life with that of the Holy Trinity.  One of the areas that I wondered about was whether the Transcendental Notions (TN) could provide any type of analogy for understanding the three persons of the Holy Trinity.  There are after all, three transcendental notions that Lonergan develops which are spiritual in nature, hence intrinsically independent of the empirical residue.  These spiritual transcendental notions are Lonergan’s transposition of the agent intellect found in Aristotle and St. Thomas, and of the Light of Being (conscience, mind, etc) as found in the Platonists and St. Augustine (as a note, Augustine was clearly not a Platonist once you get into his head more thoroughly even if he learned much from them and borrowed some notions from them).

One of the immediate difficulties of course which one finds noted in Lonergan is that in finding an analogy for the Holy Trinity, we need to deal with acts or operations, not with anything in potency.  The TN are a kind of potency, but much different than normal.  These actually have the power or capacity to bring about self-transcendence.  In St. Thomas (and Aristotle), these “lights” of the mind have the power to illumine, hence they act as agent causes.  Most potencies do not have such capabilities.  Hence the reason these lights are in a kind of actuality as well.  Notice how some of the metaphysical terms and relations get stretched (but not violated! or confused).  The TN are in a potency in relationship to the operations that arise, but in relationship to the potencies in the human subject to receive these operations they are in act.  Many would say that this imprecision of the metaphysical terms and relations is why one needs to leave out the metaphysical, and turn to intentionality analysis. That is true in part, but if one does so, one as Lonergan notes in Insight, needs to run the full circuit, and return to metaphysics, both to refine the metaphysics, but also to articulate the intelligibilities discovered as belonging to being.  To stay merely with a cognitive apprehension of conscious and intentional life leaves one ignorant of its “reality.”  So the circuit does need to be run.

The reason I mention the circuit is because if one is to transpose the analogies for the Holy Trinity found in St. Thomas, then one needs to deal with some of the metaphysical points that he makes, such as God is pure act, and hence we need to find analogies in act that help us, and this is true of the Persons as well as of God.  The Father is pure act, as is the Son, and the Holy Spirit.

Hence, are the TN in act enough for them to be used as analogies?

The TNs, though in a kind of potency, are also the “light” that makes possible the conscious and intentional operations.  This means that in some manner, they are more in act than the operations.  They underpin, penetrate, and transcend all operations.  Still, there must be a reason that Lonergan did not turn toward these as analogies. He stuck with operations (eg. apprehension of the good, judgment of value of the good, love/decision of/for the good). I suppose one could argue that these operations are in part constituted by the TN, as the TN penetrate them.  We could look at what that “penetration” means.  It of course is not physical, but spiritual.  Descriptively, it “illumines” the operation.  It is what “receives” the operation.  It is what “beholds” the operation.  The TN is not only light, but also an intentional focus, hence can be described as the “eye” of the mind as well.  I am tending to think that the TN is both light and eye (hence not distinct as these are physically in us — but I could be wrong).  I suppose one could say the “eye” is the conscious subject as awakened in a TN and thus seeking an answer, hence waiting for an operation that mediates the answer.  Then once the operation emerges, the subject as beholding the operation in the TN is an eye that beholds.  The subject is however conscious through the TN, and thus the TN constitutes both the horizon and the subject as a gazing subject.

One of the areas that I explored a couple years ago in my notebook was whether there was a sufficient distinction and set of relations between the TN to result in some kind of analogy that sheds light upon the Father, Son, and Holy Spirit.  Thus, does the TN of intelligibility have a kind of relationship to that of being/truth such that the former begets that latter.  Of course, this does not happen without an operation. And it does not happen without the subject moving (raising the question for reflection).  Likewise does the TN of goodness spirate from the TN of being?  I cannot repeat all of the reflections here, but I can say that my reflections were not conclusive.  I do intend however to start publishing these reflections in this particular sequence of blogs.

Even if I discover that those reflections do provide an interesting analogy, there is still the further question about whether the analogy is an improvement upon that of the operations as such.  I have a suspicion that they do not, but they might help to deepen my understanding of the operational based analogy (apprehension of the good, judgement of value of the good, decision for the good).  Part of my reason for this suspicion is that God as pure act is the cause of the light that is in us, which we call the TNs.  The TNs do allow us to grasp the unrestricted nature of the operations in God, but those are operations in God, not TNs.   Just a few thoughts.

More later.

