A WORLD OPEN UPWARDS IN MATHEMATICS AND THEOLOGY

 

by Professor Botond Gaal

Debrecen University of Reformed Theology

GLAUBE UND DENKEN 2002

 

I.

 

On the Nature of Mathematics from Christian Viewpoint

 

 

Pioneering Role of the Ancient Greeks

 

      Geometry of the Greeks has become a precious pearl of the history of science due to its nature carrying relative truths. Euclid’s system of axioms was free of contradictions, complete and independent so it could become a deposit of logically stable values. That was really a creation independent of cultures. While hellenic mathematicians were amazed by their own applied logical methods, however, they did not think much about the fact that if they started from evident theorems that would result in a closed system. Closeness is meant here in a postive sense as it only suggests that starting from a given system of axioms one can create a whole field of science, but that is not all. To explore another field or provide higher level knowledge, one needs a different system of evidences to start from. Of course, the Greeks were not aware of this at that time and that is why they thought their own consistent system, especially geometry, to be the top or model of sciences. They regarded it to be absolute and static and sharply distinguished the changing world of natural phenomena from it. Rendering geometry absolute this way resulted in a problem which held its position in the world of facts thought to be unsolvable for two thousand years.

 

 

Mathematicians Open the Old Closed System

 

     In our time mathematicians greatly disagree about what mathematics is or what man regards to be mathematics. A mathematician makes his own decision of how to understand his  own field of science and what functions to give to it among the laws of mind or in its relationship to reality. It is perhaps easier today to notice conceptual difficulties with which scientists of the Ancient Times were preoccupied. Do they create or discover mathematics when this discipline is described? Is it only formalism? The development in the 19th and early 20th centuries made mathematicians think and new results revealed that the interpretation of mathematics could become the object of true discussions. The case of scientists as old as János Bolyai (1802-1860), Nikolaj Lolbachevsky (1792-1856) and George F. B. Riemann (1826-1866) highlighted that mathematics did play an important role in both thinking and the description of nature.  Mathematical logic also appeared on the scene and so far unknown domains of mathematics also began to develop.  The significance of George Cantor (1845-1918) should be mentioned here first of all.

 

     In the philosophy of the modern times, it was Descartes to shock the world by identifying his idea of the  ‘infinite’ with the idea of perfection and  of God. In his opinion, the finite thinking of man alone was incapable of creating the idea of the infinite so it had to be implanted into man by God.  An old problem was re-born: Was the finite capable of perceiving the infinite or could the infinite stay within the finite? The philosophical reasoning on the topic by Spinoza and Leibniz gave rise to lots of debate, too. The coming centuries of eminent German, French and English philosophers did not result in a break-through either, but  Cantor’s activities in the field of the theory of sets, in the second half of the 19th century, did. It was well in the 20th century when the new theory of sets became an indispensable tool in keeping the diverging branches of mathematics together. The incorporation of this theory into the ‘body’ of mathematics was done amongst loud discussions. Cantor himself was in the focus of several attacks and it was soon evident that these attacks served as a fire to generate an even fiercer fire to leave its burn-traces on the face of philosophy and theology alike. Several questions made both fields re-think their ideas. Cantor did nothing but called attention to re-thinking what God, the Absolute meant and how far the intellectual perception of man with respect to the relationship of the finite and the infinite reached.  Kant was greatly respected in philosophy and especially in theology at that time.  Cantor’s ideas, however, were brand new. He claimed that the transfinite and absolute infinite, the two forms of  actual infinite were often mistaken for each other although these notions were completely different. Though infinite, the former could be expanded but the latter should basically be regarded inexpansible therefore mathematically undeterminable. He added that that type of replacement-error was seen in pantheism and that was the Achilles’ heel in Spinoza’s Ethics. Interestingly enough, philosophers since Kant had supported a mistaken idea according to which the absolute was thought to be the ideal limit of the finite. In fact that limit could be thought of as a transfinite, moreover, the minimum of all transfinites (i.e. the number beyond the minimum finite marked by the letter omega by him).

 

 

Cantor refers to „Glory of the Creator” in Mathematics

 

      This simple explanation by Cantor reveals deep depths. Offspring of Danish parents, a scientist carrying on the Jewish-Christian traditions, he came to conclusions which popped up again in 20th century theology. Let us see some more details. „With its manifold forms and configurations the transfinite refers to the absolute, the ‘real infinite’, as a rule, whose size is not affected by any expansion or reduction and therefore it should be regarded as the absolute maximum, quantitatively speaking. To some extent, he latter reaches beyond the perceiving capacity of the human mind in as much as it does not lend itself to mathematical determination. The transfinite, on the other hand, does not just gives a huge chance in learning about God but offers a rich, ever expanding field for ideal research, too. I am convinced that  it exists to some extent and in different respects in the world of creatures as well, to more effectively promote the glory of the Creator, by his own  absolutely free will, as it would be possible in the case of a simple ‘finite world’. This idea should take long to be generally recognized, in spite of the fact that it could prove valuable for theologians, it could serve as a tool to support their case (religion).” According to Cantor, mathematics makes use  of the rationality with which both the human mind and the world were supplied by God. He saw the harmony of the perceivable unity of the universe in the ‘glory of the Creator’. Consequently, he offers a chance for mathematical deductions originating from the rationality of the created world to open up new ways for further discoveries, always keeping in mind that, after all, the rational activities of man or the rational nature of the world point beyond themselves in a transfinite manner. This way, Cantor gave a telling example of the contingent rationality of the universe and let it be felt that this kind of thinking would be of great importance in the exploration of nature. Being involved in debates with contemporary philosophers he made his idea more explicit in Latin, „Omnia seu finita seu infintia definita sunt et excepto Deo ab intellectu determinari possunt!” That is: everything is either finite or definitely infinite and, with the exception of God, can be determined by the intellect.  It is evident for mathematical thinking, too, that the mind cannot interpret God in his own ontological form but refers to his existence in a transfinite manner pointing beyond itself. In other words, there is no scientific explanation which could  reason for God on the basis of the created universe, i.e. nature.

