Science
in Christian PerspectiveScience
in Christian Perspective
God and Mathematical Infinity
Brendan Kneale
4403 Redwood Road
Napa, CA 94558
The Problem
How does mathematics relate, if it does, to religion? As a Christian mathematics teacher, I have often asked myself and others that questionówithout satisfactory answers. I envied my colleagues in other academic areas. Teachers of philosophy, social studies, literature, psychologyóand even the physical sciencesócan establish rich and legitimate connections with theological, ecclesiological, and perhaps even devotional matters. But teachers in my discipline generally welcome the fact that religion in no way impinges on their field. I find that disappointing.
In answers to my past inquiries, it was hardly satisfactory to hear religiously-minded people point out that the arithmetical number three and the Trinity were related, that the number 666 is important in one book of the Bible, or that (as I once read) 153 is numerologically important because it signified Christ on the occasion when Peter at the Sea of Tiberias hauled in that many fishóprovided that one more fish be added from the fire on the beach to make 154! Nor was it much more helpful to read in Copernicus that mathematics is important in religion for setting up the liturgical calendar, in particular for determining the dates of Easter. Both he and Ptolemy, however, start their astronomy books by praising mathematics (Copernicus recalls that it is a liberal art) as a sort of propaedeutic for the abstract thinking required in philosophy. That answer is a little more promising but still rather indirect. Also, a few people have written on the ethical value of studying mathematics, but I do not find that they extend the argument beyond secular ethics to religion.
Its quantitative nature seems to make mathematics an unlikely science to connect in any important way to a religion. The involvement of higher mathematics with logic, moreover, is no help in this regard, nor does the lively history of mathematics itself seem to impact religion (or be impacted by it). What is left? Aristotle would probably have claimed that my quest for connections was futile since he believes that religion is primarily concerned with the efficient and final while mathematics is primarily concerned with the formal and material.
A Proposed Answer
As can be surmised from the title of this essay, one opening to a solution of the problem is provided by the contemporary treatment of mathematical infinity. What mathematicians have done with this topic, especially in the last hundred years, shows some relevance to religion. Of course, it needs to be emphasized that this proposed "opening" does not identify God with infinite number or with infinite space. The claim is, rather, that the careful distinctions made by recent mathematicians, and the methodology they currently use, can provide us with ways of thinking about God's infinity that turn out to be religiously useful. In other words, how they think about the infinite, more than what they think of it, can provide some guidelines in our thinking about God.
Some earlier mathematicians and nonmathematicians, from classical Greek times up to the last century, had useful things to say. A few such people will be cited in the following paragraphs.
Background
As in most matters of this kind, the Greeks anticipated, in their own fashion, the question raised here. While it is only since the time of Richard Dedekind and Georg Cantor in the 1880s and 1890s that real progress (in my opinion) has been made, the Greeks did provide insightsóas did thinkers such as Galileo and Nicholas of Cusa in Renaissance times. A quick inspection of a few of these efforts is helpful and sets the problem in the context that Dedekind and Cantor emerged from that the study of numbers elevates and orients the mind to Truth. But Aristotle argued effectively, during an extensive treatment of the role of the infinite in nature and in mathematics (see Book 3 of The Physics), that the actually infinite cannot exist, even in mathematics. He did however admit that the potentially infinite can exist in the same way that time, which is always coming into existence, is said to exist.
What came from analyzing these difficulties was the key distinction between potential and actual infinity. The counting numbers are readily seen to be potentially infinite in the sense that no matter how large an integer you name, there is a larger one. It is not so clear that, taken as a whole, the set of integers is actually infinite. The same thing can be said about geometric straight lines. For instance, Euclid uses the word "infinite" in the former sense though some readers think he means actually infinite. Aristotle notes that it is enough for geometers to have potentially infinite lines available for their theorems. And, of course, the famous Euclidean theorem about "the infinity of the prime numbers" establishes their potential infinity, not their actual infinity.
