This paper was originally presented to the Senior Seminar, Faculty of Theology and Religious Studies, King's College, University of London, February 16th, 1988, under the title, "Theology of Mathematics: The Emerging Field of Theological Investigation." Richard Kirby was representing the Order of the Academy of Christ.
I. Introduction: Purposes
II. Crises in Mathematics: Practice and Theory
(i) Applied mathematics (ii) Pure mathematics
III. A Theological response to contemporary crises in mathematics
IV. THEOLOGY OF MATHEMATICS: A THEOLOGICAL/SCIENTIFIC INITIATIVE ON FOUR FRONTS
1. Theology of Mathematics
a. Mission/Research fields
b. A resource for the Mission of the Church
2. Theology of mathematical- science
3. Mathematical theology
(ii) Mathematical theology and the life of the Church
a. Cosmology b. Mathematical philosophy c. World Mission Theology d. Pure mathematics
4. Theological mathematics
(ii) Theological mathematics and the life of the Church
5. Collective Intelligence
V. ESCHATOLOGY AND MATHEMATICAL SCIENCE
VI: CONCLUSION: Towards a redemptive encounter of theology with Mathematics.
I would like to express my thanks to Professor Gunton for the opportunity to present this paper. I am also grateful to the Faculty of Theology and Religious Studies, and to the members of the Department of Christian Doctrine and History, for their hospitality within Professor Gunton's Senior Seminar. My thanks to the participants in this seminar for their kind attention to the doctrines presented in this paper.
I would also like to express my gratitude to Dr. Parker Rossman for his companionship along the rather unfamiliar path of the systematic theology of computer science and artificial intelligence. Bishop Lesslie Newbigin and Professor T. F. Torrance have helped me clarify my ideas about missiology and theological science respectively, and it is a pleasure to acknowledge their prodigious advancement of these fields.
However, responsibility for the content of this paper remains solely mine.
I - INTRODUCTION: PURPOSES
I would like to begin by explaining my hopes in presenting and writing it. First of all, my hope is to present a survey, with some synthesizing principles, of a number of important new ideas and research fields in the Anglo-American world, especially at those points where theology and mathematics share common boundaries.
If we imagine the geography of ideas as similar to a world map, we can portray theology and mathematics as having a long, though irregular, common border: similar to (say) the Sino-Soviet border. At the present time (1988), it is the field of computer science which occupies the largest part of that common border. Computer science, and its companion discipline of information technology, has its own frontiers, in both theory (concept, idea) and practice. The notions of supercomputer and artificial intelligence, and more remotely, robotics science, occupy much of the high ground in the frontiers of mathematical computer science. In mentioning these concepts, it is not too early to make the important point that the contemplation of mathematical science is in part the study of prevalent mathematical ideas.
It is important to remind ourselves that mathematics, in addition to being (allegedly) an "exact science", is also a field of discussion and an intellectual world in which the free play of novel and experimental concepts takes place within what may be called Wittgensteinian "language-games". In other words, the advancement of mathematics occurs through the discussion of mathematical ideas and heir attendant "paradigms" as well as through the actual deployment of mathematical instruments of thought in such fields as ballistics and astrophysics.
In presenting this survey, it will be my responsibility to draw together many strands of research and of mission, as well as to mention some recent viewpoints in mathematical and theological method. Some of the names which I mention will be well known to you; others less so. However, all the protagonists in the following story deserve to be considered leaders in the advancement of true religion in the field of mathematical science.
My second hope is to give some articulation to a new field of theological inquiry: the systematic theology of mathematical science. This is not offered as a primarily theoretical exercise, nor as a hope, but as an attempt to provide a coherent paradigm, an intellectual canopy, for an already ongoing international programme of research, action, communication and technology.
Adopting the terminology of Bishop Lesslie Newbigin, we could say that part of what is described here is a missionary encounter of the Gospel with a central element of our culture: the science of mathematics and all its resultant technologies. So my second hope, then, is to define and inaugurate, on behalf of my colleagues, a programme of action by the Church (for all missionary action is the action of the Church) in relation to the foundations and frontiers of mathematical science.