 

     It was as early as 1895-97. Just a generation earlier James Clerk Maxwell (1831-1879) came to almost the same conclusions based on the laws of physics when, contemplating about the existence of matter he claimed that „thus we have been led, along strictly scientific path, very near to the point at which Science must stop. - ... Science is incompetent to reason upon the creation of matter itself out of nothing. We have reached the utmost limits of our thinking faculties when we have admitted that because the matter cannot be eternal and self-existent, it must have been created.” This opinion almost perfectly matches Einstein’s (1879-1955) ideas  saying that we cannot deduct the beginning of the formation of the universe, mathematically it should be some kind of singularity.  Essentially the same idea came in Hermann K. H. Weyl’s (1885-1955) mind about the absolute velocity of light; nature has a physical and a metaphysical feature which the mind tries to follow and research. This is the only way to understand Einstein’s intention: he was not satisfied by asking ‘how’ but kept asking ‘why’ as far as the whole of the universe was concerned.

 

     Cantor should not be treated as an exception when, in his own way, he pointed out, or better to say, hit upon the importance of contingency. Suffering a lot due to the refusal by his contemporaries he was driven mad. His theory of sets was alien for many as some of the exact terms of the new mathematical discipline were still missing at that time. Despite the clumsy terms he could clearly see tenets which were proved much later by the succeeding generations. To demonstrate how impregnating his work was and how fruitful his activities in the field of mathematics and logic turned out to be, we must briefly glimpse into Gödel’s, Hardy’s, Church’s and Turing’s ideas. Of course, there is another reason lying behind: we wish to observe the survival of the contingent approach.

 

 

 

A World Open Upwards in Mathematics

 

     Kurt Gödel (1906-1978), Austrian mathematician, worked in Princeton and published a famous essay entitled “On formally undecidable propositions of Principia Mathematica and related systems” in 1931. Originally Principia Mathematica was a three-volume work by English mathematicians North Whitehead (1861-1947) and Bertrand Russel (1872-1970) published between 1910 and 1930. Based on Leibniz’s ideas and the so-called Peano system of axioms they re-wrote each item of pure mathematics into a logically consequent formal system.  Gödel tried to demonstrate that it was not viable because there were tenets in each system of axioms which can neither be proved or denied within that system. Disregarding the wording  of mathematical logic, in a simple language, Gödel says that  if a system of axioms  satisfies certain general criteria free of contradictions, it cannot be full.  Hungarian mathematician László Kalmár (1905-1976) summarized its  importance saying that „Among ultimately general conditions, Gödel’s theory  is true for every system of axioms meant in the sensible sense of the word.”  But Gödel went further showing that a wider system of axioms was always available within which the truth of the tenet in question could be decided. In other words, the truth of formal systems can always be decided from the superior system. That is true for the axioms of the theory of sets as well so the ‘visions’ of Cantor have come true. Thomas F. Torrance (1913- ) went even further in reasoning and re-evaluating the relationship between Cantor and Gödel’s theories he came to the conclusion that “Gödel showed that the true source of the incompleteness attaching to formal systems of mathematics is to be continued into the transfinite.” Here we are given  the fact that mathematical propositions always form an open system with respect to the decidability of truth, as it was suspected by Pascal and then raised by Cantor as a theorem. By no means, this openness should be explained by the deficiency of the human mind. On the contrary, it should be accepted as a fact which is an inherent property of the universe.

 

     This openness was adored by the whole world. The story itself  is interesting, too. At about the turn of the century, David Hilbert (1862-1943), a famous scientist thought that each branch of mathematics could be axiomatized and if it worked, the falsity or truth of each proposition could be decided in principle. Some time later, Godfrey H. Hardy (1877-1947), English mathematician claimed that “There is of course no such theorem, and this is very fortunate, since if there were we should have a mechanical set of rules for the solution of all mathematical problems, and our activities as mathematicians would come to an end.”  Gödel’s theorem of incompleteness was still to come. The discovery astonished the world of mathematicians. What mathematics can ultimately achieve is an endless list of the logically  undecidable. It was then that Alonzo Church (1903-1993?) ) in Princeton came forward with a method suggesting that the truth  or falsity of certain mathematical propositions could not be decided in a series of finite steps. Alan M. Turing (1912-1954) in Cambridge in England came to the same conclusion but he used a different approach. He postulated a hypothetical machine which would do the opposite of Church’s presumption. It turned out that it would not! It would not do it because there are uncountable sets and within them there are incalculable numbers, too, which are  not infinite in the meaning that infinitely much time is required to count them but they are immanently infintie.  Moreover, their cardinality is much bigger than the cardinality of  natural numbers, therefore the decision about the truth or falsity of propositions is  pushed further to the infinite. The more one would expect help to make a decision the more likely Turing’s machine would say it was increasingly impossible. Again, this justifies Cantor’s ideas, i.e. the contingent nature of perceivability. In his specific language Cantor would have said that contingent perceivability had its influence on the mind from beyond the created order, from the absolute infinity of the God Creator. Mathematics is of such nature! In my opinion it is of such nature because it has been created and that is how it is adjusted to the nature of the universe and its contingent order. Márton Sain’s remark concerning the whole of this problem is quite suggestive: „ ... this proposition exceeds the world of mathematics and joins it with the reality beyond mathematics”.  In 1988, the secret lying in the depth of this matter was summarized by John D. Barrow as follows: “If the universe is mathematical in some deep sense, then the mysterious undecidabilities demonstrated by Gödel and Turing are part of the fabric of the universe rather than merely products of our minds. They show that even a mathematical universe is more than axioms, more than computation, more than logic – and more than mathematicians can know.” Thomas F. Torrance, theologian,  is practically of the same opinion, which is quite an accurate statement about the essence of mathematics but we cannot be perfectly sure that this statement will be accepted by all mathematicians. “In mathematics, of course, we elaborate symbolic systems as refined instruments by which we may extend the range of our thought beyond what we are capable of without them. The significance of mathematical symbolism, however, is to be found not in the mathematical equations themselves but in their bearing upon non-mathematical reality. As far as I can see, mathematics is effective in the physical sciences because it belongs to the actual contingent world, and reflects and expresses the patterned intelligibilities embodied in it, even though they cannot be captured in abstract mathematical form."

