by Rev. Richard S. Kirby, Ph.D.

After reading this essay, join its discussion with Dr. Kirby. Go to the Future of Religion forum, where a special essay page has been set up.

I WAS A MATH PART-MAJOR IN HIGH SCHOOL and have retained a love for the subject of pure math, philosophy of math...and so to my topic: theology of math: a natural counterpart to theology of science. I do think it's an important, though admittedly difficult subject. But we have to do it! Perhaps summer is a good time for such a rarefied field.

"God and the future of mathematics" -- surely a subject of small interest, or perhaps a subject of zero subject matter, like exobiology or [it is jocularly said] financial ethics: "sciences without a subject matter."

In this series of articles I have written about many areas of 'Tomorrow,' such as Animals, Children, Seniors, Language. Our subject this month is more easily comprehensible if we considered that we were inquiring first of all into the future of education, secondly into the future of mathematics education, and thirdly into the future of mathematics as a social phenomenon, that is, into the future of mathematics as a humanities subject.

If, therefore, we rephrase our subject, "God and the future of mathematics," as an inquiry into the education of the professional mathematicians of tomorrow, we penetrate some way into the field of the future of mathematics. This paper was first presented at King's College, University of London on February 16th 1988.

Only if we see the field of math as one of 'objectivity', beyond values, clean, crisp, clear and value-free, could we imagine that God and math have nothing to do with each other. The fact is, it has been known for over a hundred years that there is no such value-free mathematics! Mathematicians such as Riemann [whose math was used by Einstein], Lobachevksy [non-Euclidean geometry], Russell and Whitehead, and more recently Morris Kline have shown us that math, like any science, is at its highest level the play of free concepts in the higher reaches of human intelligence.

The future math, like the past, cannot be value free. It has always been a radically human activity. It serves human purposes. It may be rigorous, but its objectivity is subject to human values, including the [evolving] ideology of 'objectivity.' It also one of the most vibrant areas for human creativity, as new fields emerge in areas such as vector analysis, topology, calculus, set theory, and many other fields.

The study of the principles and the nature of mathematics is known as the philosophy of mathematics, or sometimes mathematical philosophy. Where there is a philosophy, a theology is never far away. But theology deals not so much with wisdom as with redemption. Math is conducted by the human mind, which has experienced a fall away from highest truth. In the following article, which was first presented in a graduate seminar in the Theology department of King's College London about twelve years, I begin to inquire into the issues which will a rise for those who pursue the study of the redemption [regeneration] of science and mathematics. If it is read with this in mind, I hope our readers and scholars will enjoy it and see new fields of mental enjoyment [and acrobatics] opening up. Thus, here we make an opening move in the emerging discussion of 'theology of mathematics', or mathematical theology.

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THEOLOGY OF MATHEMATICS

by Richard Kirby

I - Introduction: Purposes

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

A. PRACTICE

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.

B. THEORY

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 conclusion 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.

Bibliography

Kirby, R. S.; "God and the 'Star Wars' Research Program - a Theological Investigation in Three Parts - Part II: Towards a Systematic Theology of Science and Technology"; Visions; Volume 4, no. 2; 1986

Kline, Morris; Mathematics: The Decline of Certainty; Oxford University Press; New York; 1980.

Kuhn, T. S.; The Structure of Scientific Revolutions; University of Chicago Press; Chicago; 1962.

Lonergan, B. A. W.; Insight; Darton, Longman and Todd; London; 1958.

Torrance, T. F.; Theological Science; Oxford University Press; New York; 1969.

Copyright 1988 - 2000 by Richard Kirby.

Dr. Richard Kirby, International Chairperson of the World Network of Religious Futurists, welcomes your collaboration in shaping tomorrow's theology of mathematics. Your counsel and comments are invited. To share your response to this essay, go to the Future of Religion forum, where a special essay page has been set up.