Al-Kindī's Eternity of The World in Light of Georg Cantor's Continuum Hypothesis

May 2022 · PHIL 501/5122 Advanced Seminar In Islamic Philosophy

Abstract

In this paper, I deeply analyze the meaning of what infinity is in the Aristotelian tradition and what came before it. I distinguish between the concepts of actual and potential infinity and how Aristotle used that concept to build on his theory for the eternity of the world. I examine possible arguments going back and forth between both Aristotle and Abu Ya'qub Al-Kindi, the Arabic philosopher, where he rejects the theory of the eternity of the world using the three proofs he mentioned in his treatise On First Philosophy. Finally, I analyze Georg Cantor's Continuum Hypothesis, and I suggest a new theory called the Past-Future Bijection Hypothesis to disprove the Eternity of The World.

"No one will drive us from the paradise which Cantor created for us."

— David Hilbert

Outline

I. Aristotle's Definition of Infinities.

II. Aristotle's Theory For The Eternity of The World Against Al-Kindī's

III. Georg Cantor: Past-Future Bijection Hypothesis

Aristotle's Definition of Infinities

The concept of infinity is rather an abstract concept that defies intuition and human nature, for one cannot imagine nor physically conceptualize how something can have no end. However, imagining an end is not the only methodology we can use; Aristotle suggested in his physics an interesting alternative. He suggested thinking of this abstraction as something that always has another thing beyond itself. "Infinity turns out to be the opposite of what people say it is. It is not that which has nothing beyond itself that is infinite, but that which always has something beyond itself" (Aristotle 73). That makes us reimagine infinity somehow, not just as some absurd line that has no beginning nor no end, but as something that is n+1+1+1+1… long where "n" can be any number. By that, how can it be infinite if n is a finite number but at the time we do not stop adding to the finite n? This perplexed Aristotle, Plato, and the Pythagoreans before them to the extent that it made the latter two believe that infinity is an entity existing on its own and not as an accident or some other thing. "Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is nobody outside (the Forms are not outside, because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also" (Aristotle 39). This increasing abstraction of the nature of infinity made Aristotle lose faith in the nature of actual infinity more over time, so he suggested five considerations to decide on their basis to believe in the actual infinite or not:

"From the nature of time—for it is infinite.

From the division of magnitudes—for the mathematicians also use the infinite.

If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.

Again, because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.

Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody—not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought" (Aristotle 40).

By that, Aristotle is able to distinguish between two types of infinities because contradictory nature and many questions that revolve around the former definition of that infinite, most importantly, its existence as a substance or as the essential attribute of some entity (Aristotle 41). So, he suggested the separation between two different types of infinity, actual infinity and potential infinity. An actual infinity is one that exists in its entirety at the same moment, such as an endlessly huge body or, more broadly, any set with an infinite number of members. However, when a finite magnitude may be expanded or multiplied endlessly, this is referred to as a potential infinite. Aristotle, for example, believes that every finite magnitude of space or time is theoretically infinite, in that it may be divided into as many pieces as desired, with smaller divisions still feasible (Aristotle 45). From that, Aristotle can easily deny the existence of the actual infinity, both in concept and reality and depend on the existence of the potential infinite for his theories. Aristotle believed that actual infinity was inconceivable because if it existed, anything would have gained infinite magnitude and would be "larger than the heavens." Nevertheless, he claimed that mathematics pertaining to infinity was not rendered useless by this impossibility because mathematicians only need a limited, arbitrarily big magnitude for their arguments. Aristotle's influence to deduce this idea mainly depends on the notion of addition and division comes from Plato, as he mentioned the same idea: "But Plato has two infinities, the Great and the Small" (Aristotle 40). He asserted this idea in the Physics: "For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different." (Aristotle 46). He also affirms it in Book nine, Chapter six of the metaphysics: "For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately." All of this allows Aristotle to maintain his arguments for the Eternity of the World, as we will explain.

Aristotle's Theory For The Eternity of The World Against Al-Kindī's

Aristotle claims in Book I that everything that exists arises from an underlying thing. As a result, if the universe's basic matter came into being, it would come from something else. However, the nature of matter is to be the fundamental entity from which all other things emerge. As a result, the underlying matter of the universe could only have come into existence from a previously existing matter that was precisely like itself; thinking that the underlying matter of the cosmos came into existence would imply that an underlying matter already existed. Aristotle contended that matter must be everlasting since this notion is self-contradictory. In Book VIII, he makes the case that if an absolute beginning of motion is postulated, the thing to be moved must either (A) have come into being and began to move, or (B) have been in an endless state of rest before beginning to move. Option A is paradoxical since an item cannot move before it comes into being, and the process of coming into existence is itself a "movement," hence the first movement necessitates a previous movement, namely the act of coming into existence. Option B is similarly inadequate for two reasons. He determines that motion must be eternal. Most Greek philosophers agreed with Aristotle that the universe is eternal, which means that it had no beginning nor a creation moment and it will always continue to exist without an end. Finally, in his argument, he argues that the heavens must be eternal because they have a perfect, circular motion thus they are not made of any of the perishable four elements of our regular world.