If one were to ask why such a programme is necessary, the answer would be, "Because of the crises in mathematics and its technologies" - crises to be documented below. This paper - this programme of action - is possible because of the juxtaposition in time of two phenomena; the collapse of certainty in the philosophy of mathematics and in advanced computer research; and the appearance of new instruments of theological research and collaboration. In reporting to you these two phenomena, I hope to show that the new theological instruments of mission and community entitle us to postulate and initiate the emergent field, the systematic theology of mathematical science.
Thirdly, and finally, my hope is to draw attention to some events at the boundaries of theology itself, and of its instrumentalities in the domain of theological method. For example, we can discern the emergence of a "theological mathematics" to complement the "Theological Science" of which Professor T. F. Torrance has written (see Torrance, 1969).
II - CRISES IN MATHEMATICS: PRACTICE AND THEORY
To honour the principle of God's Incarnation in this world, I begin with mention of two news items from the frontiers of mathematical inquiry and the philosophy of mathematics. One is from the USA, one from the UK; one pertains to pure mathematics, the other to applied mathematics.
I) Applied mathematics.
First, the USA. A few weeks ago, Cray Supercomputers, Inc. suffered a debilitating loss in their research programme: Steve Chen, their director of research, resigned. The result, from the company's viewpoint, was near-catastrophic: the company shares fell on the stock market, and I believe that the entire Dow-Jones index fell as a result. Reasons for Mr. Chen's departure are not entirely clear, but one reason appears to be that the progress in supercomputer research had slowed down; it had encountered unexpected difficulties. At the high frontier of applied mathematics, a sudden loss of confidence had occurred.
ii) Pure mathematics.
Second, a little news item from the U.K. The Times, for February 8th, 1998, reports, "Baker's top maths adviser resigns", and continues: "The Government's plans for raising the standards of mathematics teaching in schools were thrown into disarray yesterday by the resignation of Professor Sigbert Prais, its most prominent supporter on the national curriculum working group... The 14 members of the group were appointed last July by Mr. Kenneth Baker, Secretary of State for Education and Science. Their task was to redefine how mathematics should be taught in the light of the growing evidence that British schoolchildren of average and below average ability were falling behind their counterparts in West Germany and Japan".
Here we see in the more basic levels of education in principles of pure mathematics, some confusion concerning the desired "redefinition of how mathematics should be taught".
I would like to suggest that these two apparently isolated instances are part of a much more widespread malaise in the entire field of mathematics. According to one of its most eminent philosophers and historians - an apologist for mathematics - mathematics is in crisis: the philosophy of mathematics has come close to collapsing.
Morris Kline, the well-known historian of mathematics, has been emeritus Professor of Mathematics, and is associated with the Courant institute of Mathematical Sciences, within New York University.
In 1980, a fascinating treatise appeared under his authorship, published by Oxford University Press. Its title was simply MATHEMATICS; but it had a sub-title also: The Decline of Certainty. One reader of this work described it as a history of mathematics written as a tragedy.
In this work of history of mathematics, Professor Kline focuses on the modern period; more than two thirds of his study is dedicated to the examination of the modern history of mathematics, from Newton to Poincare. Following a review of the origins of mathematical inquiry in ancient times, he records what he calls the "Mathematization of science", and then goes on in successive sections to speak of what he terms "The First Debacle", "The Illogical Development", "The New Crisis of Reason", "Disasters", and the "Isolation of Mathematics". The penultimate chapter is termed, "Whither Mathematics?"
To a theologian this "strange, eventful history" reads rather oddly. Theology is not even mentioned in the index of the book, and Fr. Lonergan's monumental study, "Insight", is conspicuous by its absence. Such instruments of thought and inquiry as prayer, meditation and liturgy are also absent from overt consideration. Nevertheless, the theological aspect appears at the very end of the story of mathematics, in a very vivid way. For Professor Kline concludes his dissertation by writing of mathematics:
"...though it is discomfiting to have to grant that its foundations are not secure, it is still the most precious jewel of the human mind and must be treasured and husbanded. It has been the van of reason and no doubt will continue to be even if new flaws are discovered by more searching scrutiny. Alfred North Whitehead once wrote 'Let us grant that the pursuit of mathematics is a divine madness of the human spirit.' Madness, perhaps, but surely divine." (Kline, 1980; page 354)
III - A THEOLOGICAL RESPONSE TO CONTEMPORARY CRISES IN MATHEMATICS
Several elements in Professor Kline's conclusion, taken collectively, constitute an invitation to theologians to begin a systematic encounter with the foundations and frontiers of mathematical inquiry, mathematical science, mathematical philosophy and philosophy of mathematics. These elements are, first, the alleged divinity in "the pursuit of mathematics"; second, the loss of security in the foundations of mathematics, and the sense which mathematicians have that they do not - to paraphrase Bertrand Russell - know where they are going, or why, or what they will do when they get there; and thirdly, the concept that mathematics is "the most precious jewel" of the human mind. This encounter with both the foundations and the frontiers of mathematical science is beginning to be known as the theology of mathematics, with some similar phrases competing at present for the status of working title in an intensifying field of research and inquiry: theology of mathematical inquiry, theology of mathematical science, systematic theology of mathematical science, etc.
