In 1884, when Cantor was in the depths of despair over the cold rejection of the theory he discovered about transfinite numbers and potency, and his theory about the infinite and its bounds, and the whole and the part of the infinite, he fell into depression. Once he had recovered from that depression, after 1885, and until his return to mathematics as the chairman of the German Mathematical Society in 1891, he searched for his own path by mixing with Catholic theologians of his own generation.
However, this was not the only period in which Cantor showed his theological side. Just before, in 1883, he considered that the concept of transfinite numbers he had himself discovered would become a mathematical model for the concept of actual infinity that had existed since the days of Ancient Greece and the Middle Ages. In other words, by becoming a mathematical model of the concept of an infinite with bounds, in the sense that the concept of transfinite numbers, the concept of infinity that its own bounds were itself, the concept of the actual infinity were completed, Cantor considered that it was possible to champion the concept of the actual infinity itself.
The actual infinity is simply a predicate of God. So let us take a look at Cantor’s theology.
The Actual Infinity
To the ancient Greeks, the infinite was an extremely paradoxical phenomenon. Anaximander, of what is considered the first philosophical group in human history, the Milesian school, considered that the indefinite (infinite) was the source of all things. He was certain that the extremities of things were unlimited, with no boundaries.
However, the ancient Greeks considered that the real nature of things, what it was, was limited by itself, was shaped as something itself that was given boundaries by other things. For them, a thing was considered as being created from infinite materials that have no limits whatsoever by being limited, and thus shaped, by something.
In particular, Pythagoras of the Ionian school, considered the Father of Mathematics, considered that the shape which created the real nature of all things was numbers or figures. Pythagoreanism, which held that the real nature of the world was expressed by mathematics, was inherited without change by the contemporary sciences, and even today it retains an immovable position as one of the few universal ideas of humans.
For the ancient Greeks, the infinite, not limited as anything, was a material, was matter, and by giving shape to them by limiting them as something, by giving them form, then and only then would they exist in this world. This “form” is simply Plato’s “idea” and Aristotle’s “eidos.” Both of these words mean forms, and are created from the root word “weid,” which means “to see.”
Aristotle considered that the actuality that is existence in this world through the limitation and completion of the potential of the infinite creates or completes the process by which existence in this world is created by being given form through the limitation of an infinite material. In other words, Aristotle considered that the infinite is not limited, it is the state of as yet incomplete potential, it is potential, and when that is limited and completed as something, thus existing in this world, it is actuality, or reality. As a result, for Aristotle it was not possible for the infinite to exist as an actuality.
In truth, Aristotle divided up the infinite as potential, or the potential infinity, and the infinite as actuality, or the actual infinity, and attempted to reconcile the contradiction to which the actual infinity unavoidably led him. That is, as we saw in the previous chapter, it is proof that if we allow the actual infinity, the completed infinity, then as a consequence it would become what for Aristotle could only be seen as an extreme logical impossibility, where the whole and the part would match. In fact, Aristotle is a philosopher known for his assertion that the whole is greater than the sum of its parts. In fact it wouldn’t even be the sum of the parts, but that a single part is the same as the whole, which would be impermissible.
Aristotle forbad the concept of the actual infinity as leading to a contradiction. As long as the concept of the actual infinity was the concept of the completed infinity, thus an infinity with the final results of something, or limits, then that concept would lead to a contradiction as seen from the way normal language is used, which is pretty evident without needing Aristotle to point it out. For nearly 2400 years, since Aristotle’s banning of it in the 4th century BC to Cantor at the end of the 19th, this prohibition of Aristotle’s was not challenged by a single mathematician. However, it is a different story for the theologians.
In the 4th century AD, the Greek and Roman world, which had produced the greatest philosophers of the Ancient World in Plato and Aristotle, converted to Christianity, some 300 years after the formation of the Christian Church. Just why the world of Greece and Rome, which boasted the wisdom of humanity, converted to the Christian religion which believed in the folly of God is a question that to this day remains of interest, but whatever the reason, Europe became Christian.
