# From the Total Gift of Self to a New Relational View of Reality; From a Mystical Insight to the Foundations of Mathematics: A Transdisciplinary Approach

The paper that we present is the fruit of the efforts of an interdisciplinary team of mathematicians: interdisciplinary within a common framework, in the sense of being mathematicians specialized in specific areas, such as logic, statistics, number analysis, mathematical physics, etc. What joins us together – besides a common general field of endeavour – is a shared spirituality based on the profound principles at the basis of a worldwide Christian Movement, the Focolare Movement^{1} which takes its inspiration from what has been called the charism of unity. As mathematicians motivated by the conviction that there is a deep relation between the material dimension and the spiritual, we have attempted to convert relational patterns experienced in spiritual life into abstract formal terms.

Concomitantly, groups of academics in a number of other areas – from philosophy to theology, natural science to economics, psychology to law – have been working to express the same deep-seated insights regarding unity according to the methods and using the terminology of their specific fields of study. The first part of our paper presents some brief theological input regarding the mystery of Christ forsaken on the cross, a fundamental theme underlying the discourse on unity, and which has particular bearing on the theme of this Congress, revealing that"I am myself…when out of love I am lost in the other.” The second part describes an attempt to render these contents – the particular pattern of relation perceived therein – in a purely formal language, within an axiomatic framework typical of our discipline. The pattern that emerges can, we maintain, contribute in a general sense to enunciating a relational view of reality.

1. Brief theological input

Christ dying on the cross is the personification of total giving. In what Paul calls *kenosis,* an emptying, he pours out his life, human and divine (cf. Phil. 2: 6-7), to the very last drop. The total annihilation of the Son of God, to the point of no longer feeling His oneness with the Father, is expressed in His cry:"My God, my God why have you forsaken me” (Mt. 27:46).

The depth and mystery of this event, central to Christianity, is of course too vast to give justice to in this particular paper, which is not meant to be theological. It is a topic which has been studied in depth by contemporary theologians and scholars such as H. Urs Von Balthasar, J. Moltmann, and S. Bulgakov, to name only a few.^{2} In the significant words of the latter, he places the abandonment of Christ on the cross into the context of the eternal *kenosis* of the Trinity :"The *kenosis *of the Divinity is so profound that before the God-Man there opens up the abyss of death with the darkness of non-being, with all the intensity of the abandonment of God… The cry from the cross: ‘Elì, Elì, lama sabacthàni, is the extreme point of the extenuation of the Divinity in the annihilation of the crucifixion.’”^{3}

In addition to theologians, religious leaders and mystics have reflected on this same event in its relationship to humanity, and saintly people have reflected something of this fathomless mystery in their own lives. We can mention two prominent figures of our times. First, Mother Teresa of Calcutta, who as a recent publication of her private writings reveals, during fifty years of a complete darkness of faith, a veritable eclipse of God, experienced an anguish similar to that of Christ abandoned on the cross.^{4} Second, Chiara Lubich who through what has been recognized as an ecclesial charism came to understand and explicate the significance of the event of Christ’s abandonment in the"economy of unity,” ie. in the perspective of the fullness of communion, both between God and man and in human interpersonal relations. In the following section we will present briefly some of Chiara Lubich’s insights on this theme.

1.1 "Jesus forsaken” and Christian oneness

In Lubich’s descriptions one can find in the event of Christ’s abandonment, which she calls"Jesus forsaken,” the key to a radical form of interrelatedness that could be termed"dynamic oneness,” that lies at the heart of Christian anthropology and at the same time evokes the inner life of the Trinity.

We could sum up the essential content of an authentic Christian anthropology in a few Gospel passages:"As the Father has loved me I have loved you… Love one another as I have loved you (Jn 15:9.12).”"May they be one as I in you and you in me” (Gv 17:11). As Lubich comments:"When we live the new commandment, seeking to receive the gift of unity in Jesus that comes to us from the Father, the life of the Trinity is no longer lived only in the interior life of the individual person, but it flows freely among the members of the Mystical Body of Christ.”^{5}

This perspective of the fullness of the Christian identity as being"one in many” is at the heart of Lubich’s understanding of what she calls"a spirituality of communion.”^{6} And over and over again as an underlying theme she repeats that unity and Jesus forsaken"are two sides of the same coin.”^{7}