Whence the phrase "theology of mathematics"? It is a natural continuation of the line of terms which have grown up over the last hundred years in relation to science itself. Thus we speak of "philosophy of science" (Kuhn, 1962 etc.), "sociology of science", "politics of science", and even "psychology of science". In the same way that we can now speak of "theology of science" (see Kirby, 1986), we can accordingly speak of "theology of mathematics" as an emerging field of research being undertaken by theologians, philosophers, scientists and mathematicians around the world. If we ask what kind of definition is employed in speaking of theology of science or mathematics, we could respond "a stipulative definition". Therefore, the concept is a kind of "performative utterance" in J. L. Austin's sense, but the definition can and should be progressively refined by its users.
IV: THEOLOGY OF MATHEMATICS: A SCIENTIFIC INITIATIVE ON FIVE FRONTS
At present, we can summarize the international research frontiers in this domain by using the following five divisions of subject matter; 1, "Theology of mathematics"
a. Mission/Research fields
"Theology of mathematics" is the more general term for all fields of investigation, by theologians, of:
a. all past, present or future branches of mathematics;
b. the psychology and sociology of mathematics including formal and informal conceptions of mathematical method;
c. the twin fields of mathematical philosophy and philosophy of mathematics;
d. the principles and practice of mathematics education;
e. the objectives, methods and results of all past, present and future branches of applied mathematics.
Although this sounds both technical and broadly described, it can be seen to have simple and immediate meaning for the life of the Church. Among other things, it implies the study and development of such initiatives as:
a. models of holiness for mathematical inquiry;
b. conceptions of ethics in applied mathematical science;
c. examination of the value of mathematical instruments - present and future- in the mission of the Church;
d. principles in the pastoral care of mathematicians;
e. ecclesiological and liturgical consideration of the circumstances under which mathematical creativity can occur in collectives such as groups or churches;
f. the examination of truth-claims and epistemological assumptions in published mathematical reasoning.
g. the development of a systematic historiography, and a consequent theological history of mathematical science, with an attendant derivation of guidelines for futurists studying desirable future developments in - for example -- computer science,
h. the formation of Christian "mathematical missionaries" who can carry out a "redemptive encounter", on behalf of the Church, with the frontiers of high technology such as artificial intelligence and supercomputers. (Such a group already exists; its mission is well under way.)
i. the employment by theologians of valuable mathematical concepts such as "Calculus", "Vec-or" and "Tensor", in their conceptual work and in communication of its significance for Church and world. (See Kirby, 1987).
j. the formation of sacred societies of mathematical researchers, cadres of Christian mathematicians, whose theology of community advances concurrently with their mathematical discoveries, in accordance with Professor Torrance's principle of "fluid dogmatics". (This would include computer scientists, and technologists of artificial intelligence, supercomputers and robotics as special cases of sacred societies of applied mathematicians.)
k. The development of joint research programmes shared in a University's mathematics, theology and computer science departments, linked in a "Wisdom Community" model (see Braxton, 1980) to the Magisterium or Episcopate, to Church Mission Boards, and, in the non-ecclesiastical realm, to governments and industrial leadership (including entrepreneurs.) In this way, mathematical creativity governed by the Holy Spirit can be allowed to result in new products which provide (in accordance with Dr, Beeby's hope) systematic alternatives to existing manifestations of applied mathematics (e.g. micro-computers and ICBMS.)