What the God of Christianity is, what God is, was of course something that any learned man at the time with any knowledge of Greek philosophy would have asked. However, “No one has ever seen God” (John 1.18). God is something that can never be understood. The Christian leaders of the time, who had been educated in Greek philosophy, or in other words the Greek Fathers, reached the following conclusion after centuries of debate. The eighth century Greek Father John of Damascus formulated this in that his great collection of Christian theology, “An Exact Exposition of the Orthodox Faith” as:
God then is infinite and incomprehensible and all that is comprehensible about him is his infinity and incomprehensibility.
God cannot be limited in his being and cannot be comprehended. This simply means that God must be infinite.
However, the infinite in Greek philosophy was no more than the potential of the matter of this world, its incompleteness. The infinite is given form, and completed, and thus creates an existence in this world. So is God a potential of incompleteness, and existing as something to be completed through being given a form? For the Christian theologians of medieval Europe, these conclusions were unacceptable. God had to be perfect, complete, and final. God needed to be an infinite that had bounds – that was perfect, complete, and final.
The 13th century Thomas Aquinas, one of the most noted Scholastics, the Christian theology of medieval Europe, asserted in his masterwork Summa Theologiae that It is clear that God himself is infinite and perfect. God must be seen as a perfect, complete, and final infinity, or in other words an actual infinity. Thomas, the most loyal to Aristotle, was forced to use the concept of the actual infinity, which Aristotle had so thoroughly rejected. This is an existence which requires the predicate that God is an infinite that has bounds. This was as far as medieval European theology got.
Modern science from the 17th century and later, and its language, modern mathematics, started not by talking about infinites with bounds, the actual infinity, in a logical way but in a practical way. Modern calculus, created by Newtown and Leibnitz, meant the use of the actual infinite as the limit to the infinite series of real numbers, and modern geometry, led by Descartes, not only assumed the infinite Euclidean space that is the direct product of the real number line, but introduced the bound to infinite space known as the point at infinity, shaping the form of the actual infinity for geometry. However, modern mathematics continued to hesitate to provide a theoretical basis for this actual infinity until Cantor at the end of the 19th century. We can only assume they feared Aristotle’s prohibition.
What was modern philosophy doing during this time? Immanuel Kant remained loyal to Aristotle’s prohibition, but Friedrich Hegel was different. Hegel’s philosophy was that the infinite as the extremity of things would go through a stage of the relative infinite differentiated from and in opposition to things made limited or finite by something, then develops into the absolute infinite, or the actual infinite, which has itself as its bounds. This is the development stage theory, or the philosophy of dialectic development. The actual infinity in Hegel’s system was once again denied that it was the opposing object to the infinity denied by having the potential infinity, its origin, being limited, and in the position where it has managed to integrate higher dimensions as the unification of an opposing object that has bounds at the same time it is infinite. Hegel boldly invaded Aristotle’s prohibition, in an attempt to position the actual infinity as the highest stage of the dialectic development where the actual infinity is integrated with the infinity and its bounds.
However, while it is very harsh to say this of Hegel’s colleagues, who hurried to catch up with him, but this dialectical “logic” of Hegel does not solve any of the contradictions inherent in the actual infinity, the contradictions of an infinite with bounds. To start off with, Hegel’s dialectical “logic,” which attempts to unify dialectically A and its negation, ¬A, does not have the substance that can demand a rough logical coherence, or in other words, consistency. A and its negation ¬A cannot both stand. This law of contradiction is a line that cannot be crossed for those who are involved with logic, and also mathematics.
To affirm the infinite with bounds – the actual infinity – without violating the logical contradiction, that was the issue the mathematician Cantor was faced with. Cantor pursued this issue by considering the whole of the infinite, or in other words, the set, and in using this to demonstrate that the bounds of the infinite are its own whole. This is Cantor’s theory of the transfinite. If it is permitted to have as an axiom that the whole of the infinite, the infinite set, exists, then that the infinite has bounds, and that the bounds of the infinite are its own whole, Cantor was able to overcome Aristotle’s prohibition by demonstrating that the actual infinity did not lead to a contradiction. When affirming the actual infinity, rather than saying something like Hegel that it is true because it contains a contradiction, would it not be much better to say that, like Cantor, it is true as it does not lead to a contradiction?