Feeding the thirst of Jesus Christ

Why does Jesus need or want us to feed him? It would seem that the only appropriate relation to him is to allow him to feed us. Very true of course. At the same time, from the Cross, he cries out that he thirsts.  He thirsts as St. Mother Theresa tells us.  Jesus is in those whom we meet, especially the poor and the destitute.  All of those who fall under the beatitudes.  He thirsts in and through them for us to give him a bit of drink and food.  It is part of the immense mystery of being a member of the body of our Lord.  He knows us.  He knows us in his divinity and he knows us in his humanity.  As he hung on the Cross, he proclaimed the thirst of his entire body, as it exists in his mind and heart.  This is the meaning of the unity of Christ and his body.  In fact, it is a unity that each of us has with each other.  When anyone thirsts, and it comes to dwell in us, it then comes to inform us as a constitutive act of meaning.  Hence another’s thirst becomes our own.  Likewise with Jesus Christ. We are his.  And we are in him.  He thirsts because we thirst.  He thirsts because he became one of us.  And as he fills that thirst, so we as part of him are to fill that thirst as well. This is the meaning of to abide and to mutually indwell.

Higher Viewpoints: Part Two From Algebra to Calculus: The Emergence of the Power Rule

[This is a reprint of a 1997 posting]

 

Higher Viewpoints: Part Two

From Algebra to Calculus:The Emergence of the Power Rule

A Thought Experiment

DRAFT VERSION 2

by David Fleischacker

Copyright © 1997. All rights reserved

September 4, 1997 (Originally written in 1992)

(First presented as supplementary notes in a seminar on INSIGHT held September 13, 1993)

The following is an exercise in creating a dynamic image which leads to the insight underpinning the power rule in calculus. This image is a particular “play” with algebraic equations and geometric graphs and definitions. Furthermore, I have intentionally set up diagrams, or symbols, in particular ways so as to illustrate the importance of images in order to get the insight. This exercise does not explicitly distinguish between the rules of calculus and the rules of algebra, but all this is not a far step once the exercises have been performed.

The general outline of the paper begins with some definitions. An understanding of arithmetic and some other basic definitions in math are presupposed. Once some key definitions are established, then we proceed to the setting up of the dynamic image and the thought experiment which leads to the power rule.

 

Part I. Some Preliminary Definitions (skip to part II)

(1) The Definition of a Point:

The definition we are using for a “point” is that any “x” and “y” on a coordinate system will define a point. The coordinate system in this case is two dimensional. Here is a general diagram of it;

y-axis
5|
4|
3|………. * (5,3)
2|
1|________________
0 1 2 3 4 5 6 7 x-axis

The y-axis is the vertical line and the x-axis is the horizontal line on the coordinate system. Technically the two lines are perpendicular and intersect at a point which we have label (0,0). Every point will be given the form (x,y) where x is the number on the x-coordinate and y is the number on the y-coordinate. Thus, the point identified by the “*” on the graph is 5 units on the x-coordinate and 3 units on the y-coordinate (5,3).

(2) Definition of Slope:definition_slope

The slope of a line is found by taking a segment of the line and measuring its rise and dividing by the measure of its run. The rise is the distance on the y-coordinate axis in a given segment on the line itself. The run is the distance on the x-coordinate axis in the same given segment (d) of the line. So, to get the slope of a line, simply select two points on that line [such as (x1,y1) and (x2,y2) in the diagram. Examine how you can figure out the slope from knowing two points on a line.].(1)

One may wonder why such numbers are used. The reason depends upon the problem one is solving. For now, let us say that we are just putting the numbers into a type of pattern, and later the reason will become clear.

 

 

(3) Definition of a Tangentdefinition_tangent

The definition (geometrical) of a tangent is a line which passes through a curve on one and only one point on that curve. Thus, to “tilt” the line one way or the other would necessarily result in the contact of a second point on the line with a second point on the curve.

 

 

 

 

 

(4) Tangent and the Curve: An important cluepatterns_tangents

Notice, when the tangent moves to points “higher” on the curve,  the slope of the tangent increases. In other words, the ratio of rise/run increases. When the tangent is moved lower on the curve, then the slope decreases.(2) This raises the question about the existence of a relationship between the slope of these tangents to the curve.

 

 

 

(5) Definition of a function:

The next step is to introduce the notion of function. Instead of a curve, one can actually figure out an algebraic function for the curve. Here, we cannot enter into the tricks of how that is performed. But we can go in the reverse direction, namely start with a function and then draw a curve using it.