l: The development of a radically new alternative to relativity theory and quantum mechanics, based on an epistemology which is neither Platonist, nor Aristotelian nor even Whiteheadian in its metaphysical assumptions and categories, but which is truly Christian. (Such a Christocentric metaphysics has yet to be developed, of course, See Kirby, 1986)
b. Theology of Mathematics; A resource for the Mission of the Church
Not only are such initiatives the responsibility of the Church, they are also opportunities to enlarge and expand the missionary capability of the people of God, For example, the use of computers, and even supercomputers and robots, to locate and reach hungry people, in order to provide them in Christ's Name with with food and drink is entirely consistent with the ultimate moral values of the New Testament, and with the actual mission agenda of the twenty-fifth chapter of the Gospel according to St. Matthew. Thus we can see that the appearance of electronic computers and the dawning of so-called artificial intelligence constitutes a crisis for missiology. The lesson of computers for the Church is that mathematical science and a missiology must draw together; in their collaboration the moral. destiny of computer science can be discovered. And to speak of destiny is to draw attention to the eschatological dimension even of mathematical science - a dimension which cosmologists have made especially their own- (see Barrow Tipler, 1985).
Indeed, from the viewpoint of dogmatic theology, one can more generally establish as an axiom of the dogmatic theology of mathematics this assertion: those mathematical instruments and actions which do not deliberately advance the mission of the Church are guided by a principle or power other than the Holy Spirit.
This assertion must, of course understood be as an axiom in the systematic theology of mathematics, and not as a potential indictment of actual mathematicians or branches of mathematical science and technology. We are here engaged in what might be called - in the thought-forms of systematic theology - the prolegomena to a systematic theology of mathematical science.
Such a venture is consistent with the ambitions of contemporary missiologists, typified by the Rev. Dr. Dan Beeby. His British Council of Churches programme, "The Gospel and our Culture", is a response to Bishop Newbigin's call for a missionary encounter of the Gospel with postEnlightenment culture. Dr. Beeby calls for a "mission to the Western world view and its fruits", and calls this "perhaps the most important single undertaking in the world-wide Christian mission today."
Theology of mathematics includes "mission to mathematics", although of course theology of mathematics is a much wider field; it includes potentially all points of intersection of the values and life of mathematical practitioners and their cultural products with the multidimensional revelation of God in Christ and the Church.
Indeed, theology of mathematics must be entirely open-ended in its contemplation of mathematical science; for it may be that by the power of the Holy Spirit, theology of mathematical science can become a midwife for the birth of new branches of mathematical science, ones more closely harmonized with moral ultimate disclosed by God in Christ.
In fact, it is not too much to say that such mathematical creativity, invention or discovery is entirely to be expected - by a kind of theological equivalent of deductive reasoning - if the Church develops a systematic theology of mathematical productivity and a spirituality and pastoral theology for mathematical practitioners.
Mention of deductive reasoning invites a passing reference to logic. Logic itself is a science, sibling to mathematics - some would say progenitor or descendant (or both)! -- and much of what can be said of theology of mathematics can and should be said, mutatis mutandis, of logic. It will take another paper to begin to develop a systematic theology of logic. Such an enterprise will in part he in response to a call from Professor Torrance, for it is he who has called for the development of a true "Theologic". His analysis of received systems of logic is an essential element of theology of logic. (See Torrance, 1969.)
The combined enterprises of theology of mathematics and theology of logic can, we may hope, be the agents by which another of Professor Torrance's hopes is realized. For he has written that Karl Barth did not invent the "appropriate cognitive instrument or instruments through which we may bring to light and represent for ourselves the profound harmonies and symmetries of the divine grace in which is enshrined the inner logic of God's creative and redemptive operations in the universe.
Historical theology has never even come up with an instrument corresponding to the four-dimensional geometries of space and time which have played such an astonishing role in the advance of our scientific knowledge of the created universe. But this is a task of the future." (Torrance, 1984, page 282.) To adapt Professor Torrance's terminology, it may be that theology of mathematics should be regarded as a search - by the entire Body of Christ -- for a "Theomathematics" within or alongside such "Theo-logic". But such a neologism may be more of a hindrance than a catalyst to true discovery.