However, mathematicians of the 19th century, as we have already seen, showed an almost emotional attitude in their denial of Cantor’s theoretical defence or basis of the actual infinity. I am forced to consider that the spirits of the mathematicians, which should normally have been free, were being chained by Aristotle’s prohibition. But the theologians were different.
For Christianity at the end of the 19th century, and for the Catholic Church in particular, Pope Leo XIII, who reigned from 1878 to 1903, propounded the idea of Neo-Thomism, which held that modern science, which appeared to be unable to coexist with Christianity, and the theology of Thomas Aquinas, or Thomism, which was at the heart of Catholicism, were in fact able to coexist and were complementary. This was an era where the location of Christianity within the modern world was being redefined. Pope Leo XIII issued his famous encyclical, Aeterni Patris, in which he directed all Catholic Churches to pursue the issue of making modern science and Thomism complementary.
According to Thomas Aquinas, God is a perfect, complete, and finished infinite, or in other words, God is an actual infinity. However, the actual infinity got caught up in the suspicions since Aristotle that it would lead to a contradiction. The German Catholic theologian Constantine Gutberlet, in his 1886 article “Das Problem des Unendlichen” (The Problem of Infinity), rather than considering Cantor’s transfinite numbers as a model of an actual infinity, attempted to shed some light on these suspicions. In other words, the actual infinity, the infinity with bounds, becomes completely consistent if Cantor’s transfinite numbers, that the infinite is its own bounds, is used as a model. Therefore, the theology of Thomas does not lead to a contradiction if the mathematics of Cantor is used as a model.
At the same time, as long as Cantor’s transfinite numbers were taken as a model for the actual infinity of the predicate of God, Catholic theologians had not forgotten to point out that it must not be used as a predicate for any target other than God. The reason was that if this world, which was not God, could be given the predicate of actual infinity, therefore predicated as a transfinite number, the same way as God, then this world itself would end up being God. That this world itself was God, the theory of pantheism, was a heresy that was quite unacceptable not only to Christianity, but to all monotheistic religions. The difference between God and this world, God is other to this world: these are the lines in the sand for monotheistic religions. Therefore transfinite numbers must not be predicated for anything other than God. Cardinal Johann Franzelin, famed for his doctrine of Papal infallibility, pointed this out in an 1886 letter to Cantor.
Cantor was very proud of the passionate support and deep interest in his mathematics shown by theologians such as Franzelin. For Cantor, thrust into the depths of despair by the disinterest and cold rejection from the mathematicians, the interest and support from the Church were without doubt a lifesaver. Moreover, Cantor had accurately forecast how his mathematics would open up new developments in Christian theology. In a letter dated 15 February 1896 and addressed to the Catholic theologian Thomas Esser, he noted as follows: For myself, Christian philosophy is what first provides the true theory of the infinite. What sort of new developments did Cantor’s mathematics offer for Christian theology?
The time has finally come to ask this question.
Theology is the study of the relationship between gods and this world, including we humans. Originally, “No one has ever seen God” (John 1.18). God is not of this world, but is the Other to this world. The essence of God, what God is, is not something that can be grasped in this world, including by we humans, and is not able to be bounded. God is infinite, and in his existence and abilities is vastly beyond our limited selves. God is nothing less than the transcendence of the infinite.
We humans are limited in both space and time, and suffer from physical diseases or pain, economic poverty, or lack of love or hope. We have a bounded existence. This world, including we humans, is also a limited existence in both space and time, and is in opposition to the infinite transcendence of God to the extent that they are both mutually incompatible.
However, the act of religion places this infinitely transcendent God next to we humans, who suffer the limits of existence, and by existing in this finite world, this world, including us, is, religion holds, saved. Religion would not be possible without having this infinitely transcendent God existing in this limited world, or at least the monotheistic religions, including Christianity, would not be.
An infinitely transcendent God is limited and inherent. In terms of normal daily language, this statement is a complete logical contradiction. After all, religion is a logical contradiction, although that assertion would end this discussion right here. But that does not mean that we can just turn around and say it is religion because it is a logical contradiction – that would be very illogical. For example, even if it is a religious statement, if there is a logical impossibility there, then it is just a meaningless statement.