Abu Ya'qub Al-Kindi, a famous Arabic philosopher and a translator of the works of Aristotle, Greek mathematicians, and Neoplatonists into Arabic. He wrote many treatises, but his most accomplished work is On First Philosophy, where he refutes Aristotle's theory for the Eternity of The World being heavily inspired by Philoponus. Al-Kindi essentially admits that the heavens are formed of an ungenerous and everlasting fifth element – but nonchalantly adds that they were created at the beginning of time (Al-Kindi 40). Of course, Al-Kindi, like Philoponus before him, managed to notice the contradictions of Aristotle where he rejects the existence of the concept of actual infinity, the infinity that adds on itself from both ends, and the same time suggests the eternity of the world which suggests the actualization of the same actual infinity he rejects. An example of this is the argument of the body of the cosmos where the body of the cosmos cannot be infinitely large because the body of the cosmos is finite in spatial magnitude where it expands infinitely, but that does not mean that within that expansion, we can reach a certain limit. Thus, as nothing cannot come before its predicated agent, and time is predicated on the body of the cosmos itself, and the latter is finite, then time has to be finite as well, therefore, eliminating the possibility that the world can be eternal. Despite the observation that Al-Kindi made about Aristotle's contradiction, it failed to take note of something significant, which is the differentiation between Actual and Potential infinity. By that, Aristotle would say that any magnitude of time and space, by the rule of division and addition mentioned before, can be divided into smaller and smaller parts infinitely, making it potentially infinite. That would make the result of this still a finite body, yet such a body can be infinite in its increase, like in the case of the body of the cosmos, and infinite in its decrease like in the case of the division of any finite body would still give a finite amount of parts resulting from such a division. Al-Kindi may reply to the event of counterargument by Aristotle using the Traversal of Infinite Argument where something cannot come before its predicator. So, to reach the current, present moment, an infinite number of seconds or moments should have passed since the beginning of time but, the infinite cannot be traversed. By that, something to come into existence by having infinite things to exist before it is impossible. Therefore, the world cannot be eternal. The idea that this argument is valid or not is questionable because Aristotle himself may not allow for the argument to choose a distant point at the beginning of time, affirming that any point we choose would give a finite number of years nevertheless.

Georg Cantor: Past-Future Bijection Hypothesis

Now, in modern Set Theory, Georg Cantor, the German Mathematician, revolutionized modern mathematics and defied all intuition by introducing the Continuum Hypothesis. First, we have to define both concepts of Cardinality and Bijection. Cardinality is the number of elements that any kind of set, finite or infinite, can contain. Bijection, or one-to-one correspondence, is the equality of any two sets by connecting all of their members together where every member from the first set is connected to only one set from the second set and there are not any members left in either the two sets. A practical example of that is a class of ten students consisting of five males and five females; if every male is able to stand in front of every female with no reminders, then there exists a bijection and the two sets (groups) are equal. Going from there, we can conclude three types of sets, finite sets, countable infinities, and uncountable infinities. Finite sets are, for example, the set the students we mentioned above. It is finite, limited, and can be counted. The second type is countable and uncountable infinities. Let us take the set of integers, for example, ℤ = {… −1, −2, −3, 0, 1, 2, 3 …} This set contains the positive and negative non-rational numbers and the zero. Also, the set of Natural numbers ℕ = {1, 2, 3, 4 …} that contains all the positive nonrational numbers only. We can show their bijection by connecting each element to the other, regardless of the identity of the other element infinitely as follows. ℕ × ℤ = {(0,1), (1,2), (−1,3), (2,4), (2,0) …} or as in the picture below.

By that, we manage to prove their equality using bijection and countability, making both of them countably infinite sets where we can, supposedly, list all of their elements.

However, there exists another type of infinity, the uncountable infinity which is for example the set of real numbers ℝ. Real numbers are numbers that include both rational and irrational numbers. Rational numbers such as integers (-2, 0, 1), fractions (1/2, 2.5), and irrational numbers such as √3, π (22/7), etc., are all real numbers. We can never be able to create a one-to-one correspondence with neither the Naturals nor the integers from the real numbers because simply there would be always a rational number that we could not list. Even the infinity of rationals between the number 1 and the number 2 is larger than that of the integers and the Naturals that we mentioned before because there would always exist a rational number that we were not able to list.

This type of infinity is the infinity that neither Aristotle nor Al-Kindi in their calculations for the eternity of the world had. I believe that Georg Cantor provided us with the true actual infinity, which is the uncountable infinity. The uncountable infinity presents us with unlimited possibilities within time itself, all its rational fractions in both the distant past and the upcoming future. The uncountable infinity of time tells us that there is no way that we can create a one-to-one correspondence between both the past and the future in any case due to the fact that there would always be rational fractions of time that we did not count. By that, we can assume the following. If the past is infinite, and the future is finite, then we cannot create a 1-1 correspondence between both of them. If the future is infinite, and the past is finite, then we cannot create a 1-1 correspondence between both of them. If both of them are infinite, we still cannot create a 1-1 correspondence because we will have one point left, the present moment! Thus, the world cannot be actually infinite and must have been created at a certain point.