Infinity: A Window on Divinity

2009-07-01 00:00:00

David HilbertThe infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.

The concept of infinity has always fascinated humankind. The excitement and awe that are affiliated with the very idea of this concept inspire us. Poets, philosophers, artists, astronomers, mathematicians and ordinary people have pondered the questions related to infinity over the centuries. Is the universe finite or infinite? Is matter or space infinitely divisible? Is there eternal life? Where does a finite being get the idea of infinity from?

In this article, we first consider the concept of infinity from a mathematical point of view and then try to explain how that is connected to Divinity and the Attributes of God. An incorrect understanding of infinity in mathematics leads to seemingly irresolvable paradoxes; likewise, an incorrect understanding of the Attributes of God leads to an incomplete or incorrect understanding of creation.

Any serious consideration of infinity inevitably leads to mathematics, where the concept has its deepest roots. Mathematicians use the symbol ∞ to denote infinity. To many people infinity is a sort of number that is larger than all other numbers. However, that is not an adequate description of infinity. Strange things can happen in the realm of infinity. Many intuitive ideas fail when infinity enters into the picture. This leads to apparent paradoxes. Here is one example that is attributed to David Hilbert (1862–1943), the famous mathematician whose quotation appears above.^{1} Consider a hypothetical hotel with infinitely many rooms. On a stormy night all the rooms of the hotel are occupied, and a wet and miserable couple shows up asking for a room. The concierge comes up with this clever idea to make room for these people. He says, "The hotel is full but here is what I can do for you. I can move the guests from room 1 into room 2, the guests from room 2 into room 3, and so on. The guests in room n will move into the room n+1, for each n = 1, 2, 3, ...".

So, if we can have a hotel with infinitely many rooms it is possible to accommodate an incoming guest even if the hotel is totally full. But then we see that by a similar reasoning we can accommodate any number of guests, and in fact infinitely many more guests! Clearly, this is not possible in ordinary hotels with finitely many rooms that we are familiar with.

To get more sense of what might happen in the realm of infinity let us consider a couple of other examples. You would have no trouble accepting that two sets have the same number of elements if there is a one-to-one correspondence between them. For example, the set {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} has seven elements because it is in a one-to-one correspondence with the set {1, 2, 3, 4, 5, 6, 7} given by Monday↔1, Tuesday↔2,Wednesday↔3, Thursday↔4, Friday↔5, Saturday↔6, Sunday↔7. Note that there are other possible ways of setting up a one-to-one correspondence between these two sets but what matters is the existence of one such correspondence.

There is absolutely nothing surprising about this example. We all know that there are seven days in a week. However, if the concept of infinity pops up in a similar situation, counterintuitive conclusions are obtained. You may be surprised to find out that the set of all positive integers {1, 2, 3, 4, 5, 6,...} is in a one-to-one correspondence with the set of even positive integers {2, 4, 6,...}. One way of setting such a correspondence is

1 2 3 4 5

← ← ← ← ←

2 4 6 8 10 ...

Therefore, there are as many even (positive) integers as there are of all (positive) integers (this is a mathematically correct statement), even though the set of even integers is a strictly smaller subset of all integers. Such a situation is not possible with finite sets.

There are other well-known examples of "paradoxes" involving infinity such as Zeno’s paradox, and Gabriel’s horn.^{2}

Treating infinity as if it is like other numbers, even with the assumption that it is greater than all other numbers, is a major reason for these "paradoxes." Their resolutions are obtained with a proper understanding of the nature of infinity and the concept of limit. For example, adding or subtracting one element, or any finite number of elements, from an infinite set does not change its size. In some cases adding or subtracting infinitely many elements from an infinite set does not change its size either!

Even mathematicians struggled with the idea of infinity a great deal before rigorously formulating the concept and the rules about it. The fundamental notion of modern calculus is limit, which is intimately connected to infinity, in terms of either infinitely small quantities referred to as "infinitesimal" or infinitely large quantities. It took mathematicians a long time to formulate the modern rigorous definition of limit, the rules involving infinity and infinite quantities, and sets of infinite size.

