Thursday, November 19, 2015

Theology and Mathematics by W. H. KENT 1910


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"THE pure intellect usually exhibits to the full its astonishing capabilities, I think, only on two subjects—pure mathematics, which are its creation, and in which it legitimately claims absolute supremacy; and dogmatic theology, in which it submits contentedly to the only position allowed it on the field of morals and religion, the humble and dutiful subserviency to the spiritual nature."

These weighty words of Dr. William George Ward, in his once famous work The Ideal of a Christian Church (Chapter V., page 28), may be said to serve a two-fold purpose. For, while they bring before us very forcibly the high intellectual delights that may be found in the study of dogmatic theology, they remind us at the same time that the Catholic theologian must bring something else besides mere intellectual power to this sacred study. But, apart from its immediate purpose, the comparison has a curious interest for its own sake. And it may well set some readers wondering why these two sciences, which on this showing would seem to have much in common, are so seldom associated with one another in actual experience. The sciences themselves appear to move in different spheres, so that they never come into contact. There is thus none of the hostility which too often arises between theologians and professors of other sciences. But, on the other hand, there is no mutual help or friendly co-operation; and those who are masters in one of these high temples of knowledge very often know little and care less about the other.

To some extent, we suppose, this fact can be explained partly by the diversity of natural gifts and tastes and aptitudes, and partly by the exigencies of professional education. Without adopting to the full Dogberry's doctrine, that reading and writing win by nature, we may safely say that most children are born into the world with special fitness, or maybe unfitness, for certain lines of learning; and even at an early age the bent far science or literature may be plainly discernible.

This may be illustrated by the words of a French song on the boyhood of Napoleon:

A genoux, a genoux, au milieu de la classe, l'enfant mutin! Avec un cerveau en feu pour l'Algebra, et la glace pour le Latin.

The first stages of a schoolboy's education are rightly made on broad and general lines, for the mere rudiments both of letters and science may well be within the capacity of most children. But there will always be many who cannot go beyond a certain point, and in some direction they will soon reach their natural limit. Thus, even with those who have leisure and opportunity for a full development of their powers, and love learning for its own sake, many must fain content themselves with elementary mathematics, and cannot hope to reach the higher regions of this science. But, in any case, comparatively few are allowed their choice or opportunity of full development on their own lines. The great mass of men meant for active life have but small scope for intellectual culture of any kind. And those destined for some learned profession will soon have to specialize their studies at the expense of other fields of knowledge outside the province of their own profession.

It is true that in many cases time might be found or made for other studies. But most men need some other stimulus to study besides the love of knowledge for its own sake. For this reason theological and biblical learning is generally left to the clergy. And the study of higher mathematics will be confined, for the most part, to those who require it for their work in life, or for the purpose of an examination at the outset of a professional career. In the latter case it will generally be relinquished when once the object in view is achieved. In the same way students of theology, when they come to the parting of the ways in their educational course, not unnaturally take the more literary and classical line, which seems more closely connected with their own sacred science. And unless they happen to be schoolmasters or have some special gift and taste for mathematical studies, they will generally drop them and, sooner or later, lose the little they have learnt in their schooldays. In this way it may well be that accomplished theologians will be at a loss if called upon to discuss the metaphysics of the infinitesimal calculus, and many of them even may never have heard of any quaternions save those appointed to guard the Apostle in his captivity.

It would be unreasonable to complain of this dissociation of theological and mathematical studies. For, in the case of the generality of students, it is natural, not to say unavoidable. Many would only waste their time in attempting to combine the two studies. And in other cases there may be other branches of science more practically useful to the theologian. Yet here as elsewhere it may be that the division of labor in the field of science has been carried a step too far. And, as theology stands to gain from the wider culture of its professors, it might surely be an advantage if those who have a natural taste and capacity for mathematics were to cultivate this branch of study and note its analogy with their own sacred science. The old Schoolmen, it may be remembered, conceived of all the sciences as an ordered system or hierarchy, wherein theology was the queen and the others the ministering handmaidens. Looking at the matter in this light we may well expect to see some signs of connection or of sympathy between mathematics and the sacred science of theology. And though at first sight it may seem that the two sciences lie far apart, and belong to wholly different regions of thought, a broader and deeper study of their literature and history will reveal many points of contact.