However, it is clear that at least a part of theology of mathematics must be the quest for such instruments of thought as that to which Professor Torrance alludes. Thus, theology of mathematics can be portrayed as an heuristic exercise by the whole Body of Christ. As such, the growth and evolution or development of theology of mathematics will entail what Professor Hollenweger of Selly Oak terms "liturgies of intellectual inquiry." It may be that the liturgiological dimension of mathematical inquiry will need to proceed concurrently with its Christology; we cannot know at the moment. For this is a matter within the technical domain of systematic theology itself. Similarly, it is too early to assess whether the theology of creation, redemption and sanctification should be studied in that order, or another order, or all at once, in relation to mathematics, logic, and computer science.
2. "Theology of mathematical science", is a subset of the more general term: it refers to the theological investigation of:
a. the aspirations of mathematics and its exponents to be counted as an independent science; and the place of mathematics among the sciences.
b. the realities of the scientific method as evidenced in learned mathematical discourses;
c. the metaphysical, epistemological, moral and eschatological validity of the ostensible and implicit assumptions of self-described "mathematical science";
d. the Christian moral excellence of the products of mathematical science and its attendant technologies,
e. the development of Christocentric alternatives to existing branches of mathematical science, and the development of technologies pursuant to such Christocentric alternatives.
"Theological investigation and theological method" are terms which must be elucidated in the context off the present programme. Here we must acknowledge our indebtedness to two great Jesuits, both deceased 1984. Karl Rahner's twenty-plus volume of theological investigations invite continuation in the fields of science and culture itself. Lonergan's Method in Theology yields an approach to theological science which in some ways complements that of Professor Torrance, Bishop Newbigin, Dr. Rossman and their colleagues. To speak of the theological investigation of mathematics is therefore to imply an a devout exploration, by theologians and other Christians, of the facts and possibilities of mathematical science.
However, we can expect theological method and theological science themselves to evolve under the pressure of such an encounter, A part of that evolution must be in the direction of mathematical theology as described above. A second part of that evolution must be towards a greater collegiality of theological research. Dr. Parker Rossman, formerly of Yale University, suggests the term "Collective Intelligence" as an appropriate epithet for theological research-in-community. At present, Collective Intelligence (a systematic Christian alternative to Artificial Intelligence as a scientific, mathematical, ecclesiastical and cultural goal) is being developed within church computer networks. But there is reason to suppose that the evolution of the concept of collective intelligence will proceed in other areas of theological and ecclesiastical activity, such as in liturgiology and ecclesiology. At the present time, the developers of the theory and practice of Collective Intelligence are testing a system of theology entitled Vector Theology. This is to be understood as a successor, in several ways, to Process Theology, for it is based upon a metaphysics of direction and power rather than of process. (See Kirby, 1987). (Also see section IV 5 below).
3. Mathematical theology
Mathematical theology can be defined as (a) that branch of theology which systematically employs mathematical instruments of thought in order to potentiate the mission of the Church. Alternatively, (b) mathematical theology can be considered as an approach to all branches of theology. It can therefore assist theologians in any or all of the traditional divisions of systematic, philosophical and dogmatic theology: fundamental theology, doctrinal theology, practical theology, etc,
(ii) Mathematical theology and the life of the Church
It is especially valuable to uphold the necessity of a systematic mathematical theology at the present time. The appearance of such concepts as "physical eschatology", "the intelligent universe", and "the anthropic cosmological principle" bespeak a need by cosmologists, astrophysicists and their colleagues to engage in truly learned, expert dialogue with theologians concerning the meaning of mathematical facts born of astronomical research. Such dialogue requires the appearance of a truly mathematical theology and indeed of a cadre of mathematical theologians. This in turn implies an agenda for mathematics education in Church and State,
b. MATHEMATICAL PHILOSOPHY
However, the potential contribution of mathematical theology to the Church is far wider than in the facilitation of dialogue with astronomers. More generally, mathematical theology facilitates the encounter of the Gospel, of Christ Jesus with the thought-worlds of mathematical philosophy and philosophy of mathematics. (See Russell, 1919.) It will be an essential instrument of thought in the moral advancement, even redemption, of computer science and technology. It can be regarded as an essential element of the conceptual instrumentalities of the mathematical dimension of missiology.
