Beauty of Math in Christianity 1 – BEAUTY OF THE INCARNATION OF CHRIST AND THE DIRAC DELTA FUNCTION
THE DIRAC DELTA “FUNCTION”ENLIGHTENS THE THEOLOGY OF THE INCARNATION
The Incarnation of Christ is the most beautiful and yet mysterious event in the history of the human race- God enters into Creation as a human being. Here we wish to use a Spirit-inspired mathematical entity to enlighten this beauty, to reveal its simplicity in its finiteness. This mathematical beauty, conceived in the mind of a human, but stirred up from impulsive intuition shows us one aspect of the link between the theological depth of the Incarnation and the wonder of the Delta function.
Why the Incarnation of Christ?
There can surely be no theological assertion in which East and West are so united as the statement that the Incarnation happened for the sake of man’s redemption on the Cross[1]
Into the creation of creations, there occurs the most miraculous of events that could ever have been conceived by the human person. From the Trinitarian Personhood of God, the very essence of Being and Love, the minus of infinity to the plus of infinity, the Father sends the Second Person to this earth, to enter into the time of this visible world. This “sending” is not in any spatial sense, that is, from this place to that place, e.g. heaven to earth. Rather, Jesus enters into the timeline, the created time is now given a new meaning, a new focus. Jesus’ Conception through the Holy Spirit in Mary establishes the definitive starting point in time, the time from which all is counted, both forward and backward, the anno Domini nostri Jesu Christi. As both God and Man, Jesus plunges into this densely packed timeline of humankind and instantaneously places Himself as the Norm, the Reason, the Mission for the lives of all on that line in the mind of God.
God entered the history of humanity and, as a man, became an actor in that history, one of the thousands of millions of human beings but at the same time Unique![2]
He immediately stands as the Light that penetrates the darkness that existed before his coming, a Light that extends form the minus infinity to the plus infinity.
No one has ever seen God; it is the only Son, who is close to the Father’s heart, who has made him known. (Jn 1:18)
Dirac and His Impetuous Impulse
This Jesus is then not an ordinary man, made flesh into history, incardinated into the timeline of humankind, into our entire history but the God Man, and here is where mathematics may provide additional insight into a way of expressing the uniqueness and specialness of the Incarnation of this person.
Paul Dirac
The calculus describes an entity with a very mysterious behavior which can be seen to characterize Jesus’ mysterious presence among humankind. A man by the name of Paul Dirac was inspired to publish the beauty of this entity, not in the 15th or 16th century but in 1958. Dirac, born in 1902 was an atheist as a young adult but in later years following his fame as a quantum physicist, he spoke extensively of the beauty in mathematics.
What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature.[3]
This is rather remarkable for a physicist to talk so eloquently and even passionately about “a quality which cannot be defined”. He is even quoted as once declaring: “God used beautiful mathematics in creating the world.” Dirac is most famous for his “Dirac Equation” which we will introduce at some later point. Here we will speak of his impetuous almost foolish, totally heuristic relationship which is called the Dirac Delta “Function”, the latter word in quotes since it really is not a function. Since everything is created by God, even this relationship, could perhaps be of some use to us in probing more deeply into the mystery of the Incarnation.
The Delta Impulse
So let us see if we can open this thing of the calculus, unfold the mystery of this entity of Dirac, and reveal its relevance. We begin by defining the entity as zero everywhere on the timeline except at the time of zero where it is equal to infinity. Note that we easily now speak of time of zero (actually approaching zero but never quite getting there) where we have previously said such a concept is not conceivable or measurable yet here we simply define it! Such is mathematics. We can already see the strangeness of this thing. But then we are told to believe even further that when we take the sum from minus infinity to plus infinity of this strange thing standing at zero time over a very tiny interval of time, essentially zero, we get the number equal to one. This is what we are told to believe: that something that is infinitely high when multiplied by its width which is infinity small gives an area of one. (Recall area equals width times height.) This is not derived from any cause-and-effect relationship nor any rational line of logic but is presented as a pseudo-function to be accepted and believed and indeed, so it has been. Here is this Providential impetuousness of Dirac!