In the first place, it is a significant fact that, though ordinary students of the one science may neglect the other, it has been otherwise with many of the great masters. For reasons already suggested this fact may have attracted little attention. But those few who happen to be familiar both with theological and mathematical literature, know that many theoogians have done good service to the science of mathematics, and not a few of the first masters of mathematical science have achieved some distinction in the field of theology. Even outside Christian literature we meet with minds naturally disposed to speculate both on divine mysteries and on numbers, even if they do not combine them in a curious numerical mysticism. It will be enough to mention Pythagoras, Plato, and Plotinus, and Proclus, who, through the writings of PsuedoDionysius and the Book on Causes, had considerable influence on medieval theology.

In the scholastic period we find a conspicuous instance of the association of mathematics and theology in the person of the Doctor Profundus, in other words, Thomas Bradwardin, Archbishop of Canterbury, whose name will be familiar to readers of Chaucer. This remarkable man, one of the most original minds among the later Schoolmen, has left some treatises on mathematics, among them being one, De Quadratura Circuit. And some traces of his predilection for this science may be seen in the pages of his great theological work, De Causa Dei Contra Pelagianos. For, instead of following the fashion of contemporary scholastics, he anticipates Spinoza in the application of mathematical method in the field of religious philosophy. The book is probably little known to students of the present day. But, as Thomassinus surmises, it had considerable influence on the course of later theological controversy.

It is in some ways more remarkable to meet with instances of this kind in the later period, after the great movement of the Renaissance and the Reformation. For science and learning extended themselves more and more among the laity; and there was, moreover, a general tendency to greater specialization and division of labor. Yet even here we may find some of the most important mathematical work accomplished by members of religious orders, by theologians or amateurs in theology. Thus, it may be said that the first important step in the making of modern higher mathematics was the discovery of the method of indivisibles by Father Bonaventura Cavalieri, a member of the Jesuate or Hieronymite order. On this point it may be enough to cite the emphatic words of Carnot: "Cavalerius fut le precurseur des savants aux quels nous devons l'analyse infinitesimale; il leur ouvrit la carriere par sa Geometrie des Indivisibles" (Reflexions sur la Metaphysique du Calcul Infinitesimal, n. 113).

The merits of this religious mathematician are not, perhaps, so widely known as they deserve to be. But no student of mathematics is likely to forget how much the science owes to the painstaking analysis and ingenious suggestions of Rene Descartes. And if the father of analytical geometry and the inventor of the method of indeterminates was not exactly a theologian, his new presentment of the ontological argument betokens an intelligent interest in natural theology. Another great name in the history of mathematics fills a larger place in theological literature. And the Catholic theologian will naturally agree with the mathematician in wishing that Pascal had given to the science, which was his own peculiar province, the time and labor he bestowed on theology and Jansenist controversy. But those who are perplexed by the problems of apologetics would be loath to part with his Pensees, and the lover of literature could scarcely spare that delicate irony in the Provincial Letters. Curiously enough, one of the victims of that irony, "notre docte Caramuel," was a master of mathematical science as well as a moral theologian, and he gave some practical proof of his scientific gifts by his work as an engineer at the siege of Prague.

Before coming to the great names of Newton and Leibnitz, it may be observed that Newton's teacher, the Anglican Bishop, Isaac Barrow, was illustrious as a master of mathematical science before achieving distinction in the field of Protestant theology, and anti-Papal polemics. At the present day, no doubt, he is best known by the memory of his voluminous theological writings. But there can be little doubt that he rendered a more real and enduring service to scientific literature by his Latin edition and adaptation of the works of Archimedes and Apollonius. Newton himself, the master mind of modern mathematics, can scarce be accounted a theologian. But it will be remembered that he took a keen interest in some theological subjects, notably the interpretation of prophecy; though it may be safely said that his writings on these matters are only remembered for the fame of their author in other fields. A far higher importance attaches to the theological efforts of his great rival, Leibnitz. That truly universal genius has left much that is of permanent value in most of the varied sciences which engaged his attention. Yet it may be averred that the volume containing his theological writings is next in importance to the mathematical works that form the chief foundation of his fame.