c. WORLD MISSION THEOLOGY
An example can be helpful here. On almost the simplest mathematical level, David Barrett's World Christian Encyclopaedia represents -- in the form of non-inferential, descriptive statistics - the mathematization of mission. Such mathematization is a beginning; it is the task of mathematical theology to advance from that starting point. The development of a truly experimental dimension to theological science also awaits the development of mathematical theology, especially in the form of mathematical instruments such as the inferential statistical tests employed by behavioural scientists. However, we cannot anticipate the instruments which mathematical theology will develop; for the creativity of mathematical theology must follow upon an obedient "waiting upon God", in order that He may reveal to us those instruments of thought which are most germane to His holy purposes for Church and world.
d. PURE MATHEMATICS
The codification of theology of mathematics represents, potentially or perhaps embryonically, a new beginning -in Christ - for arithmetic, geometry, mechanics, algebra, logic, computer science and all the branches on the tree of mathematical inquiry.
4. Theological mathematics
To achieve an adequate preliminary definition of "Theological mathematics", we must pause for a moment to consider some issues in the philosophy of language. Philosophers from Wittgenstein to Derrida, from Austin to Riceour, linguists from Chomsky to Lyons, have taught us the labyrinthine complexity of even the simplest phrases. To parse a sentence, or even a phrase, is to undertake a task of boundless complexity, as layer after layer of deep structure and cultural presuppositions are exposed.
Thus, in dealing with an apparently simple concept such as "theological mathematics", we must examine the possible syntax of the phrase.
a. The most obvious structure of the phrase would present "mathematics" as the noun, and "theological" as adjectival. Thus, the entire phrase denotes mathematics with a theological colouring or outlook,
b. If, however, we consider the sibling phrase "Theological Science" as deployed by Professor Torrance in his book of that name (Torrance, 1969), we can see that it is syntactically deceptive. What is actually implied is the reverse structure: it is really the (implied noun) Thology which is the nounal element, and the apparent noun, "mathematics" is really adjectival in function. The "Theological Science" described by Professor Torrance is as much "Scientific Theology" as it is "Theological Science", and this is equally true of "Theological Mathematics".
Therefore, "Theological mathematics" can be provisionally defined as:
(a) The use by the Church and its members of mathematical knowledge and methods in order to advance the Mission of the Church;
(b) (more colloquially) The mathematical facts, statistics, etc. which pertain to the mission of the Church.
(y) The mathematical elements - past, present and future - of theological method and theological science,
(S) (Syntactic type (b) above) Branches of mathematics, and instruments of mathematical science/art, which are being used by the Church and/or by theology, for specifically Christian purposes, or towards specifically Christian ends.
(ii) Theological mathematics and the life of the Church
The interaction of the four elements implied in these four definitional components lead to deduction about the role of "theological mathematics" in the life of the Church.
It is Theological Mathematics which will allow the global theological community to employ for Christian purposes the power of mathematical models set within the hypothetico-deductive method of experimental science. In this sense, Theological Mathematics is a kind of "archaeopteryx" by which Theological Science mutates into Experimental (Scientific) Theology, with branches in fields ranging from Ecclesiology to Homiletics. For example, in the field of missiology, Theological Mathematics permits descriptive statistics to mature into inferential statistics; models of mission can be tested and compared with exact mathematical instruments such as factor analysis.
5. Collective Intelligence
Parker Rossman's "Collective Intelligence" can be considered as - among other things - a branch of theology of mathematics. For it implies a collegiality of research in central mathematical fields such as intelligence and artificial intelligence, computer science, hypertext, set theory, mathematical groups, etc, Collective Intelligence is both a theory or model of research-in-community, and a subject to be investigated in its own right. Here theology of mathematics implies two research agendas for the Church:
a. The mathematization of the results and methods of Collective Intelligence, with the instruments of psychometrics;
b. The intersection of the conceptual apparatus of Collective Intelligence with that of Ecclesiology or, more generally, the doctrine of the People of God,
Collective Intelligence implies a concept of community for all branches of theology of mathematics: for it represents the Pentecostal experience (the forming of community from separate elements) within scientific research (including mathematical research) itself.
VI: ESCHATOLOGY AND MATHEMATICAL SCIENCE
Finally, we cannot conclude this survey without mentioning - as systematic theology must - a consideration of the meaning of eschatology for mathematical inquiry, science, and technology.