But Galileo is quite clear in his Dialogue on Two New Sciences that, if we are careful, we can talk, pace Aristotle, about the actual infinity of the counting numbers. What is entailed in such thinking is a willingness to give up familiar rules of arithmetic. For example, there is a sense in which the whole is no longer greater than the part, and this is illustrated by a new definition of "size." Thus, if we take the positive integers as a whole, {1, 2, 3, 4, 5,..., n,...} and call the set Z, we can also take its subset of even integers, {2, 4, 6,..., 2n,...}, call it E and claim that even though E is a part of Z, the two sets are the same "size." The new definition states, quite reasonably, that two sets are the same size if their members can be put into one-to-one correspondence with each other. Since, in the case of Z and E, the even numbers can be put into such a correspondence simply by doubling each member of Z (or by halving each member of E), it follows that the whole of Z is the "same size" as the whole of its subset E. Modern mathematicians use this insight to define an actually infinite set: An actually infinite set is one the members of which can be put into one-to-one correspondence with members of one of its own proper subsets. Thus, not only is Z an actually infinite set, so is E since its members, too, can be put into such a correspondence with, say, every fourth integer. A sort of corollary to this analysis is that combining two distinct infinite subsets does not produce a "larger" infinite set. Similarly, Galileo produces a number of geometric paradoxes to show that if we take a line to be made up of an infinite set of points, we must abandon some of our usual assumptions.
These kinds of reasoning about infinite sets, we know today, can be carried out with logical consistency, though it requires us to take great care not to confuse our intuitions about the finite with the counterintuitive rules about the infinite. I believe such analytical efforts to be salutary exercises for all people interested in the philosophy or theology of the Infinite. Mathematicians learn to treat the actually infinite with great respect, a respect that seems to me readily transferable to a reverence for God.
Galileo was a genius at clarifying the suggestions of his predecessors. One of these was Nicholas of Cusa, an early renaissance thinker much admired, it is said, by Cantor. One argument reiterated by Cusa was that we know the infinite first and the finite only derivatively. Later Descartes rediscovers this idea. His famous line in the Third Meditation puts it: "I see that there is manifestly more reality in infinite substance than in finite, and therefore that in some way I have in me the notion of the infinite earlier than the finite..." Cantor liked this reasoning and switched it to mathematics. He argued that finite lines and finite numbers are embedded in the infiniteóso that we can, for instance, proceed indefinitely far with the counting numbers only because they are like footsteps on the preexisting path provided by the actually infinite.
A recent contemporary philosopher, Karl Jaspers, selected Cusa as one of the five most important thinkers of all time. One characteristic of Nicholas of Cusa was that he liked mathematical analogies. For example, Cusa felt that the infinite is where opposites are reconciled. To show this he noted that "the straight" and "the curved" are opposites, but in a circle of infinite radius the curved circumference becomes straight. In theology likewise, we have to reconcile such opposites as the transcendent and the immanent, and we do so in God. Cusa also praised the role of mathematical symbols. Since we cannot know God directly, we must resort to symbols, as it were. The value of mathematical symbols is that they tend to be univocal and part of a system that carries human certitude to its limits. Moreover, he recalled with approval that some of his predecessors compared the Trinity to an infinite triangle.
Toward the end of the renaissance, in the second book of his Essay Concerning Human Understanding, John Locke rediscovers the important notion which Boethius had stated centuries earlier in these words:
"If you compare the duration of a moment with that of ten thousand years, there is a certain [mathematical] ratio between them, however small, since each is finite. But ten thousand years, however many times you multiply it, cannot even be compared to eternity. Finite things can be compared, but no comparison is possible between the infinite and the finite" (emphasis added).
This insight, it seems to me, can be turned into a profound religious one.
In the middle of the seventeenth century Blaise Pascal made a, perhaps superficial, application of mathematical infinity to religion. In his PensÈes there is one entitled "Infinite Nothingness," an entry which is generally called "Pascal's Wager":
We know that infinity exists, but we are ignorant of its nature. Since we know that it is false to say that number is finite, it must be true that there is infinity in number...We cannot say that it is even, or that it is odd...[Similarly] we may perfectly well know that God exists, without knowing what He is...Let us now speak according to natural lights....Let us examine this point and declare: "Either God exists or He does not." To which view shall we incline? Reason cannot decide for us one way or the other; we are separated by an infinite gulf. At the extremity of this infinite distance a game is in progress, where either heads or tails may turn up. What will you wager?...A bet must be laid. There is no option; you have joined the game. Which will you choose then?...Let us estimate the two possibilities: if you win, you win all; if you lose, you lose nothing. Wager then, without hesitation, that He does exist....There is the infinity of an infinitely happy life to win, one chance of winning against a finite chance of losing, and what you stake is finite. That removes all doubt as to choice: wherever the infinite is, and there is not an infinity of chances of loss against the chance of winning, there are no two ways about it, all must be given...
