# Al-Kindī’s Mathematical Metaphysics

Al-Kindī uses infinity to disprove the eternity of the world. His first step can be summarized in his statement:

[I]t is not possible, either for an eternal body or for other objects which have quantity and quality, to be infinite in actuality, infinity being only in potentiality.[1]

He is saying that actual infinity is impossible. Why? And how? Al-Kindī constructs an argument that is heavily based on mathematics:

First, he thought it is necessary to appeal to some of the axioms in mathematics. He calls these axioms: “the true first premises that can be comprehended without any mediation.” By this he probably meant that the mind could grasp their truth immediately, without demonstration or need of proof. These premises need no mediation because they are self-evident, and these we can call axioms. He lists the axioms necessary to disprove the eternity of the world as follows:

- all bodies of which one is not greater than the other are equal;
- equal bodies are those in which the dimensions between their limits are equal in actuality and potentiality;
- that which is finite is not infinite;
- when a body is added to an equal body it becomes the greatest of them, and greater than what it had been before that body was added to it;
- whenever two bodies of finite magnitude are joined, the body which comes to be from both of them is of finite magnitude, this being necessary in (the case of) every magnitude as well as in (the case of) every object which possesses magnitude; and
- the smaller of every two generically related things is inferior to the larger, or inferior to a portion of it.[2]

## Al-Kindī’s argument against the eternity of the world

After stating his thesis—actual infinity is impossible—and supporting it with six mathematical axioms, al-Kindī presents his refutation of Aristotle’s thesis that the world is eternal in a logical argument not unlike a mathematical theorem. The following is the argument as al-Kindī states it:

Now, if there is an infinite body, then whenever a body of finite magnitude is separated from it, that which remains of it will either be a finite magnitude or an infinite magnitude.

If that which remains of it is a finite magnitude, then whenever that finite magnitude which is separated from it is added to it, the body which comes to be from them both together is a finite magnitude; though that which comes to be from them both is that which was infinite before something was separated from it. It is thus finite and infinite, and this is an impossible contradiction.[3]

## The structure of al-Kindī’s argument

If there is an infinite body, then after separating a finite magnitude from it, the remaining body will (A) either be a finite magnitude, or (B) be an infinite magnitude.

Al-Kindī refutes both as necessarily contradictory. Let us start with A.

- The remaining body is a finite magnitude.
- If the remaining body is a finite magnitude, then whenever that finite magnitude, which is already separated from it, is added back to it, then the body, which comes into existence from both of them together, is a finite magnitude (axiom number 5).
- However, that which comes to be from them both (the remaining body with the separated body) is that which was infinite before something was separated from it.
- Therefore, it is finite and infinite.
- We know from axiom number 3 that that which is finite is not infinite.
- Therefore, a finite infinity is an impossible contradiction.
- Thus, actual infinity is impossible.
- Therefore, the existence of an infinite body in actuality (such as an eternal world) is self-contradictory.

———————————————

- Therefore, the world is not eternal.

Let us move now to the second part of the argument (part B). Al-Kindī refutes this part in the following:

- The remaining is an infinite magnitude. If the remainder is an infinite magnitude, then whenever that which was taken from it is added back to it, it will (B1) either be greater than what it was before the addition, or (B2) be equal to what it was before the addition.

For the rest of the argument, please see Dr. Al-Allaf’s book: *The Essential Ideas of Muslim Philosophers, *2006

[1] Ibid., p. 68.

[2] Ibid. Numbers were added.

[3] Ibid., pp. 68–69.