And beyond their intrinsic merits, both alike have historical significance. For, on the one hand, much of all that is best in modern mathematics owes its origin to the suggestions of his genius, and in his first tentative essays we may see the forms of this science elaborated and elucidated by later writers. And, on the other hand, his efforts in irenical theology, his Protestant approximation to Catholic orthodoxy, seem to foreshadow the great movement of Catholic revival. This position of those two great mathematicians may remind us that in the eighteenth century the leadership in science was no longer left in the hands of Catholic ecclesiastics, and in the age of the encyclopedists and the revolution it seemed to belong to men yet further removed from Catholic orthodoxy. Yet even in those days some excellent work was accomplished by religious writers. Thus, it is pleasant to note that one of the best editions of Newton's Principia was edited in Rome with illuminating commentaries and appendices by Fathers Jacquier and Le Seur of the order of Minims. The value of this edition may be gathered from the fact that it was reprinted in Glasgow in the nineteenth century. By a curious confusion the editors of this reprint speak of Jacquier and Le Seur as Jesuits, in spite of the fact that the title page tells that they were Minims— an order which somehow seems more appropriate in connection with the method of fluxions and infinitesimals.

In these later days, when in every branch of learning there is an increasing tendency to greater specialization, we can hardly look for so many instances of a literary association of theology and mathematics. Yet the nineteenth century can boast some conspicuous examples of men who were masters in both realms of science. Thus, readers of this review will naturally recall the name of the late Father Bayma, the mathematician and religious philosopher, some of whose best work made its appearance in the early numbers of The Catholic World. A somewhat different association of the two sciences may be seen in the pages of that singular volume of mathematical theology or mythical mathematics, Ler Gott des Christenthums als Gegenstand streng wissenschaftlicher Forschung, published at Prague some thirty years since, by Doctor Justus Rei—a book which irresistibly reminds us of Pope's line,

"See mystery to mathematics fly."

It may be hoped, however, that it does not fulfill the other half of the couplet. A more searching and systematic survey of the history and literature of theology and mathematics might add many another name to the list of those who have achieved distinction in both these realms of science. And it must be remembered that, besides those who have written on both subjects, there are many more whose published work is confined to one alone, while the other has still remained a favorite theme of study. We have an instance of this in Dr. W. G. Ward, whose words were cited at the opening of this article. His writings bear witness to his proficiency in theology, and to most men he is mainly known by the part he took in a great theological movement. But it is only when we turn to his biography that we find that his attainments as a mathematician were scarcely less than his merits as a theologian. And, though his active work in this science was confined to the days of his tutorship at Oxford, to the last he found delight in that fascinating study.

This personal and historical association of theology and mathematics may well suggest the thought that there must be some objective connection between the two sciences, or that the same mental powers are called into play by both. And if this be so, the cultivation of mathematical study should be of some service to the theologian, both as a mental exercise and as a source of argument, or illustrations on suggestive analogies. Thus, to take an obvious instance, the aforesaid association and the comparison made by Dr. Ward may serve with some as an argument in defence of theology. In an age of materialism some men are apt to regard nothing but hard facts and objects that fall within the range of their sciences. And the purely intellectual speculations of theologians and philosophers are often dismissed as idle dreams without any solid foundation. The evidence of the senses is naively accepted, but it is doubted whether the reason can arrive at truth and certitude. But this shallow scepticism is confuted by the fact that the purely intellectual speculations of mathematicians arrive at results which can be safely tested by the evidence of the senses.