 

     If we look back upon the extremely long way covered by mathematics from the Ancient Times till the 20th century paying special attention to the change in approach, we must admit that Euclid was right to tell King Ptolemy I that „There is no royal way to geometry!”  From the axiomatic world of thinking, which neither inductively or deductively binds with the close world of experience, we have come to an openly structured mathematical approach which, according to our best knowledge, is a means of theoretically grasping the contingent universe and is therefore congruent to it by nature. It must be admitted that there are physical phenomena which cannot be described using mathematical notions as yet, but, at the same time, mathematics has plenty of notions, procedures, equations which have not been used by physics yet. Physics has not been fitted with all of the ‘mathematical truths’ yet nor has mathematics been matched with ‘physical truths’ everywhere. Nature is open to that extent and so is the mind perceiving it which is an integral part of its contingent character.

 

      A question has been left unanswered: can theology be considered as a science with this opennes?

 

 

II.

 

On the Nature of Theology from the Viewpoint of Mathematics

 

 

Long-lasting Effect of the Greek Heritage

 

      Science must always start from reality and hand in hand with experience it should reach the highest level of abstraction, that is, the setting of basic laws or axioms. The Greeks were very good at it. Looking at their crystal-clear system, especially in geometry, we can but wonder even today. They constructed a field of science whose theorems were true within that specific system in that period. They are, were and will always be correct carrying relative truth.  Because we are familiar with the theory of relativity and Gödel’s theory, too, this knowledge enables us to claim that the geometry of Euclid has become one of the greatest value and most influential idea of the Ancient Times. Euclid’s axiomatic system, as we have already mentioned, was consistent, complete and independent so it could become a deposit of logically stable values. It was not true of the approach-dependent conclusions relating to physis; they always had to be harmonized with reality.  That kind of adaptation was unnecessary in geometry: its further development and rise to higher spiritual levels were the only tasks to be fulfilled.

 

      Greeks did not aim at changing the world but understanding it. That happened when they created mathematics, too. Their logical method was so succesful that, in their opinion, no one else could reach beyond that spiritual height. Satisfied, they made their mathematical system and the truth of their deductions ansolute. This absolute is not meant in the religious sense. It is public knowledge that Pythagoras founded a separate cult and Sokrates and Plato were also profoundly religious. But there is no proof that the greatest Greek mathematicians ever paraded their results or claimed to having reached God by means of those results. We can only see that Greek thinking was greatly determined by stories and contemplation about the deeds of the gods but they were always wrapped up in some complicated and mystified mythological past. Greek religious thinking did not have dimensions projecting to the future, their world did not have any specific purpose, it was almost immobile, it did not change much, it performed just a little motion. Scientists dealing with geometry, too, developed a similar, closed theoretical system and surrounded themselves by a static, timeless and unhistorical world from which they could observe the unchangable universe. Experiments, empiricism, practical application were out of the question as they could have changed the direction of the future. Everything reached its summit in understanding. These spiritual efforts were independent of the gods. The thinking of the scholar or educated Hellenic citizen was always centered around self-generated dualistic constraints and there was no escape from this trap. The ultimate reality for them did not mean divine deeds or the surrounding world but the truth figured out by the mind. They were confirmed by the approach of many talented mathematicians. These facts led them to be conceived geometry absolute which became a generally accepted notion upto the 19th century. Even Kant reinforced this thought and died believing in this ’certain misconception’. Gauss did not dare to doubt this idea either but Bolyai and Lobachevsky did.

 

      One can definitely feel from the above how highly the Greeks appreciated geometry. They were so sure of the absoluteness of their geometry that they regarded it to be a model for other fields of science, too. This idea was involved in a later expression, ’more geometrico’. From this one can see, that the geometry of Euclid is an adequate expression of the Greek spirit. Two thousand years later, Spinoza also intended to create ethics in the spirit of ’more geometrico’. So the application of this scientific model was promising for other fields of science, too, and, as it could be observed later, for a very long time, scientists did not even think that this highly praised scientific approach resulted in the strictest kind of rigidity in free thinking, presenting a major obstacle for development. From the Middle Age theology also worked under the influence of the ’more geometrico’.