A function equates variables to one another through the familiar operations of addition, subtraction, multiplication, division, roots, and powers. plotting_functionSo, in the equation Y = X2,(3) the function uses equality and the operation of “powers” in order to relate two variables, namely “x” and “y.” In this equation, “x” and “y” are fixed, such that if you know “x” or “y” you can calculate the other (I do not wish to discuss imaginary numbers or other problems which arise in this activity, for we are staying with real numbers). So, if x equals 2, then y equals the square of 2, or 4. If x equals 3, then y = 9. One can set this up in a graph (see “PLOTTING A FUNCTION.”)(4)

In the diagram to the right, the function is plotted as a curve. One simply plots a point where “x” and “y” meet on the coordinate system. In addition to the six numbers plugged into the function one could include many more. This curve then approximates to the function, and the more points one calculates and the more dots one marks, the closer the approximation (If one could plot the infinity of points on the curve, one would have a continuous curve which would entirely represent the function, but since the curve is material and imaginative, it only approximates, hence the imagination struggles to keep up with intelligence).

 

Part II: Image and Insight underpinning the Power Rule

(1) The Slope of a Tangent

Before we move to the actual image that leads from algebra to calculus, we need to discuss how one arrives at the slope of a tangent of a curve. You may ask why, and again you will have to wait and see. It is simply another way of organizing the data or numbers for the purpose of understanding the Power Rule. The following set of diagrams will reveal one way to approach the slope of a tangent.

The exercise is to locate the point on the curve through which the tangent line passes, call it point A. Then choose any other point on that curve (5,25) and draw a line from point A to your chosen point. Since you have two points, you can figure out the slope (m2).

Then select a point closer to point A. Perhaps move to the other side (2,4). Although you cannot tell from the diagram, slope m1 is closer to the tangent slope (m3) than is slope m2. As one gets closer to point A, you will find a convergence upon some slope. From this convergence, you can actually approximate the slope of the tangent (m3).(5)

An example will reveal this convergence. We shall use the function “y = x2.” Let us say that we are interested in the slope of the tangent of this function at point (3,9). So, we need to approach the slope by drawing lines through points on the curve which are increasingly closer to (3,9). As the points approach (3,9) from both sides of the point, the lines drawn from (3,9) to those points will increasingly approach the slope of the tangent at (3,9).(6)

 

Destination(3,9)(8) Selected Point(x1,y1)(7) Calculation(9)of Slope (m)rise/run = (9-y1)/(3-x1) = m
a. (3,9)b. (3,9)

c. (3,9)

d. (3,9)

e. (3,9)

f. (3,9)

g. (3,9)

h. (3,9)

i. (3,9)

(1,1)(2,4)

(4,16)

(2.5,6.25)

(3.5,12.25)

(2.75,7.5625)

(3.25,10.5625)

(2.95,8.7025)

(3.05,9.3025)

(9-1)/(3-1) = 8/2 = 4/1 —thus 4 is the slope(9-4)/(3-2) = 5/1

(9-16)/(3-4) = -7/-1=7/1

(9.00-6.25)/(3.0-2.5) = 2.75/.5 = 5.5/1

(9-12.25)/(3-3.5) = -3.25/-.5 = 6.5/1

(9-7.5625)/(3-2.75) = 1.4375/.25 = 5.75/1

(9-10.5625)/(3-3.25) = -1.5625/-.25 = 6.25/1

(9-8.7025)/(3-2.95) = .2975/.05 = 5.95/1

(9-9.3025)/3-3.05) = -.3025/-.05 = 6.05/1

Notice: As we moved from step “a” to step “i” you can see that the point (x1,y1) approaches the point (3,9) and the slope (m) approaches 6. So, perhaps the slope of the tangent at 3,9 on the function “y = x2 is 6. It at least approaches that number. If one continues to bring the points closer to (3,9), one will find that the number likewise continues to approach 6.(10)

The basic question is “what is the relationship between the slope of a tangent line and the curve itself?” A clue was given earlier, when we noticed a correlation between the location of the point on the curve and the slope of the tangent through that point. Obtaining an insight into this will be gained through a series of hypotheses about this relationship that serve as the playground for our inquiry.

(1) Hypothesis Number 1

In the next pieces of data, let us say that we have performed the above activity for the points (4,16), (5,25), (6,36), (7,49) on the same function and found the various approximations to slopes.

