The Kalaam Cosmological Argument

The Kalam Cosmological Argument
[from http://www.philosophyofreligion.info]

The temporal, kalam cosmological argument, dates back to medieval Muslim philosophers such as al-Kindi and al-Ghazali. It has recently been restored to popularity by William Lane Craig. Like all cosmological arguments, the kalam cosmological argument is an argument from the existence of the world or universe to the existence of God. The existence of the universe, such arguments claim, stands in need of explanation. The only adequate explanation, the arguments suggest, is that it was created by God.

What distinguishes the kalam cosmological argument from other forms of cosmological argument is that it rests on the idea that the universe has a beginning in time. Modal forms of the cosmological argument are consistent with the universe having an infinite past. According to the kalam cosmological argument, however, it is precisely because the universe is thought to have a beginning in time that its existence is thought to stand in need of explanation.

This argument has the following logical structure:
The Kalam Cosmological Argument

(1) Everything that has a beginning of its existence has a cause of its existence.
(2) The universe has a beginning of its existence.Therefore:
(3) The universe has a cause of its existence.
(4) If the universe has a cause of its existence then that cause is God.Therefore:
(5) God exists.

The first premise of the argument is the claim that everything that begins to exist has a cause of its existence. In order to infer from this that the universe has a cause of its existence the proponent of the kalam cosmological argument must prove that the past is finite, that the universe began to exist at a certain point in time.

The crucial premise of the kalam cosmological argument, then, is the second: “The universe has a beginning of its existence”. How do we know that the universe has a beginning of its existence? Might not the universe stretch back in time into infinity, always having existed? The proponent of the kalam cosmological argument must show that this cannot be the case if his argument is to be successful.

Advocates of the kalam cosmological argument claim that it is impossible that the universe has an infinite past. In support of this claim, modern advocates of the argument often appeal to modern science, specifically to the Big Bang theory. Modern science, they say, has established that the universe began with the Big Bang.

Traditionally, however, it is mathematics that has been used by proponents of the kalam argument in order to establish that the past is finite. There are a number of ways of doing this; I’ll outline three mathematical arguments for the finitude of the past.

Maths and the Finitude of the Past


The kalam cosmological argument rests on the idea that the universe has a beginning; its second premise states as much. Advocates of the argument offer two kinds of argument in favour of this claim: scientific and mathematical. Here three mathematical arguments for the finitude of the past will be outlined.

The first argument draws on the idea that actual infinites cannot exist, the second on the idea that actual infinites cannot be created by successive addition, and the third on the idea that actual infinites cannot be traversed.

If any of these arguments is successful, then the second premise of the kalam arguments will have been proven.

The Impossibility of an Actual Infinite

The first mathematical argument for the claim that the universe has a beginning draws on the idea that the existence of an infinite number of anything leads to logical contradictions. If the universe did not have a beginning, then the past would be infinite, i.e. there would be an infinite number of past times. There cannot, however, be an infinite number of anything, and so the past cannot be infinite, and so the universe must have had a beginning.
Why think that there cannot be an infinite number of anything? There are two types of infinites, potential infinites and actual infinites. Potential infinites are purely conceptual, and clearly both can and do exist. Mathematicians employ the concept of infinity to solve equations. We can imagine things being infinite. Actual infinites, though, arguably, cannot exist. For an actual infinite to exist it is not sufficient that we can imagine an infinite number of things; for an actual infinite to exist there must be an infinite number of things. This, however, leads to certain logical problems.

The most famous problem that arises from the existence of an actual infinite is the Hilbert’s Hotel paradox. Hilbert’s Hotel is a (hypothetical) hotel with an infinite number of rooms, each of which is occupied by a guest. As there are an infinite number of rooms and an infinite number of guests, every room is occupied; the hotel cannot accommodate another guest. However, if a new guest arrives, then it is possible to free up a room for them by moving the guest in room number 1 to room number 2, and the guest in room number 2 to room number 3, and so on. As for every room n there is a room n + 1, every guest can be moved into a different room, thus leaving room number 1 vacant. The new guest, then, can be accommodated after all. This is clearly paradoxical; it is not possible that a hotel both can and cannot accommodate a new guest. Hilbert’s Hotel, therefore, is not possible.

A similar paradox arises if the past is infinite. If there exists an infinite past, then if we were to assign a number to each past moment then every real number (i.e. every positive integer) would be assigned to some moment. There would therefore be no unassigned number to be assigned to the present moment as it passes into the past. However, by reassigning the numbers such that moment number one becomes moment number two, and moment number two becomes moment number three, and so on, we could free up moment number one to be assigned to the present. If the past is infinite, therefore, then there both is and is not a free number to be assigned to the present as it passes into the past.

That such a paradox results from the assumption that the past is infinite, it is claimed, demonstrates that it is not possible that that assumption is correct. The past, it seems, cannot be infinite, because it is not possible that there be an infinite number of past moments. If the past cannot be infinite, then the universe must have a beginning. This is the first mathematical argument for the second premise of the kalam cosmological argument.

The Impossibility of an Actual Infinite created by Successive Addition

The second mathematical argument for the claim that the universe has a beginning draws on the idea that an actual infinite cannot be created by successive addition. If one begins with a number, and repeatedly adds one to it, one will never arrive at infinity. If one has a heap of sand, and repeatedly adds more sand to it, the heap will never become infinitely large. Taking something finite and repeatedly adding finite quantities to it will never make it infinite. Actual infinites cannot be created by successive addition.

The past has been created by successive addition. The past continuously grows as one moment after another passes from the future into the present and then into the past. Every moment that is now past was once in the future, but was added to the past by the passage of time.
If actual infinites cannot be created by successive addition, and the past was created by successive addition, then the past cannot be an actual infinite. The past must be finite, and the universe must therefore have had a beginning. This is the second mathematical argument for the second premise of the kalam cosmological argument.

The Impossibility of an Actual Infinite that has been Traversed

The third mathematical argument for the claim that the universe has a beginning draws on the idea that actual infinites cannot be traversed.

If I were to set out on a journey to an infinitely distant point in space, it would not just take me a long time to get there; rather, I would never get there. No matter how long I had been walking for, a part of the journey would still remain. I would never arrive at my destination. Infinite space cannot be traversed.

Similarly, if I were to start counting to infinity, it would not just take me a long time to get there; rather, I would never get there. No matter how long I had been counting for, I would still only have counted to a finite number. It is impossible to traverse the infinite set of numbers between zero and infinity. This also applies to the past. If the past were infinite, then it would not just take a long time to the present to arrive; rather, the present would never arrive. No matter how much time had passed, we would still be working through the infinite past. It is impossible to traverse an infinite period of time.

Clearly, though, the present has arrived, the past has been traversed. The past, therefore, cannot be infinite, but must rather be finite. The universe has a beginning. This is the third mathematical argument for the second premise of the kalam cosmological argument.