 

 

Nature of the Contemporary Jewish Way of Thinking

 

      For the Jews, absolute reality meant the existence of Yahweh; without him man and the world surrounding man was unimaginable. God and the universe created by him form reality together; reality is nothing but  the adoption, learning, developing and realizing of the will of  God-Creator in all fields of life. That is what they meant by saying “The fear of the Lord is the beginning of wisdom”. While the Greeks were preoccupied with getting to the deity by means of speculation through the interpretation of nature and a self-consistent logical system, the Jews immersed in religious life and its socialized and moral practice in the strict sense of the word. At the same time, they attributed every single historic event and natural phenomenon to the final cause, God. For the Jews, God was the ultimate reality – and we must take it for granted – but, with time progressing and the moral norms changing, in the centuries immediately preceding Christ, Israel got to a stage when faith could accept the deeds of God as real only if God made them legitimate through signs. But it was not the live faith of the fathers any more. It was best illustrated by the demand “let him now come down from the cross, and we will believe him.” This is an example of the Jewish approach which  also renders empirical truth, or evidence that can be understood by the mind, dependent on the justifying action of God who is regarded as reality in the Jewish way of thinking.  In other words, the truth, as they thought of it, was no longer the result of less and less contingent, self-consistent speculative work dependent on the absolute. Only those things were accepted as truth which could be justified by the traditions of the past or via some divine sign.  That is why one should not wonder hearing that Jewish thinkers did not  attribute much content to human truth.  The Book of Wisdom has an excellent sentence to illustrate it saying that ‘Truth is immortal’. In other words this means that, looking at affairs from the revelation of God, the laws, wisdom literature, the prophets and the promises all look forward to free choice in faith on behalf of man, but what happened was a rigid, and rationally systematic interpretation of law and its word-for-word execution and application in practice. It was only experienced and heard from Jesus that it was not man who was made for the law, but the law was made for man. He addressed the scholars of Israel strictly, saying: “Ye are they which justify yourselves before men; but God knoweth your hearts.” The original laws and their extensions became axiom-like theorems whose truth was beyond doubt, because they all could be traced back to some law of divine origin, or rules of the holy historic path legitimized by divine signs. This way the Jews lost their openness, that is, the contingent nature of their approach to the world and their existence before God. Owing to their moral system they brought about a traditional and absolute world under the control of laws, and, in both faith and intellectual activities they tried to monopolize the living, personal and omnipotent God, creator of the universe, in this world. Moral practice originating from the interpretation of the law and traditions became more important than the personal message of God to man, as it was customary in Abraham, Isaac and Jacob’s time.  In all cause-and-effect matters, the final explanation and all-deciding authority was God himself, due to the preference of morality in the foreground. The significance of the idea that man can do  something, actually quite a lot, for the world had shrunk into an insignificant factor in people’s eyes. This attitude was not in favor of the human desire to search and learn about the created universe using the mind. It is here that we can see that Jewish thinking followed a different path in its religious life. Christianity belived that all the promises of the prophets were fulfiled by Jesus Christ.

 

 

The Appearance of Christianity – New Conception in the Order of the World

 

      Christian theology met Hellenic science or philosophy at three points: in the philosophy of Plato, Aristotle and Stoics. It is evident that Greek thinking had a huge burden, the dualistic approach to the universe, which was inevitably intensified by a permanent axiomatic approach. They tried to get close to everything with preconditions in mind, and in doing so, unfortunately, they attributed too much importance to the logical system of axiomatic structure in mathematics and its role. This preconception was further strengthened by the results of geometry. They did not notice how far its area of truth expanded and what the ways of development were. They believed to have obtained absolute reality. The only way out of this world was through the destruction of these fundamental tenets of Greek thinking, and yet by saving its values new tracks would be laid down for scientific thinking. Christianity played an invaluable role in this process.

 

     According to Christian teaching, world order cannot be described either through the categories of chance or necessity as understood in the Greek way. A firm conviction of Christian thinking suggests that the world can be understood in an inherent manner, that is man can perceive it.  The explanation lies in the tenet that God created the world from nothing, consequently man and the world surrounding him belong to the same created universe. They are subordinate to the same world order. That is why there is a relationship of common origin between the creaturely rationality of the human mind and the rationality of the order of nature. Both are immanent within the universe. Human mind and Logos are congruent with the created world and God, respectively. That is how we can speak of God’s transcendent rationality and the world’s immanent rationality, as it we did earlier. Getting even closer to the heart of the matters: according to the Christian teaching about the world and the Christian interpretation of the universe, the laws of nature are to be regarded as immanent structures, introduced by God himself, together with creation in this spatial and temporal world.  The laws, representatives of order, are neither timeless nor necessary but are open structures, unconcluded knowledge, so they are imperfect by themselves, and since they belong to the created world of God, they depend on him. God was not under pressure creating the world,  but he could create it freely, therefore the universe has no self-contained cause  which would explain why it is as it is. Though it has rational order, that is not a self-explanatory principle.  Therefore it is not self-sufficient, and that implies the created and immanent order for which Christian thinking used and applied the term ‘contingence’. Christian people regarded the laws of nature as the contingent form of the immanent rational world-order, which can really be perceived by man and can be researched via experiments and accessible by the human mind. But just because of its divine origin, it points beyond itself, at the transcendent basis of this perceptibility, God’s creative Logos. The preexistence of this Logos ensures a permanent and unchanged basis upon which the immanent rationality of the world is built. Apostle Paul expressed it in the language of the Hellenic culture when he called the incarnated Logos, Christ, the image of God of whom later he said that he “is the image of the invisible God, the firstborn of every creature and he is before all things and by him all things consist”. That is why the world is contingent, because the final basis cannot compare with the axiomatic thinking of the Greeks. Moreover, there is no dualistic approach here nor is there any empirical legitimation, as expected from God by the Jews; rather, we have a biblical concept of God, according to which he maintains and governs this world and its order owing to his power – theologically speaking: he holds it in his love.