In particular, the commandment of love is"lived out and measured against Jesus’ love for us, to the point of abandonment.”^{8} He who was God emptied himself on the cross. This event illustrates an essential attitude necessary for oneness among human beings, generated by being open to others, making room for others, even to the point of emptying oneself completely."To welcome into ourselves the one who is everything, we must, like Jesus forsaken, become nothing … We must be nothing (Jesus forsaken) in the presence of each of our neighbors in order to embrace Jesus in him or her.”^{9}

Reflecting on the mysterious cry that Jesus addressed to the Father before dying,"My God, my God why have you forsaken me?” Lubich explains:"There may be those who think that to affirm self is to struggle against all that is not self, because what is not self is perceived as limit and, what is more, as a threat to the integrity of the self. But Jesus forsaken, in that terrible moment of his passion, tells us that while the awareness of his subjectivity appears to be diminishing because it seems he is being annulled, in that very moment it *is* in all its fullness.”^{10}

Based on this example, she draws out important implications for the philosophy of being:"Jesus forsaken shows us, by his being reduced to nothing, accepted out of love for the Father to whom he re-abandons himself (‘Into your hands I commend my spirit,’ Lk*.* 23:46), that I am myself not when I close myself off from the other, but when I give myself, when out of love I am lost in the other.”^{11} In this description we find the dynamics of relationality that is at the basis of the true fulfilment of the human person.

1.2 Jesus forsaken and the inner dynamics of the Trinity

Moreover, in his *kenosis* Christ reveals the interpersonal dynamics that lies at the very heart of the Trinity. We have already quoted Bulgakov who has written profound thoughts on this topic. In considering the mysterious relationship between the abandonment of Jesus on the cross and the Trinity, H. Urs Von Balthasar, wrote:"The gesture with which the Father expresses and gives away His entire divinity (a gesture which He not only accomplishes, but"is”), in which he generates the Son as Other than self, this gesture is inevitably at the same time the eternal necessary condition for and the triumph over every shade of division, suffering and alienation in this world….”^{12}

Lubich expounds on the same theme extending it to the three Persons of the Trinity, in as much as it is intrinsically connected to the concept of love:"Three form the Trinity, yet they are one because love *is* and *is not* at the same time.”^{13}"In the relationship of the three divine Persons, each one, being Love, *is* completely by *not being*, each one mutually indwelling in an eternal self-giving.”^{14} In God, love is identical with being and simultaneously with the self-emptying that is a total gift of self. The *kenosis *of Christ on the cross is thus the visible emblem of the eternal mutual giving and receiving of love which is the very essence of the Trinity. In the words of Moltmann:"The ‘boundless sacrifice of love’ that the Son reveals on Golgotha, is implicitly present from all eternity in the exchange of substantial love which constitutes the divine life of the Trinity.”^{15}

2. From a Christian mystical insight to the foundations of mathematics

From this perspective, reflecting as mathematicians, we felt challenged to describe, using a language and method proper to the foundations of our discipline, the abstract pattern of relatedness that we discerned in the theological discourse outlined above. And since as mathematicians we portray patterns of relatedness empty of all contingent details, it follows that our results could also be interpreted as describing an ontology.^{16}

2.1 Introductory remarks

Two introductory remarks may be helpful. The first concerns the general question as to whether it is proper to consider a spiritual topic as a source of inspiration for a field as far removed from theology and spirituality as mathematics. This same question could be posed from the opposite point of view: has mathematics anything to offer to theology? For example, can we draw profit from attempting to employ the language of mathematics to describe the dynamic theological pattern outlined above, which goes beyond the realm of empirical science, and can of course never be fully encapsulated in human expressions?

Let it suffice here to quote the renowned Italian mathematician Ennio De Giorgi, recipient of the prestigious Wolf Award in 1990. Deeply convinced of the possibility of a fruitful interaction among all the various branches of knowledge, which he considered to be commonly rooted in divine Wisdom, De Giorgi wrote:"Every branch of knowledge can be a source of inspiration for mathematics and vice versa. Mathematics can offer a valid contribution to every other branch of knowledge both towards understanding things which might otherwise be incomprehensible, and towards expressing with clarity intuitions which might otherwise be inexpressible.”^{17} It is worth noting that elsewhere in his writings De Giorgi explicitly places theology in the arena of fields of knowledge relevant to mathematics. More generally Howell e Bradley in"Mathematics in a Postmodern Age”, describe various ways in which a Christian prospective can enrich our understanding of mathematics, and vice versa, ideas from mathematics can increase our understanding of matters of faith.^{18}

A second premise concerns the very object of our discipline which has been a source of discussion among philosophers of mathematics through the years. Although the debate is still alive, an understanding has gradually emerged that, contrary to common opinion, the object of mathematics does not consist in the abstract entities such as points, lines, numbers, or sets in themselves, but rather in the systems of relations which determine these entities or by which they are connected.^{19} In this perspective the fundamental object of mathematical study concerns relations.