The Delta as a Metaphor of the Incarnation
Of what use is this apparently useless mathematical figment to us? Jesus can be seen in this mysterious entity as presenting Himself to us on the timeline of humanity as positioned at the zero focal point as a person extending from zero in this worldly time upwards to infinity, that is, back to the oneness of God, specifically of the Father. The zero time is the Incarnation, the Conception of Christ in the Virgin Mary; mysterious, not comprehensible on human terms but really no more so or less so than Dirac’s function. This Delta Impulse reveals in one brilliant manner the uniqueness of this God-Man, existing both on earth with a human existence as well as extending to infinity which is the Triune God, expressing the Divine existence.
We should immediately and forcefully state that this not to reduce our Lord to a line. It is simply to demonstrate from a new perspective the uniqueness and power of the divinity and personhood of Jesus Christ. The Dirac Delta Impulse is then a metaphor for the indescribable reality of God made Man in the Incarnation. We can then say: “The Lord is my Delta, my Infinity, my Infinite closeness!”. In terms of this functional beauty, this coming, this Jesus as God incarnated into the human race is linked infinitely to God, i.e., He never leaves his Trinity, his Divinity. He is also linked to his humanity in the infinitely small incarnational width of time with his presence among us on the human timeline. Yet his infinity to the Father and his infinity to us are bound together in a unity, in a oneness. There is no other person among the infinities on our human timeline that has these divine/human characteristics. This is truly the revelation of the Infinite in the Finite. Here is a sketch to illustrate this mysterious behavior.
Grace of God in Dirac
The marvelous grace of God, the inspiration of the Holy Spirit is revealed in this insight given to Paul Dirac. His intuition, his audaciousness, is marvelous and reflects the Beauty of the Trinity, the infinite Trinity enclosing finite area. We have already spoken of this marvel; that infinite strings of numbers can have finite sums.
Leiva-Merikakis is right then when he states that
in Jesus’ person and life, eternity and time truly intersect and coexist in perfect harmony[5].
Mouroux affirms that this intersection of times is a link back to the original creation of time:
The time of the world, now and forever, in every part and in its totality, is rooted, established, and measured by Jesus Christ….The entrance of the Eternal (Jesus) into time means that the Eternal has assumed time. Time has been recreated…[6]
The meaning of eternity and time intersecting is made clear through the Dirac function – Christ intersects time at t=0 and yet intersects that time with his infinite binding and relationship to the Father in the Trinity.
Intuition Seizes truth
This intuition of Dirac as with the other intuitions mentioned herein is not to be casually disregarded but has been the focus of much philosophical thought. Consider Bergson who in speaking about the motion of a pendulum says that when it is brought back to its “normal position: space, time and motion shrink to a mathematical point. Just so, human reasonings are drawn out into an endless chain, but are at once swallowed up in the truth seized by intuition…”[7]
Intuition seizes truth! Mathematics is shot through with intuition which can point to truth which surpasses reason! Christianity announces Truth, a reality that God exists that there is a Love that transcends this universe, and that Truth is intuitive to the very heart of every human. So, our deeply rooted intuition about the presence of a Being links us at the level of the gift of human intuition to the mathematics of intuition. We have already seen examples of this and will see much more yet.
How is it possible that the area which is given by width times height can be defined by a width of zero and a height of infinity? But this shines on us as for the first time a supernatural light that helps us to see that we can be loved and can speak with God, more specifically the Father who is infinitely beyond us and yet at the same time is infinitesimally close to us, so high and so close that the product is unity! This helps to further support Ratzinger’s certainty of how we can even speak with the Father: “I express my certainty that God, despite the infinite difference between him and me, is such that I can speak to him, may even address him familiarly as ‘thou’ [German du]”[8].
Lacordaire in his Conference on the Inner Life of Jesus Christ proclaimed:
“…at the same moment when He opens the Infinite to us by His look, He folds us in His arms…”[9]
Beauty! To open up the Infinite and enclose the Infinite infinitely close is the Incarnation of Christ and is an insight of truth that only God can reveal to Lacordaire and also to Dirac. This is Jesus Christ- at one and the same instant, at one and the same forever, the unique never repeated in any other being, union of the infinite and the finite. This strange upward pointing arrow towards infinity rooted at t=0 now has a new beauty, a new meaning that extends to the very heart of Creation and as we will see later extends throughout history.