We need to look for a way to explain how an infinite transcendence can be inherent, with limits, at the same time, without involving ourselves in logical contradictions. Cantor’s set theory, especially his theory of the infinite bounded by itself, or in other words the theory of transfinite numbers, appears in this aspect of theology. If we can permit God to be predicated as a transfinite number, or in other words, the infinite set, then it becomes possible to assert that God is an infinity bounded by himself, and is both infinitely transcendent and limited and inherent, without transgressing into a logical contradiction. Cantor’s mathematics provides us with a language to explain one of the most fundamental issues in theology without transgressing into a logical contradiction.
This section, which discusses how Cantor’s mathematics gave a language to explain without contradiction the core issue that in Christian theology God was an infinite transcendence at the same time as being a bounded inherence, and the previous section, which discussed how Cantor’s mathematics became a language to explain without contradiction the issue of actual infinity may seem to indicate a tendency towards some form of discrepancy. In both cases, the way they use Cantor’s mathematics to explain without contradiction the very contradictory way in which God can be a bounded infinite. However, whereas in the previous section God was described as a perfect infinite and yet an infinite with bounds, in this section he is described as an infinite that is inherent in this bounded world and yet must be an infinite without bounds. In other words, in the previous section, God is complete and yet has bounds, and in this section, God must have bounds as this world has bounds.
For this world to have bounds is for we humans to have bounds to our existence and abilities, as we see them, and these are the same as our weakness, our suffering. However, God is complete and yet has bounds. This implies that God shares with us our weakness, our suffering, through being inherent in this world, including we humans, by being complete and yet with bounds, and by being complete and yet bounded. As the word of God notes, “My power is made perfect in weakness”. (2 Cor.12.9)
God is an actual infinity, a complete infinity, and is an infinity that has himself as bounds, yet is inherent within our bounded, limited weakness, and can share our suffering. God is weak precisely as he is complete, and God is complete precisely as he is weak. This core paradox of Christianity is explained using Cantor’s mathematics in a completely consistent way.
So what does it mean for theology to use mathematics as a model? We all know the rhetorical device of the metaphor. A metaphor is the application of a predicate that is not in the appropriate predicate category for the target that is being given a predicate. This method allows a previously unseen essence, or hidden attribute, of that target to be made visible by creating a category mistake.
For example, people say “God the Father.” This is a metaphor. God can not be, in the biological sense, the father of humans, and is not even male. By giving the object that is God the predicate of the father, we are making a clear category mistake. However, by using God the father as a metaphor, can we not see some previously unseen attribute of God a little more clearly?
Or we might say that God is an actual infinity. This again is a metaphor. Since we cannot know the nature of God, whatever God might be, then even if we give him the predicate of infinity, it still does not change the fact that it is a category mistake. However, by using the metaphor that God is an actual infinity, or infinity with bounds, perhaps we can get a glimpse of the hidden nature of God that shared our sufferings with us.
We must rely on metaphors not just when making assertions about God, but when humans attempt to attribute predicates to the unknown. By comparing an unknown object with something else, we can get closer to what that object is. In the agency we call science, we call the thing to which we liken this unknown object to, the object’s model. It goes without saying that mathematics provides this model to almost all sciences.
Therefore, the way mathematics becomes a model for theology is the same as giving the clearly categorically mistaken concepts of mathematics as a predicate to the objects of theology, of predicating the concepts of mathematics as a metaphor for the objects of theology. In other words, this is the same as using the concepts of mathematics as a metaphor for the objects of theology. By doing this, theology is able to explain whether its own objects are logically consistent or not. This makes possible theology as the apologetics of the religion that is Christianity.
- OCHIAI Hitoshi is a professor of mathematical theology at Doshisha University, Kyoto. He has published extensively in Japanese. All books are written in Japanese, but English translations of the most recent two books (Kantoru—Shingakuteki sūgaku no genkei カントル 神学的数学の原型 [Cantor: Archetype of theological mathematics], Gendai Sūgakusha, 2011; and Sūri shingaku o manabu hito no tame ni 数理神学を学ぶ人のために [Those Learning Mathematical Theology], Sekai Shisōsha, 2009) are available from the author at email@example.com.