While the concept of infinity is essential in mathematics, it is not the only field where it is so important. Another field in which the concept is very important is theology. The concept of infinity is intimately connected to the concept of Divinity. In mathematics, compared to infinity all numbers are equal, and equally small. This fact is mathematically expressed as a/∞ for any number a, whether it is 7 or 17 million. This is parallel to the theological fact that compared to God we, and the rest of the creation, are infinitely small. Nobody can claim to be greater than any other human being or any other being in the universe, no matter how powerful or wealthy he or she might be. Therefore, we cannot attribute to anybody greatness to the degree of regarding him or her as divine. Nursi expresses this view in the following words:

"O human! It is a Qur’anic principle that you must not consider anything other than God Almighty as greater than you to the degree that you worship it. Nor must you consider yourself greater than anything and thus claim greatness before and dominion over it. For, just as all creatures are equally far from being the Object of Worship, they are also equal in being creatures."^{3}

We have seen that treating infinity like other numbers (even with the understanding that it is greater than all other numbers) leads to seemingly irresolvable paradoxes. Similarly, treating God like humans leads to an incomplete and incorrect understanding of God and the creation. God is not only greater than all that is created, He is also not bound by the kinds of restrictions that created beings are subject to. With His infinite power, creating one flower is as easy for Him as creating all the creatures on earth every spring. Creating all the trees in the world is no more difficult for God than creating one piece of apple. He can create countless things at the same moment. Creating a trillion things is no more difficult for Him than creating one thing.

Unfortunately, it is all too common to see the evidence of attributing incorrect characteristics and restrictions to the Almighty. For example, some "try to free God from the burden of special acts of creation"4 by attributing the development of species to evolution, natural selection, and random phenomena. They do not seem to understand that it is no "burden" for God to create anything, whether it be a tiny individual or billions of species. He can create a great number of things at once. He is not forgetful or incompetent. Nothing He does interferes with another one of His actions. He continues to create every moment everywhere in the universe and He has power to do so. He is closer to every one of us every moment than we are to ourselves: Assuredly, it is We Who have created human and We know what suggestions his soul makes to him, and We are nearer to him than his jugular vein (Qaf 50:16). Therefore, there is no burden for God in dealing with special acts of creation.

Since we are subject to many restrictions as humans, we may fall into the error of attributing similar restrictions to God. So, we need to be more careful when we consider the Attributes of God. One of the restrictions by which we are bound is time. Unlike us, God is above and beyond time. Many people have difficulty comprehending the idea of destiny (or qadar, or divine decree in Islamic literature). If we understand that God is not restricted by time, it becomes much easier to understand this concept. Since He is beyond time, He does not need to "wait" to see the consequences of events in the universe. While it is not our purpose to have a full discussion of this important article of faith here, we want to point out one of its connections with the topic. A detailed explanation of divine decree can be found in Said Nursi’s Twenty-sixth Word.5 Those who do not understand that God is not restricted by time ask questions like, "What was God doing before creating the universe?" This question is meaningless with a proper understanding of the Attributes of God.

Attributing to God deficiencies by which we are constrained, such as forgetfulness, tiredness, and incompetence, is a great mistake, one similar to treating ordinary numbers like infinity. It leads to paradoxes that we cannot resolve. It prevents us from having a proper understanding of God and the creation. We can only understand the mystery of the creation by a proper understanding of its Creator, and we need to be very careful when talking about the Attributes of this Creator.

*Nuh Aydin is an associate professor of Mathematics at Kenyon College, in Ohio, USA.*

- Maor, E. To Infinity and Beyond: A Cultural History of Infinity, Birkhauser, 1987. See also http://www.brainyquote.com/quotes/quotes/d/davidhilbe181572.html; http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Hilbert.html; http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
- Ibid.
- Nursi, Bediüzzaman Said, The Gleams, Seventeenth Gleam, NJ: Tughra Books, p. 160.
- Collins, F. S. Language of God, Free Press, 2006, p. 140.
- Nursi, The Words, Twenty Sixth Word, NJ: Tughra Books, pp. 479–494.

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