For example the Pythagorean theorem gives an explicit description of the relation between the hypotenuse and the sides of a right triangle, and this relation is generalized for more sophisticated entities, such as elements of kinds of spaces that have nothing to do with line segments or the angles formed by lines.

One of the principal activities of a mathematician thus consists in identifying and studying patterns, relational configurations, from the most simple to the most complex.

From a more dynamic point of view these relational patterns can describe a process, either a process that is evident in the real world, or an abstract process. In this perspective the concept of number can be considered as referring to an implicit process of counting, and simple geometrical entities refer back to processes like tracing a line, measuring, etc. .^{20}

In the section that follows, Jesus forsaken will be the object of our investigation not in His quality of being God (which would be a source of theological or philosophical considerations), but rather as a process or event indicating a relational pattern that a mathematician can attempt to describe by way of analogy in abstract terms. So we ask: what is the relational pattern indicated in the event-Jesus forsaken and how can it be depicted in the abstract language of mathematics? Moving beyond this descriptive phase, we will also try to understand the prospects for further research in our field opened to us by the event of Christ’s abandonment considered as a total gift of self.

2.2 Jesus forsaken the Point

Let us take as our point of departure a passage of Lubich’s mystical intuitions describing Jesus forsaken as"the Point.”

"

Love must be distilled, until it is nothing other than Holy Spirit. It is distilled by passing it through Jesus forsaken. Jesus forsaken is Nothingness, He is the point, and through the point (= Love reduced to the last extreme, having given away everything) can pass only the simplicity that is God: Love.”^{21}

The phrase is in the form of a direct metaphor. Just as a plane has two dimensions and a line one, a point is said to be of null dimension. With reference to the quality of a point of having null dimension, of being"nothing” in the sense of having no extension, Jesus forsaken is depicted as being the point. Since a point is of null dimension, the idea of an extreme distillation is rather straightforward: nothing having a higher dimension can be contained in it, nor – in a sense -"pass through”.

Any analogy however is always partial and incomplete, it is not an identity or isomorphism (structural identity). Also the analogy at hand of the event-Jesus forsaken as the point indicates something else, not inherent in the classical concept of point. The point represented by Jesus forsaken is in fact qualified by another phrase which indicates a dynamic property not expressible in Euclidean geometry. The event-Jesus forsaken coincides with the operation of self-emptying, of giving away everything, as the extreme expression of the Love that is God’s very essence. Going back to our original analogy, it is as if the point were to project itself outward, dissolving itself completely to give rise to another:"being and not being” at the same time, so to say.^{22}

If we want to use a formal expression to represent the sense of *kenosis*, the process of a total self-emptying gift, which is the essence of the event-Jesus forsaken, the idea of a fixed point will not do. A dynamic image is necessary. So if we wanted to use a point to represent this process, it would have to be a point that is dynamic, open and self-emptying, purely"direction from”.

In section 3 we shall try to formalize this pattern using a particular abstract expression, admitting of course that various other approaches could be used. Indeed the idea of"being and not being at the same time” poses a challenge to classical logic, with its principles of non-contradiction and of the excluded middle, which underlie our discipline, and there have been abundant attempts to formulate alternative forms of logic (cf. multi-valued logic, fuzzy logic, supervaluationistic logic, intuitionistic logic, paraconsistent logic), none of which however face the question from a purely relational point of view.^{23}

However before continuing on in this direction of a formal proposition, it will be helpful to review some simple ideas regarding the concept of point, as it has been handed down to us through the centuries; it is a concept not devoid of serious questions, still waiting for an adequate response.

2.3 Atom yes - Atom no?

We have seen that the prospect opened by the description of the event-Jesus forsaken offers elements for considering the possibility of defining a new kind of point, which we might depict as a dynamic point.

It is generally thought that a line is"made up of” its points. In modern terms a line is defined as a set of points satisfying certain conditions. Greek thought, however, was far from espousing this atomic view.

It is well known that mathematics as a deductive science began in ancient Greece, in close synergy with philosophical thought, which was growing up there around the same time. Most people in our western world think of a point in terms of the description which has come down to us through Euclid. In the first lines of his monumental work *Elements*, written in Alexandria around 300 a.C., he defines it as follows*:"A point is that which has no part.” *This would seem to refer to an atom.