Slope (m) at (x,y)

(x,y)(3,9)

(4,16)

(5,25)

(6,36)

(7,49)

rise/run (m)6/1

8/1

10/1

12/1

14/1

Are there any patterns? Examine the numbers in both columns. There are many relations which could be examined, but to move toward our goal, notice the relationship between the “x” in the left column and the slopes in the right (each is boldfaced below).

(x,y)(3,9)

(4,16)

(5,25)

(6,36)

(7,49)

rise/run (m)6/1

8/1

10/1

12/1

14/1

What is the relationship? The relationship appears to be 2*x or 2x(11) (“*” means multiply, and in 2x, the multiplication symbol is implied).

x * 2 = m

3 * 2 = 6
4 * 2 = 8
5 * 2 = 10
6 * 2 = 12
7 * 2 = 14 (12)

Let the “2x” be named the “slope function” because it is the equation which relates the “x” to the slope of the tangent which passes through the point on a function at (x,y). Once again, we could ask whether this has significance. To ascertain this significance, return to the original equation of the function. It is “y = x2.” Do you see any pattern?

Both the square(13) in the function, and, on the other hand, the “slope function” have two’s in them. Perhaps the relationship between the slope of the tangent and the function involves the power which in this case is 2. To get the slope of any tangent on the function at any point (x,y), you simply multiply the power of the function by the “x”.

 

(2) Hypothesis Number 2

Let us turn to another function that is not complicated, such as “y = x^3”. If you perform all the suppositions and operations done on the earlier function, this is what you get

(x,y)(1,1)

(2,8)

(3,27)

(4,64)

(5,125)

rise/run (m)3/1

12/1

27/1

48/1

75/1

The pattern is not exactly the same. The relationship between “x” and the slope of any tangent is not 2x. In looking at the first point, (1,1), maybe it is 3x. But, in trying to multiply the x-coordinated in the second point (2,8) times 3, the number is six, not twelve which was the approximated slope of the tangent at this point. Let us draw up a quick list, placing 3x alongside the (x,y) and the slope (m)

(x,y)(1,1)

(2,8)

(3,27)

(4,64)

(5, 125)

rise/run (m)3/1

12/1

27/1

48/1

75/1

3x3

6

9

12

15

Disappointed? The relationship between the function and the slope of its tangent is not simply multiplying the power by “x.” Look at the numbers again for a pattern. Try another function. Perhaps “y = x4” and add 4x alongside so that it will be consistent with the two earlier diagrams. This will keep things simple.

(x,y)(1,1)

(2,16)

(3,81)

(4,256)

(5,625)

rise/run (m)4/1

32/1

108/1

256/1

500/1

4×4

8

12

16

20

Set this up in the same manner as the first two sets because keeping a consistency in the setups improves the chances of recognizing patterns. Sit back again, and look at the numbers.

Look at the “y = x3” data again.

(x,y)(1,1)

(2,8)

(3,27)

(4,64)

(5, 125)

rise/run (m)3/1

12/1

27/1

48/1

75/1

3x3

6

9

12

15

Notice that if you multiply the “3x” by the “x” again, you get the slope.

3x * x = m

3 * 1 = 3
6 * 2 = 12
9 * 3 = 27
12 * 4 = 48
15 * 5 = 75

Then turn to the “y = x4” data.

(x,y)(1,1)

(2,16)

(3,81)

(4,256)

(5,625)

rise/run (m)4/1

32/1

108/1

256/1

500/1

4×4

8

12

16

20

Notice that the pattern does not follow when you multiply 4x times x.

4x * x does not equal m, except when “x” is 1.

4 * 1 = 4 does follow the pattern

8 * 2 = 16 does not equal 32, which is the slope

12 * 3 = 36 does not equal 108

16 * 4 = 64 does not equal 256

20 * 5 = 100 does not equal 500

 

Look at the numbers again. Notice that if you multiply the outcome of what you just did (4, 16, 36, 64, and 100) with “x”, you get the slope.

4 * 1 = 4

16 * 2 = 32

36 * 3 = 108

64 X 4 = 256

100 X 5 = 500

Now let’s see. To get the slope of the tangent when the function was “y = x2” then the “x” was only multiplied once, by the power. When the function was “y = x3” then the “x” was multiplied twice, once by the power and then by itself. When it was “y = x4” then the “x” was multiplied three times, once by the power and twice by itself. If you carry out the same activities with the function “y = x5“, you will find a similar pattern.This time the “x” was multiplied four times, once by the power and three times by itself.