 

      The world has a divine order which was generally described with the words oikonomia and pronoia as early as the apostolic times. These words became theological terms and the first one has been used in the same way by theology since then. This order is of contingent nature independently of whether we mean the whole world or only parts of it. It does not necessarily exist, but if it exists, it is not necessary for the world to exist in its present form or to have laws exactly like the present ones. Beyond all this, the contingence of the world is not only valid for self-existence alone, but also for its relationship with God as God might have created the world in a different way, too. If one starts from the world’s own order, he cannot make deductions concerning the origin of the world, that is the reason and essence of its existence can only be understood from beyond itself, from its relationship with the God-Creator. It will be seen that according to the Christian interpretation this is not a closed world, but an open one, as far as the essence, its inherent features are concerned. In this upward openness, each stage of the process of recognition is immanent but not independent of the transcendent. This does not readily mean that any scientific explanation about the world should be based on Christian theology. Neither should axiomatic principles be involved in the description of the laws of the world to the smallest extent at all. But, based on Christian principles, we may obtain a certain view of the world which might help us get positive knowledge of the world if the theoretical and empirical factors are seen and used in their mutual relationship. The world does have its contingent reality as far as both the theoretical and the empirical knowledge are concerned. We choose the right way to learn about the world if we examine the world according to its internal nature and in harmony with its immanent perceptibility, as it was so often emphasized in Christianity – kata physin. Any cosmic reality can and must be revealed according to its essence. – Christianity did offer this chance.  The Greeks were convinced that the essence of the empirical world was in what they thought of it –  what is more, they regarded this idea to be divine. They were wrong.  Mathematicians believed their knowledge to have reached its final limits because they were unable to find the way to take them any further. It turned out that a system of presupposed truths would result in a closed world of ideas. The Jews thought that the divine deeds of the historic past would be absolutely valid in modern times, too, therefore the strict moral rules of everyday life could help them change the reality of God into his earthly presence. Man proved to be too sinful and mentally finite to achieve all this. The free citizens of Rome were busy with the real activities of state organization, therefore they interpreted ’truth’ at a legal level instead of dealing with the manifestation of nature. Christianity brought about a significant change only in the field of their social life.

 

 

A World Open Upwards in Christian Theology

 

      19th century protestant theological thinking was shaped by many outsiders, but the most significant among them were Kant, Hegel and Schleiermacher. The dominant effect of one or the other philosopher is seen in the work of theologians. Generally speaking, theology accepted Kant’s aprioristic approach as unshakable truth, Hegel’s philosophical synthesis had enormous effect on both spiritual and natural sciences and parallel to them, due to Schleiermacher’s activities, ’culture-Protestantism’ developed, which was a kind of harmonized union of Christian theology and philosophy. It is worth to cast a glance at the reaction of theology, strongly influenced by early 19th century philosophy, to the challenges of the late 19th century and the years at the turn of the new century.

 

      Highly respected Albrecht Ritschl (1822-1889) and his disciples brought back the idea of Kant’s moral imperative and based each part of their system on it. In the spirit of Kant’s apriorism, Adolf von Harnack (1851-1930), who also belonged to this school, said that the historic teaching about Jesus was different from the teaching of the gospel. Ernst Tröltsch (1865-1925) went further with theological liberalism and came to the conclusion that  a priori  existed not only in the moral but also in the religious sense, which helps man to develop religion. Based on these examples it could be seen how much protestant theology was under the influence of the contemporary philosophy. The situation was righteously and tauntingly described in Sören Kierkegaard’s (1813-1855) strong criticism saying that „theology was sitting in its Sunday best by the window, currying philosophy’s favor and offering its own charms to philosophy.” As it can be seen from the example of Kant, Schleiermacher and their followers, on the model of  more geometrico they only needed a single sure principle or  safe clue to faith and they based their theology on it. Analysing the Christian thinking of the 19th century, Thomas F. Torrance drew the conclusion that theology had gone stray from the nourishing source of Christianity. Theologically speaking, it „lost its objective basis hidden in the Word of God and controlling it, and had essentially become anthropocentric.” In this situation appeared Karl Barth (1886-1968) who went back to an important tenet which was emphasized by the theology of the Early Church and reformation. There is unity and an authentic relationship between God and his revelation. It was a problem for the man of the period of enlightenment because the revelation in the Bible sometimes only discusses who God is in his eternity, then one can also read about what God did to his people and in other chapters the two can be seen together. Thus ontological elements and direct practical deeds are found within the same idea. This conceptual problem was solved in modern theology presuming that God could be recognized from the Word, that is, Jesus Christ was the incarnated Son of the God-Father through his love, thus letting the universe know about his essence and revealing himself this way. Theologically speaking, it means that God and the Son incarnated in Jesus Christ are one. This was the Christian dogma of ’homoousion’ that was the central doctrine of the Nicaean-Constantinoplean Creed around which the theology of the ancient Catholic Church turned. In other words, God made contact with man and the world in this manner, via his divine intellect, his transcendent rationalism which we traditionally call Logos, a word from the Greek language. This Logos is not identical with physis (the world) and here we have come to the point when Karl Barth’s famous statement is thought to be appropriate and justified ’Gott ist der ganz Andere: God is completely different’. This leads us towards very essential statements in the relationship between theology and mathematics.