In fact, it is often thought that for the Greek world the point represents a basic element, an atom, from which all the rest, lines, curves, surfaces, etc. are made. In effect, for Pythagoras and his School (VI century a.C.), until the crisis of their system, the concept of point was indeed"atomic” in this sense. However later, as an unsettling consequence of the Pythagorean theorem, they concluded that this interpretation could not be valid. If the points were"atoms” of a line, one would be able to express any length as a quotient of two whole numbers, since each line segment is considered as a simple sum of points. The Pythagoreans came to realize instead that the hypotenuse of a right triangle is not always commensurable with its sides. For example, the diagonal of a square with sides of measure one, has the measure Ö2, a quantity which cannot be expressed as the quotient of two whole numbers. It is a result which led to the collapse of the entire philosophical system of the Pythagoreans.

Following this fundamental discovery, men of thought in ancient Greece continued to discuss and reflect upon the not easily definable relationship between points and lines. The philosopher Zeno illustrates this indirectly with his famous paradoxes, showing that whether we consider space as"atomic”- made up of many points -, or as continuous – an infinitely dividable whole, we run into serious difficulties.

Aristotele, reflecting on the idea that"every continuous line can be divided into parts that are again always dividable into more parts,” ^{24} reaches the conclusion that a line"is not made up of points.”^{25}

On the other hand, in Euclid’s definition"a straight line is a line that lies evenly with the points on itself,”^{26}a rather enigmatic expression that ended up being interpreted through the ages as if the line were"constituted” of its points. This is the common conviction which has come down to our times.

In reality, as Giusti points out, in the historical context described above, this could not have been Euclid’s intention. As Aristotele clearly stated, a line cannot be thought of as being constituted of points. According to Giusti’s analysis, a solution to the question can be found in realizing that Euclid’s definitions do not refer to abstract entities but rather to concrete processes. The points Euclid speaks of in describing a straight line as a line that lies evenly with the points on itself are only the two fixed points, at the beginning and end of the act of stretching a string or rope from one location to another. What lies in between the two extremes is the line determined by the process of extension and is not thought of as"points belonging to the straight line.”^{27} Of course, in the straight line under consideration, other points besides the two extremes can subsequently be defined, or fixed, which in their turn determine the ends of sub-segments of the original segment. However, nothing precise can be said about what has not yet been determined or fixed.

2.4 An atom that splits

The mystery of the *continuum*, that a line seems to be made up of points and yet cannot be, continues to intrigue mathematicians, whose studies in this area often lead to surprising and counter-intuitive results. One of these will be the object of the application of our formal approach, as we shall see later.

Before proceeding to our axiomatic formulation, however, it will be well to consider again and more deeply the pattern of relation expressed in the event-Jesus forsaken. A second source providing a different analogy, which opens new perspectives for our research, is Chiara Lubich’s description of Jesus forsaken as an Atom that splits, an undividable Oneness which in opening reveals an underlying trinity of relations.

"

Jesus forsaken is the divine Atom: a Oneness that splits, a nothingness that conceals God! Behind this emptiness we discovered the Trinity: it is a nothingness that is an open door to the Trinity.”

This passage leads us unexpectedly from the event of Jesus forsaken to the Trinity. What is being affirmed is that since Jesus forsaken is a tangible expression of the Love that is God, that is God’s very nature, then the underlying reality of this event sheds light on the mystery of God who is One and Triune, on the divine Essence in which"Oneness implies that each of the Three be truly nullity, a divine nothingness”, an emptiness of love.

As we mentioned above in the theological section of this paper, the pattern of relationality manifested in the event-Jesus forsaken seems suitable, by way of analogy, to describe the patterns of inter-relatedness within the Triune Godhead. We can think of it as being descriptive of God the Father, who in generating the Son out of love, goes completely out of himself, so to speak, making himself in a certain sense"non-being” out of love. It is also descriptive of the Son, who in returning as an echo to the Father, is a nothingness of love in Him, and of the Spirit who is pure relation between the Father and the Son, the emptiness of love in which Father and Son meet and are one.^{28}

In the passage about the"divine atom” quoted above, Jesus forsaken is described as nothingness but also as One, the Oneness. It is evident that the One or Oneness referred to is not the number one at the basis of arithmetic, nor the one that is the neutral element for multiplication, but the oneness of God’s essence which is also, paradoxically, the essence of Jesus forsaken. It is a one which emerges from the dynamism of three co-essential relations, each of which demonstrates the same basic pattern of relation revealed in the event-Jesus forsaken.