To substitute this kind of mathematical rationalization for true faith in the living God of Abraham, Isaac, and Jacob is, of course, open to criticism. Nevertheless, by taking actual infinity (both in mathematics and religion) meaningfully, Pascal supplies us with a valuable connection of the sort I have been seeking. On the other hand, infinity is only one of God's traits. An undue emphasis on it (in this paper, for instance!) is dangerous since it might lead to the evil of an "apophatic" religion, one which dwells exclusively on the remoteness of God.
There remain difficulties of these matters. The careful and meticulous thought of Immanuel Kant bears this out. He made a famous list of antinomies about infinity and concluded that the universe is neither finite nor infinite. Perhaps he meant that the universe is potentially infinite. He found the word "transcendental" useful and introduced the category of "the sublime." In this regard, he found the mathematics of number and measurement a hindrance since they dealt with the finite. Thus it became apparent that the time was ripe for mathematicians to develop a good theory of mathematical infinity.
Georg Cantor and Richard Dedekind
As mentioned earlier, we are indebted to two late nineteenth century thinkers for systematizing and rationalizing the treatment of actual infinity in mathematics. Without reviewing here how they did it, I will use their general conclusions to make several claims. The chief claim is that mathematical ideas about actual infinity are often justified and, more to the point, that they are useful in religious thought.
Cantor, in particular, felt that his work had theological value. He is quoted as writing in 1896, "From me Christian philosophy will be offered for the first time the true theory of the infinite." He got in touch with prominent theologians in order to be of service to them in this matter.
Of course, his work provoked vigorous opposition. Leopold Kronecker, a confirmed finitist in mathematics and one of his most famous teachers, opposed his ideas in writing. Henri PoincarÈ did likewise, citing Hermite on his side and calling Cantor's results "a disease." He divided mathematicians into two kinds, constructivists and Cantorians, and supported the former, claiming that our talk about the infinite in mathematics is unconsciously colored by our experience of the finite and is thereby unreliable. Hermite's view apparently was that Cantor was creating only a putative mathematical object; mathematicians are supposed to discover such objects, not create them. Bertrand Russell was also very skeptical.
Today, the opponents of Cantor and Dedekind are very few. The rigor and care with which infinite or "transcendental" numbers are treated have persuaded almost everyone that they constitute proper objects of study. Graduate students are routinely taught Cantor and Dedekind's methods and results. Paradoxes and antinomies are taken seriously and are handled with appropriate definitions and axioms. My admiration for this intellectual achievement leads me to make the following applications.
Take an instance. When mathematicians, speaking about infinite sets, use the expression, "all but a finite number of members," they are saying (somewhat like Boethius) that the infinite "all" is still the same size "all" even if we take away some finite set. The removal of billions of members from an infinite set in no way diminishes its infinity. For me, such an insight helps to solve familiar problems in, say, the theology of Providence. For God as infinite can extend his care simultaneously to billions of details, since "billions" after all are only finite.
Applications
The least that can be said is that the infinity of the mathematicians supplies us with a useful metaphor: many of the assertions we make about God and about a mathematical infinity are similar. Pascal's "wager," described above, is a case in point.
Moreover, Pascal uses the argument from Boethius, Galileo, Locke, and many othersóthat the infinite and the finite bear no ratio to each other. This seems to me a key mathematical principle, one with important religious applications. For example, I find in it a basis for hope. As Pascal says, human beings should by rights disappear (he says, be "annihilated") in relation to God but in fact do not disappear. Indeed, a major part of the "good news" of Christianity is that human beings are important (indeed, have the grace to be sons of God). Hence, in spite of the infinity of God there is hope for us as individuals. Another way to put it is: the individual "one" counts as much with the Infinite as does any finite "all" no matter how numerous. On the other hand, our share in the ratio, being "infinitesimal" should keep us appropriately humble!