Notice the pattern? Not only do you have to multiply the x more times when the powers of the function increase, but the times you multiply happen to be exactly one less than the power. You compile the pattern as follows;

if y = x2, then the slope of a tangent on that function is 2 times x or 2x.

if y = x3, then 3 times x times x or 3x2.

if y = x4, then 4 times x times x times x or 4x3.

if y = x5, then 5 times x times x times x times x or 5x4.

What this pattern solves is the slope of a tangent on a function by finding what was called the “slope function.” If you think about it more, a simple rule can be devised from the original function. Let the power = n. Then if the curve is defined by the function y = xn, then to get the slope of the tangent along this function simply multiply “x” by “n” and give the “x” the power of “n-1.”

xn ———–> nx(n-1)

Examine more functions and try out the rule. It should work in every applicable case. Basically, it gives you a new way to figure out the slope of the tangent on a curve at any point you would like to examine. Simply carry out this rule, and then plug in the “x” of the point on the curve which you would like to investigate. It makes this task much easier. Instead of performing the rather involved task in finding the slope which we did earlier, now we just follow this simple rule. Not only that, but the rule is not an approximation like the slope found on page 7 (although it is still a “serial analytic principle”–see ch. 9 of INSIGHT).(14) One thing that should be noted in the applicability of this rule is that it only works for simply functions like x2, x3, x4, x5, etc.. Functions like “x2 + 2x + 3″ do not work with this rule. Finding tangents on those more complicated functions will require more work.(15)

What has been named the “slope function” in this example is, for those who have studied calculus, the derivative. The rule developed in which xn ——-> nx(n-1) is the familiar power rule. The process of applying the rule to a particular problem is called derivation.

This rule is only a first step in developing the mathematical viewpoint of calculus, and it, like arithmetic and algebra, has an analogous deductive and homogeous expansion.

Reponse?

1.The rise of a slope is equal to the distance on the y-axis, which, regarding segment “d,” is y2-y1. Likewise, the run of a slope is equal to the distance on the x-axis, which, regarding segment “d,” is x2-x1. Hence the algebraic definition of slope.

2.In practice, you would probably examine many curves and tangents to see if there is a pattern, not just one curve like we are doing. Using terms like “up” and “down” are really only relevant to the curve and tangents we are using. Furthermore, we are only drawing and staying in one quadrant of the coordinate system. The larger coordinate system extends into the negative y- and x-axis. These extension are not need though, for our concerns.

  1. Notice the “apt” symbolism. If “x” times “x” were used, it would not have the same probabilities of leading to insight into the power rule. Pay attention to the next few sections to verify this claim.
  2. We are staying with a simple function. This is all we need for a basic grasp of the power rule.
  3. It is easier to see the convergence of slopes of the lines drawn from point A to the points 1, 2, and 3 upon the slope of the tangent to point A.
  4. This exercise uses certain rules. One can compare these to the higher rule which eventually emerges from this kind of “play,” namely the power rule.
  5. Chosen point.
  6. The is the point at which the tangent contacts the curve.
  7. This is calculated from the algebraic definition of a slope on page 1.
  8. There is actually an algebraic equation which can be used to solve this problem definitively, but it is rather complex, and it is not needed for our purposes.
  9. In standard notion, although a multiplication sign is not used here, in “2x” what this means is 2 times “x.” In other places in this paper, multiplication may be signified by the capital X.
  10. Since any number divided by “1” is equal to the number, the slopes listed in this chart do not have the form “m/1.” So, that does not mean that we have eliminate the “run.” Instead, you should just assume the “1” is there.
  11. This means the power of 2.
  12. Proving this requires utilizing the algebraic equation used to solve the slope of a tangent on a curve. Not only is this equation more difficult to learn than the approximations we performed above, but it has many limits to its use. There is much guesswork which has to be waded through in order to solve problems using this method, whereas with calculus, the rules are very systematic.
  13. In calculus, the next step is usually the chain rule. In the same way that this present “thought” experiment was set up, so one could be performed with this second rule. It would be more difficult though.

***In step “a.”, draw a line from point (3,9) to (1,1). In calculating the slope of this line (under the third column), carry out the operations within parentheses first. So, in the above equation, first carry out (9-1) and (3-1), which will result in two numbers, 8 (the rise) and 2 (the run). Then divide the first number obtained with the second, resulting in a rise/run ratio of 4/1

^ The “^” means “to the power of.”