 

      From a disciplined scientific theological point of view it really is a problem that many think the ’origin’ in Barth’s theological epistemology is merely represented by the statement ’God is completely different’. This is only half of the statement. János Bolyki, Hungarian theologian, once said nicely, „It is not only God that is completely different from the world but also our mind is completely different from the Word. That is the starting point of Barth’s epistemology. Our mind is identical with the created world whereas the Word is identical with God.” Thus, unlike with Kant or Schleiermacher, it is out of the question that we should build theology on a dogma thought to be evident by us – as Kant put it in his famous statement „the starry sky above me and morals in me”. This would jumble things a lot. The starting point for theology could be that God’s revelation is true and that revelation had something essential to say about God himself as well as the realtionship between God and the world. Consequently, Kant’s statement which says that we cannot understand the internal essence of things, only their external features can serve as important information – cannot be true in theology. The revelation in Christ refers to God himself, therefore, for theology the knowledge of God is really possible. If theology wants to remain theology it should get rid of the dualistic rationalistic approach which was borrowed from philosophy, and should accept that the immanent rationality of the world and the transcendent rationality of God are two different things. The connection between the two can only be brought about by God.

 

      These lead us to further considerations. Basically all these meant that the ancient more geometrico or theologia naturalis by analogy of the former offered a solution upon which a certain kind of theology could be built up. It was nothing alse but the ’axiomatization’ of theology, that is, finding points in nature and thinking which refer to the existence of God somehow. Instead, as Torrance referred to it, Barth thought to have found the proper way in which the biblical Jewish and Christian way of thinking about the existence of God was followed and, simultaneously, the ’Christological correction’ done by the Reformation in the ’Western Church’ was considered. Now we can ask the question why was rejected the axiomatization of theology by the protestant theology in the 20th century so sharply? Or, in other words, is it possible to bring about an axiomatic system at all for Christian theology in the manner of more geometrico?  

 

      Creating an aximatic system for theology means that it has to be fulfiled a threefold requirement of the formal systems, that is, it should be complete, independent and consistent. It is easy to recognize that none of these requirements cannot be fulfiled in cultivating of a disciplined theology. There are no finite number of biblical statements which would be enough to build up an entire theological system because they contain the revelation of the infinite God and, at the same time, biblical statements related to the God’s act for the salvation of mankind are not independent from each other because all of them refer to the core of the gospel which is the opus salutis of Jesus Christ. As to the consistency of the proposed axiomatic system, it was and it is a crucial question for a biblically based theology. The statements of the biblical revelation must not be considered as axioms because their truths are not perfectly conceivable for man. Each of them contains truths which are partially over the capability of human comprehension. For example, God is love! It is really a basic statement of God in its ontological sense but no one can tell us who God entirely is in his eternal being based on this biblical passage. Human intellect is not sufficiant to do that, only to step forward continuously in knowing the ultimate divine reality. Or, another good illustration, taking it from the Old Testament: The world and everything in it was created by God! It is true but it does not fit to the principle of the free of contradiction and, therefore, it is not appropriate to choose as one of the axioms in order to build up a perfect system of theology in the manner of the formal logic. Otherwise, we have to acknowledge that evil was also created by God or we should deny its existence. Therefore, this statement is valid only together with another biblical thought that the world was made by God as a good creation and man was created in it according to the order of the freedom. And what is more, creation also has to be understood from a christological viewpoint „for in him all things were created, in heaven and on earth, visible and invisible,…”(Col. 1,16). That is, considering this question from the viewpoint of mathematics, in a strictly logical sense, theology has only one axiom which is Jesus Christ. But, Christ can not be a logical result of the human mind. He is not a principle, but he is living person for the Christian belief. Thus, the axiomatization of theology impossible at all for it is not possible to bring about a complete, independent and consistent formal system for the science of theology. That means that a biblically based theology is ab ovo an open system making sure the free and direct connection between the living God and man as we learned it from the struggle of Martin Luther for „finding a gracious God”! The ground of theology is not a theoretical axiomatic system but it is the living faith in Christ – as Apostle Paul put it: „… being rooted and grounded in love, may have power to comprehend with all the saints what is the breadth and length, and height and depth (of goodness of God) and to know the love of Christ which surpasses knowledge, that you may be filled with all the fulness of God”(Eph. 3,18-19).    

 

      Cantor indicated a possible development in theology by studying the transfinite infinity which refers to the ’real infinity’ as Absolutum. In his theory of sets he raised a very serious thought about the two kind of actual infinities and used a special mathematical method in comparing of the infinite sets. The size of the absolute „is not effected by any expansion and reduction and therefore it should be regarded as the absolute maxiumum, quantatively speaking.” Cantor did emphasize that human intellect is not able to grasp God in his ontological being but with the help of the transfinite it can refer beyond itself to the existence of the deity. At this very point we have to ask a question: to what extent can be used Cantor’s idea of the theory of sets in Christian theology? As far as I know, there were not too many reactions to his suggestion. This topic has also a long history.

 

      It has already been a difficult problem for the Greeks when they tried to bind together ’deity’ and ’infinity’. In their philosophy both of them were taken into consideration as an epistemological question. Philosophers stated that they grasp ’deity’ in their thoughts and, at the same time, infinity was an unknowable ’field’ of their mind. Infinity, therefore, was category of nothing to do with it. It was not even a biblical phrase. On the contrary, church-fathers used it more frekvently, not for getting knowledge of the world but expressing some sorts of ontological properties of God, like his wisdom, power, eternity, omnipotence, etc. Great Reformers, in the first place Melanchton and Calvin, did the same emphasizing the right knowledge of God although they did not take part in the debate of philosophy about infinity. After the Reformation many philosophers were wrestling with this problem, namely Descartes, Spinoza, Leibniz, Kant and others, but they did not exceeded the Euclid’s principle ’the whole is greater than the part’. For the contemporaries of Kant „the absolute was thought to be the ideal limit of the finite” – as Cantor put it. That is, the infinite was absolutely unthinkable. In this situation Cantor’s idea was completely new. Infinity could already play an important role in developing a new way of thinking in various fields of sciences including theology. Roman Catholics reacted first positively but they became recently moderate saying that it is not wise to interfere in this philosophical debate. They pointed out that God’s infinity is incomparably different from the mathematical infinity and the later is not appripriate to express God’s perfections because they are not measurable. In the Roman Catholic theology God became an ’infinite set’ of his own infinite perfections. It is very similar to the Cantor’s idea indicated about the absolute as a ’real infinity’ which is not comprehensible by man.