Eschatology and Mathematical Science
This combination sounds a little unusual, though not, I hope, Jarring. For the field of cosmology has recently seized upon the concept of eschatology and begun to deploy it in the interests of philosophical and mathematical cosmology. I refer, of course, to the well-known treatise "The Anthropic Cosmological Principle," and to the notion of "physical eschatology" of Freeman Dyson. To the theologian, the annexation of the concept of eschatology by cosmologists is a little bizarre: for eschatology is a moral concept. However, one may "counter" this move with the postulation of the equally novel field of mathematical eschatology, the contemplation of which yields some surprising possibilities for mathematical science.
Here - in a speculative mode of thought ._ we must be careful to study the syntax of the phrase "mathematical eschatology", for by it must be meant, not simply a branch of cosmology, but rather the study of those factors which pertain to the eschaton in mathematics!
In other words, the consideration of the possible end. termination or completion of mathematical science is the topic to be considered; and from the viewpoint of systematic theology this is a natural and logical completion of the study of the theology of mathematics.
To the theologian, eschatology is a science in its own right; it has exegetical, hermeneutical, cosmological, Christological, ecclesiological and cultural dimensions or departments. It ramifies into the study of history, or war and peace, and of apocalyptic. But it is a young theological science, and although it can in a tentative way be applied to problems of many kinds - not just "religious" problems! - in order to provide a clarification of thought and options, for the moment we must restrict ourselves to a few remarks about the meaning of eschatology for mathematics.
Eschatology -- the so-called science of the Last Things - deals with concepts - and one hopes realities - such as end, termination, completion, fulfilment, consummation, closure. We are entitled - as Christian believers that history has an end, a climax, a fulfillment - to inquire how this affects or is affected by mathematics. One possible answer is that computer science - governed by the Church - can so accelerate and potentiate the mission of the Church that the Matthean agenda for mission will actually be completed. Then, indeed, "the end will come," (Mat, 24;14). But to disentangle the apocalyptic from the missiological element in Biblical eschatology, and to set both in the context' of cosmological conceptions of "the end of time" is something remaining to be done by an inter--disciplinary group of scientists, New Testament scholars and theologians,
Or, perhaps, mathematical eschatology implies a consideration of the intervention of God in mathematics. This in turn leads us to consider the spirituality of mathematical research, and circumstances under which mathematicians can receive the Holy Spirit in the context of their work.
Lastly, relativity theory and quantum cosmology themselves intimate an agenda for mathematical eschatology; and upon it Barrow & Tipler, among others, have made a start. Their beginning invites us to a deep meditation on the ultimate boundaries and limits of metaphysics, or relativity theory, and of all the conceptual apparatus of astrophysics. Here indeed, at the very limits of time and space, we must draw even science fiction writers into our orbit. But this would lead us from mathematics to metaphysics, and this must await another lecture.
VI. CONCLUSION: TOWARDS A REDEMPTIVE ENCOUNTER OF THEOLOGY WITH MATHEMATICS
To sum up: whatever the peregrinations of theology of mathematics, its main result should be nothing less than the coming of the living God, in the power of the Holy Spirit, into the lives of mathematicians all over the world. And that advent guarantees a festival of mathematical creativity.
"Theology of mathematics", then, intimates the hope of the beginning of the systematic redemption of pure and applied mathematics and its attendant technologies with their cultural penumbrae.
A beginning is all it is: it is the hope of a science, or cluster of sciences, and their attendant technologies, in Christ. 1t is not even the end of the beginning, it is only the beginning of the beginning. But is what well begun in Christ is guaranteed a holy outcome so long as it walks in the way of the Spirit, along the path of ascetical theology, by faith and not by sight.
Our closing hope, therefore, must be for a systematic spirituality and concept of holy community for the nascent venture of mathematical theology. It is here that Dr. Rossman's concept of "Collective Intelligence" awaits a commission of ascetical theologians and liturgiologists to refine and develop his thought. In short, we must pray without ceasing that the aborning world community of sacred Christian mathematicians and logicians will be truly in Christ Jesus; that they will unceasingly seek the will of God for their work; and that, with the support of Church and State, they will, by the power of the Holy Spirit, continue, accelerate, and complete it to the greater glory of God the Father in heaven.
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