What we have been describing to this point has been on a theological rather than on a scientific level. However since relation is at the heart of mathematical research, the pattern which emerges opens our inquiry to the possibility of expressing the pure form of relation described heretofore in an abstract, formal language.

3. The event-Jesus forsaken as a model for a new form of relation

Setting all phenomenological and descriptive terms aside, we now turn to the axiomatic method proper to our discipline. The point of departure consists in the identification of one or more primitive concepts, from which an entire system can be derived using axioms to determine other elements. In our case, the entire edifice will be built on the primitive concept of relation. The intuitions that flow from the description of the event-Jesus forsaken outlined above prompted us to define a new type of relation (tr relation) capable of describing a total self giving in abstract terms. In other words, it permits a formal and non-contradictory way of defining an ontology of"being and non being at the same time.” We proceed from here to introduce a dynamic (non standard) identity (DIT – Dynamic Identity Triple) composed of three distinct co-essential tr relations.

What we have summed up above represents the subject matter of a recently published research paper in the field of mathematics’ foundations.^{29} Since it is a technical research paper, it is not necessary to go into all the details here, but we shall try to give a partial illustration in intuitive, non technical terms. Some of the central axioms have been transcribed in the Appendix for the interested reader.

3.1 The primitive concept of relation

We begin our discourse examining the primitive concept of relation. As mentioned above, mathematics can be rightly interpreted as the science of relations, in as much as it identifies, describes in abstract terms, and studies structures and processes.

To give a simple example, in Euclidean geometry a point, which indicates a *locus*, has meaning only in relation to something else, at least one other point. It is true that some recent theories in branches of mathematics called topology or mereotopology begin not from points, but rather from other primitive realities in terms of which points are then defined. However in all these theories, the quality of"relation” remains essential. Whether one begins from single points to define geometry or defines points beginning from some other primitive, one cannot do without relations.^{30}

So our attempt to describe in formal terms the underlying relational pattern of the event-Jesus forsaken leads us to examine first of all the primitive concept of relation.

For a mathematician or logician, the concept of relation is reduced to minimum bare essentials. He or she does not discuss relations of friendship, conflict or love, as in humanistic sciences, nor a relation of attraction between bodies, or relative motion, as in physics. Rather what is examined by a logician is relation stripped of every particularity, in its nude, most fundamental reality.

Indeed in logic or in the foundations of mathematics the concept of relation is generally considered to be a primitive concept, remaining undefined. For Aristotle relation is one of the primitive categories of substance. Intuitively, when we speak of relation it is implied that we are speaking of at least two objects that are mutually referential. In fact, the primitive idea of relation presupposes two objects or entities upon which we fix our attention and a joining entity, ie. a relation between them termed binary relation. Without difficulty, from this point of departure, we can extend the concept to ternary relations, quaternary relations, etc.

Ennio De Giorgi et al. in a PrePrint entitled"Towards a system of axioms for 2000 in Mathematics, Logic, and Computer Science” presents"a first attempt of an axiomatic foundation very simple, clear and ‘natural’, on which it is hoped can be engrafted the various branches of mathematics mentioned in the title as well as some other fundamental concepts of other scientific and humanistic studies.”^{31} This article was of great interest for our research not only because it aims at a possible dialogue among the various disciplines, but especially because it is based on the two primitive ideas of relation and quality. In particular"relation” is a key concept for our purpose.

3.2 Primary relations

In De Giorgi’s formal system for any given relation (binary, ternary, quaternary) there is a corresponding Fundamental Relation which describes the way in which the given relation is joined to the entities related. He notes for example that for a binary relation, in considering the system of"binary relation + 2 entities related” we have a system of three entities and therefore corresponding Fundamental Relation is ternary. Similarly the Fundamental Relation which joins a ternary relation to the three entities related is quaternary, and so forth. Generalizing, the Fundamental Relation is always of one order higher than the relation considered.

Moving in the opposite direction, it would seem natural to call primary (or unary) relation a relation whose corresponding Fundamental Relation is binary. Although the concept of primary relation is not intuitive, the basic definition is quite straightforward, and has appeared in the literature.^{32} It can be shown that the notion of"quality” (or subset) corresponds to a primary o unary relation. Also the classical identity relation is shown to be appropriately described as a primary relation.^{33}

3.3 tr relations and the Dynamic Identity Triple

Using the concept of primary relation as a basis we were able to define in axiomatic form a specific relational form useful for our purpose of describing the kind of pattern we discerned in the event-Jesus forsaken described above, which could be portrayed as a total self out-pouring. We called this new relational form *tr relation*.

A tr relation s a relation whose co

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