 

      We can agree that we must not consider God as an infinite ’person’ in the manner of the infinite ’sets’. Neither was it Cantor’s desire. But, if we inquire about what Cantor’s intend was, indicating his new imaginations as means, in getting knowledge of the glory of the Creator, his idea could be a creating one in theology. Let us see the meaning of the trinity which was one of the most difficult problem for the Early Church. They could not express the essence of the trinity with the help of the Greek philosophy, thinking of the ’whole’ as greater than its ’parts’. Jewish monitheism was not even sufficient to do that. They had to choose a completely different way. We can discover its analogy with Cantor’s theorems.  

 

      The main idea of Christian trinity is that God revealed himself in the shape of three persons which can be distinguished from one another but are inseparable as far as the unity of their essence is concerned. The words ‘person’ and ‘essence’ are suggestive of man’s incapability of precisely expressing the secret of trinity through his set of ideas. It has always been impossible to find a suitable vocabulary to depict these teachings perfectly.  But it is self evident that we mean monotheism in the strict sense of the word. This is difficult to notice because even 20th century theological interpretations contain elements suggesting that the Holy Trinity is put together from its component parts, the Father, the Son and the Holy Spirit.  All of the theologians put special emphasis on explaining this idea and their enormous efforts often seem to be brilliant human mental achievement to understand trinity.  We are not going to follow this direction from ‘downstairs’ to ‘upstairs’. It is much simpler to start from and interpret the biblical meaning itself. In their complexity, biblical revelations aim at presenting the secret that God is One-God before all, that is, the trinity of God exists in a prae-eminent manner, in its fullness. We would not say that fullness is congruent here with the ’real infinite’ or ’the absolute’ used by Cantor in mathematics. In spite of that fact, it can be considered as something similar to this biblical term. Christianity ontologically perceived God as if he prae-eminently existed in absolute trinity in its eternal being and only revealed himself as the fullness of the Holy Trinity in the form interpreted this way. The ‘punctum saliens’ here is that the same trinitarian divine fullness is revealed in the Father, the Son and the Holy Spirit, each. That is how the biblical statement, appearing twice in the Bible, saying that in Christ ‘all the fullness of God dwells’, can be understood or is worth interpreting at all. When in the 4th century the church fathers wanted to express the equality of the divine persons, that is of the Father, the Son and the Holy Spirit, and introduced the concept of  ‘aequalitas’ for the persons of the Holy Trinity,  they actually wanted to make it clear that everything started with the absolute tri-unity of the Holy Trinity One God existed prae-eminently, which served as a basis for any other theological tenet and, also, this doctrine could not be disregarded even in the Christian approach of the universe. The Bible, however, does not attempt  to make clear how the prae-eminent fullness in the Father, the Son and the Holy Spirit should be understood. Each trinitarian person owns the divine fullness and, in spite of that fact, we must not say that they could be a ’part’ of the Triune God! That is beyond human reason like the Creator in the term of Cantor’s ’absolute infinity’; God is also a hiding God(Deus absconditus) and God cannot be determined by the intellect – as Cantor also stated in mathematics.

 

*    *    *    *    *

 

      As we can see it, theology cannot be started with an axiomatic system and cannot be either ended in an Absolutum of the human logic. Both of them are the product of the human intellect, therefore these terms and methods are not appropriate to express the infinite divine truths revealed in the Word of God. We live in a world open upwards and we are able to comprehend our universe because the nature and our mind were created by God to be identical and, thus, appropriate to do that. And, at the same time, we live in a world open upwards in which we have received the gift of faith to conceive the meaning of the creating and redempting work of God making us appropriate to do that through Jesus Christ. The world of God in his revelation is open for us to accept and approach the final reality of Christ in faith!

 

         

 

           

 

 

Cf. Emmanuel Levinal, Transzcendencia és megértés. (Transcendence and Understanding.) In: A modern tudományok emberképe, Gondolat, Budapest, 1988. Edited by Krysztof Michalski, 227.

Cf. Végtelenség a matematikában és a filozófiában (Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Ed. Zermelo, Berlin, 1932, New edition by Hildesheim, 1962) by Georg Cantor. Filozófia Figyelő, ELTE, Budapest, 1988/4. 56-87. Hungarian translation by Zoltán Szabó; edited and notes written by Imre Ruzsa.

  Cf. ELTE Filozófia Figyelő, op.cit. 80.  Ernst Zermelo, a mathematician, wrote comparative remarks on the ideas of Kant concerning the finite and the infinite and Cantor’s interpretation of the problem. It was also published by Imre Ruzsa, op.cit. 83.  Zermelo defended Kant arguing that he applied these two notions on the whole of the universe saying that “the human mind was forced by its inherent nature to equally accept the world as something limited and unlimited, and finite and infinite.”

  ELTE Filozófia Figyelő, op.cit. 79-80.

  ELTE Filozófia Figyelő, op.cit. 82-83.

  Cf. ELTE Filozófia Figyelő, op.cit. 71.

Lewis Campbell and William Garnett, The Life of James Clerk Maxwell, London(1882), Johnson Reprint Corporation, New York, 1969. 359. Cited on the basis of a university lecture by Maxwell.