The poets and prophets of the Old Testament appreciated the force of the principle that the Infinite and the finite do not compare. "What is man that Thou art mindful of him?" Almost the whole of Psalm 90 is a song to Infinity. And II Peter 3:8 quotes Psalm 90:4: "A thousand years in Thy sight are but yesterday when it is past or as a watch in the night." More to the point, Isaiah 40 says: "All the nations are as nothing before him; they are accounted by him as less than nothing and emptiness." A mathematician can appreciate the observation of Sirach 42: "He is from everlasting to everlasting. Nothing can be added or taken away." Job notes in chapter 9: "His works are great, beyond all reckoning, his marvels beyond all counting." Again, in Quoheleth 3, we read: "God has put eternity into man's mind, yet so that he cannot find out what God has done from the beginning to the end. I know that whatever God does endures forever; nothing can be added to it nor anything taken from it; God has made it so, in order that men may fear him."
It seems to me that the difficult notion of adoration (one of the four kinds of prayer) is made easier by thinking also about actual infinity. And speaking of prayer, I have come to believe that the prayer of children and my more sophisticated prayer are only finitely different from each other, so that vis-ý-vis Infinity they are indistinguishable.
Another insight is that the infinity of God requires the divinity of Christ, since the only possible atonement for sins committed against the Infinite is by way of an infinite redeemer. Psalm 50 calls attention to this necessity in words put into God's mouth: "I do not ask for more bullocks from your farms...for I own all...all that moves in the field belongs to me." No finite sacrifice is enough.
Again, it seems to me that we need not concern ourselves with any apparent insignificance in our careers. Measured against an infinite scale, all careers are infinitesimal. The humblest occupation, if it is in accord with God's will, is just as significant as the most glamorous. There is an ancient saying that we should look at things sub specie aeternitatis; another version could be that we measure things sub specie infinitatis.
Let me give a final eschatological application. To people who might worry that heaven will become boring, we might mention that the finite bears no ratio to the infinite. Even the potential infinity of heaven cannot exhaust the actual infinity of the vision of God. I like to think of heaven as an endless sequence of "peak experiences," each one larger and richer than the preceding oneólike the sequence of natural numbers embedded in the actually infinite set of all numbers. This notion helps me appreciate, for example, the final stanza of the famous hymn, "Amazing Grace":
When we've been there ten thousand years,
Bright shining as the sun,
We've no less days to sing God's praise
Than when we'd first begun. Amen.
Summary and Conclusion
Theological and religious reflection is aided by modern developments in the mathematics of infinity. Contemporary mathematicians have worked out rules and principles for careful reasoning about infinite sets. Some principles that have direct or indirect application to religion include the following:
The finite, the potentially infinite, and the actually infinite can be properly and usefully distinguished, and can be made free of internal contradictions.
Reasoning about the infinite requires abandoning many of the familiar rules that apply to the finite.
The actually infinite bears no ratio to the finite, no matter how large the latter is.
The actually infinite provides a basis for resolving some problems that arise in ordinary matters and, perhaps further, a basis for a sense of wonder.
Notes (The original article does not locate the Notes)
1. A work that disagrees with this paper is Logos: Mathematics and Christian Theology by Granville Henry, Bucknell University Press, 1976. It seems to be written from the perspective of process theology rather than of classical theology.
2. An accessible source for details about Cantor's development of transcendental numbers and of the surrounding circumstances is the article by Joseph Dauben in Scientific American 248, no. 6 (June 1983). He also has a full-length treatment, Georg Cantor: His Mathematics and Philosophy, Cambridge, MA: Harvard University Press, 1979.
3. Many of the surviving classical Greek sayings and writings about infinity are collected in Matter and Infinity in the Presocratic Schools and Plato by Theodore Sinnige; Holland: Van Gorcum and Co., 1968.
4. Karl Jaspers included his book-length account of Nicholas of Cusa in his The Great Philosophers; New York: Harcourt, Brace and World, 1966.
5. Henri PoincarÈ's attack on Cantor is in his Mathematics and Science: Last Essays, New York: Dover Publications, 1963.
6. Bertrand Russell's reservations about infinity are in his Introduction of Mathematical Philosophy, Allen and Unwin, 1953.
7. The writer is indebted to the editor and his reader-consultants for recommendations regarding the text, and to a referee for the reference to Descartes.