Dirac, of course, never imagined this of his function and yet here God has revealed this truth of his infinite distance and closeness through a human who loved the beauty of mathematics but, as an atheist of sorts, presumably failed to see the Other, the “du” in his life. This is the Beauty of the Mathematics in Christianity.
The fullness of this Incarnation of Christ is revealed in the reality that by this event, the entirety of the Christian mystery is set into motion. The life, death on the Cross, Resurrection and sending forth of the Spirit in the Church all is contained in this Incarnation, this Delta input of the Infinite inserted into the finite. Here is its visualization.
Incarnation of Christ – Joining of Infinite and Finite
Absolute Zero Time
We now have the opportunity to set in front of us a meaningful time scale with a time of zero that is readily identified, a reference point, a kind of “absolute zero” has been established: the Incarnation moment of Christ into the earthly world. The negative time is counted backward into the past from the Incarnation, the Conception of Christ in the womb of Mary and the positive time into the future begins then as well. Recall that this mystery of this plunging of Christ into our world is into an infinitesimally small time interval of zero so that both negative time and positive time begin at that point.
The timeline into which the Incarnation is inserted now is seen to be populated from some distant past up to the entrance of Christ in this strange Incarnational Delta insertion. God reveals Himself to a people who have largely lived and died and a small number of people who live at the time of His appearance in Jesus Christ. So, then this line of time stretching over multiple tens of thousands of years is drenched with billions of conceptive zero starting points. It is crucial to see however that this timeline of human conceptions does not stop at the “now” but extends towards an infinity. This infinity will also have to be defined which we will do later but what we can anticipate now is a definitive end of time given by the new presence of the returning Christ.
The Beginning of Time Not Relevant
The placement of the first human, named Adam, is also now referenced to this zero starting point. What one sees then is a line of individual beginnings, not just a collective beginning, but unique beginnings extending from a distant past to a distant future but referenced to this time as a unique whole. There is no need then to endlessly debate the beginnings of the human race since it is irrelevant here. It is also now clear that the line of time considered here is the time of the earth and its humans. It is not the timeline of the universe so there is also no need to debate the beginning (or end) of that universe and its expansiveness.
We are also fully aware of the criticism of Ratzinger of Cullman’s timeline as being devoid of a sense of history, of drama, of suffering, forgiveness, of all that makes for human living.[10] Timelines are by nature “lines”, an infinite series of points which only apparently seem to have no heart or warmth. Cullman responded by making his line wavy! But it is precisely in these infinity of points that individuality is expressed. For God, each point = each life, is so unique, so unlike any of the other points, that the love of the Divine can penetrate directly into the point dimension of the line and “see” what we cannot see.
We can now see again that the issue of when time began is not of relevance here. Rather our interest is focused on the reality that at some time in the past, a time that is referenced from Christ’s Incarnational event, the hand of God touched a creature and created a human being. WE have previously referred to this “touching” as the communication of God, the acting of God in moving from the Existent to the Created. The creation is of a person, a being, a human who was in the mind of God and is now made visible and begins life at a zero time.
Then it is as if, the billions before and after the Anno Domini turn towards this Man, this God who appears among them. Some do not see due to a veil of individual transparency, others see and turn away in a rejection of the sight and understanding of this Love and then there are those whose very beings reflect the Light, a reflection rooted in the grasping of the Love presented, a Love that calls forth from each person a life of giving of one’s self to the other in fulfillment of their mission.
[1] Von Balthasar. Mysterium Paschale. pg. 20
[2] John Paul II. Redemptor Hominis. I.1.
[3] Dirac, Paul. The Relationship Between Mathematics and Physics. pp.112–129.Lecture delivered on the presentation of the James Scott Prize, 6th February, 1939.
[4] As quoted in Pagels, Heinz R. The Cosmic Code: Quantum Physics As The Language Of Nature. Pg. 295.
[5] Leiva-Merikakis. Fire of Mercy. V.2, pg. 369.
[6] Mouroux, J.The Mystery of Time. A Theological Inquiry. Pgs.99, 100.
[7] Bergson, Henri. Creative Evolution. Loc. 4591
[8] Ratzinger, Co-Workers of the Truth, pg. 103
[9] Lacordaire, P. Jesus Christ, God, God and Man, pg. 16.
[10] Ratzinger, J. Principles of Catholic Theology. p. 188.
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