  Albert Einstein, The Meaning of Relativity, Princeton, 1953, 129. Cited by T.F. Torrance in Einstein and God, Reflections, Center of Theological Inquiry, Princeton, Spring 1998. 14.

Cf. Márton Sain, Nincs királyi út! (There is no Royal Way!), Gondolat, Budapest, 1986. 776.

Cf. Márton Sain, op.cit. 643.

Cf. László Kalmár, A matematika alapjai. (The Basis of Mathematics) Vol.2. book 2. University Textbook, Tankönyvkiadó, Budapest, 1976. 428-462.

László Kalmár, op.cit. 451.

Thomas F. Torrance, Creation, Contingent Word-Order and Time, Manuscript. 11.

John D. Barrow, The World within the World. Oxfrod University Press, Oxford, 1990. 262.

  Cf. John D. Barrow , op.cit. 263-266. Such uncountable numbers are the irrational numbers, or among them, the transcendent ones. In other words they are called sets of uncountable cardinality. What Church and Turing had already realized but Cantor did not was the set of incalculable numbers within the set of uncountable numbers. In the case of the former there is no procedure to calculate individual numbers to arbitrary accuracy. Cf. Isten gondolatai (The Mind of God) by Paul Davies. Kultúrtrade, Budapest, 1995. 96-98. In the case of  p, for example, there is such procedure.

  Márton Sain, op.cit. 806. (Márton Sain is a historian of Mathematics in Hungary.)

John D. Barrow, The Mathematical Universe. Natural Science, 1989. May, 311.

Thomas F. Torrance, Creation, Contingent Word-Order and Time, op.cit. 9

Márton Sain, op.cit. 147.

Cf. Károly Simonyi, A fizika kultúrtörténete. Gondolat, Budapest, 1986. 24.

Cf. John D. Barrow, op.cit. 240.

‘More geometrico’ is an expression of Latin origin meaning ‘like in geometry’ or ‘by way of geometry’. The word ‘more’ comes from the Latin stems  mos, - moris meaning tradition, moral or means.

Cf. Károly Simonyi, op. cit. 24.

Ps. 111,10; Prov. 9,10

Matt. 27,42.

Book of Wisdom 1,15 (Képes Szent Biblia, op.cit)

Luke 16,15

‘Subdue it’, a famous command of the Genesis (Gen. 1,28) is also known as the command of culture. It referred to the knowledge and researching of all aspects of the world.

When Athanasius interpreted the empirical world and empirical phenomena, he treated them as reality changing in time and used the term endechomenos to express his idea. Over one and a half century later, the ‘last Roman philosopher’, Boethius (480-525), a renowned eclectic neoplatonic thinker translated the word ‘endechomonos’ or rather its verbal form ‘endechetai’ into Latin as contingit and that is how he established the words contingent and contingence in the western world. The word endechomenos is the participle of the verb endechomai, meaning possible, probable. The verb has other meanings, too: think something possible, accept or allow something. The Latin word contingit was used in the meaning of ‘happen’, ‘occur’ or ‘something lucky takes place’ by Boethius.

Col. 1,16

Col. 1,15 and 1,17. Eikon tu theu tu aoratu= The image of the invisible God. Was also used by Plato, referring to the sensory world.

In Eph. 3,9 Paul said that God hid his mystery in this world at creation and it had an order of realization and in 1Tim. 1,4 he speaks  of  an edifying question. Both are expressed through the world oikonomia.  Of course, the word has other meanings, too, such as household, measure, order, etc.

The word  pronoia means care or providence, and in Latin it was translated as providentia.

Plato often used the term kata physin, which means ‘according to the law of nature’ or ‘following the laws of nature’.  Christians also used it and quite often meant the ‘essence of matters’. At the same time, another interpretation of physis made it possible to distinguish between the ‘human’ and the ‘supernatural’. An excellent illustration of this is found in Gal. 4,8 in the Bible.

When Apostle Paul is taken to the benevolent-looking Festus of Rome and talks about his faith, he is told, “Paul, thou art beside thyself; much learning doth make thee mad.” (Acts 26,24). Certainly, Festus could not understand Paul’s Christian way of thinking and judged Paul’s situation from the point of righteousness.  The same story also disclosed that the Jews wanted to kill him for those things.

Tamas Nyíri, A filozófiai gondolkodás fejlődése. Szent István Társulat, Budapest, 1977. 440.

The Legacy of Karl Barth (1886-1968) by Thomas F. Torrance, Scottish Journal of Theology, 1986., Vol. 39. 290-291.

One of the difficulties in understanding the biblical revelation can be seen with the problem of ‘I am that I am’(Ex. 3,14) when man can see and feel God’s deeds and, at the same time, wants to know who God is actually. This parallel is seen all through the Bible and the problem is partly solved – but this is only a partial solution – when Jesus declares: “I and my Father are one”(John 10,30).

János Bolyki, A természettudomány és a teológia Barth Károlynál. (Science and Theology in the Work of Karl Barth.) Theologiai Szemle, 1976/9-10. 280.

ELTE Filozófia Figyelő, op.cit. 82.

Most of the theologians explain the unity of the three divine persons with the help of the old term ‘trinitarian perichoresis’ which were really very useful for a long time. Persons penetrate into and walk though each other.

  “For it pleased the Father that in him should fullness dwell” (Col. 1,19) and “For in him dwelleth all the fullness of  the Godhead bodily” (Col. 2,9).  In both quotations, ‘him’ refers to Christ.

Athanasius’ universal view recorded this idea word for word, several times.

“Verily, Thou art a God that hidest thyself, O God of Israel, the Saviour.” (Isa. 45,15) This was mentioned earlier in the chapter entitled ‘Intellectualism of the Jews of ancient times’.