In this way the analogy of higher mathematics may rebuke the sceptic and the materialist and show how intellectual speculation and discursive reasoning may be a sure means of reaching a certain knowledge of necessary truth. But may not some theologians and apologists in their turn find wholesome lessons in mathematical analogy? There are some of us, it may be feared, too apt to conclude that a line of argument with which we happen to be familiar, or which appeals to us most powerfully is the one only and necessary way.

Thus on the great question as to the arguments for the existence of God, we have, on the one hand, the familiar scholastic arguments set forth by St. Thomas and his followers, the ontological argument of St. Anselm, and Newman's argument from the testimony of conscience—to name but these few. And, unfortunately, we find that many, who very naturally prefer one or other of these lines of argument, are almost as anxious to demolish the other arguments as to defend their own. Some who take their stand by St. Thomas roundly reject the arguments of St. Anselm and Cardinal Newman as fallacies. Others, who agree with Newman in preferring the argument from conscience, go on to say, what Newman never said, that the scholastic proofs are invalid and unconvincing. Here the student of mathematics may find some help in the analogy of his own science. For are there not many mathematical truths that can be firmly proved by many and various independent lines of argument, by geometry, by ordinary algebra, by the method of indeterminate coefficients, by the differential calculus? The modern mathematician may remember, moreover, that though he may see the force and cogency of all these lines of argument, the old masters knew nothing of the last two methods, and there must still be multitudes to whom they are unknown, and some who would in any case be unable to appreciate them. For this reason he will be disposed to welcome a like abundance of independent arguments in natural theology, and though he may find one more helpful and satisfying to himself, he will have no desire to demolish the others. Nay, even though he may fail to see their force and cogency, he may modestly surmise that the fault lies in himself and not in the argument.

Cardinal Newman, it may be remembered, incidentally touches on the analogy of the differential calculus in illustration of his own attempt at a new method in his Grammar of Assent. But the remark is merely made in passing, and he does not, apparently, think it worth while to pursue the subject. It would seem likely, however, that a careful comparison would show not a few curious points of analogy between the new methods in mathematics and theology. In this connection it is important to observe that though the infinitesimal calculus at first sight seems to be content with probability and approximation, as Carnot has shown in his admirable reflections on its metaphysics, it really issues in rigorous accuracy. And the same may be safely said of Newman's methods in religious apologetics.

Another point in which some help may be found in the analogy of mathematics is the present tendency to deny discussion and synthetic reasoning, and to exalt the method of intuition and analysis. The classic instance of this in mathematics is the proof of the celebrated Pythagorean proposition. Euclid (I. 47) established it by an elaborated argument, based on several preliminary propositions, resting, in the last resort, on the primary axioms and definitions. In the modern method, discussed by Schopenhauer (Die Welt als Wille und Vorstellung, B. I., sect. 15) we take instead the particular case of the isosceles triangle and the truth of the whole proposition is seen at a glance. As the philosopher remarks, it is superfluous to prove it by other propositions or axioms. For its truth is so evident, that one who denied it might just as well deny the axioms themselves. At first sight this seems to support the current rejection of synthetic reasoning. But a further examination of the mathematical example will serve to correct this impression. For it must be observed that Euclid's arguments are not rejected as invalid, since they do in fact arrive at the same truth which is seen more speedily by the other method. The point is that the longer way is needless and superfluous. And the most strenuous advocate of discursive reasoning would not wish to waste words in proving a self-evident proposition. But, on the other hand, it must be remembered that many important truths are not attainable by the direct and intuitive method, and most of us must be content to take the humbler path instead of the "high priori road." Some minds, it may be added, can see more at a glance than others; just as some have the power of seeing a large number, as thirty or forty, without having to count it. We have an instance of this in Archbishop Temple, who once remarked after a confirmation that there were forty-three boys present, and being asked if he had counted them, he said: "No; I saw them." The high powers of intuition possessedby some great mathematicians, may remind us of the scholastic distinction between understanding and reasoning, and of the teaching of St. Thomas, that angels in one idea see what men can only see in many. And that is another instance of the sympathetic harmony of mathematics and theology.

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