the Kalam cosmological argument: Defending Premise two

Seth Baker

 

The Kalam Cosmological Argument: First Defense of Premise Two

“The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought…The role that remains for the infinite to play is solely that of an idea.” – David Hilbert, On the Infinite

“The more plausible assumption, which does not produce bizarre consequences, is that the number of past events or moments is finite.” – Richard Creel, Philosophy of Religion

Questioning Premise Two: Did the Universe Begin to Exist?

    The Kalam Cosmological Argument (KCA), as was pointed out in the last post1, is a temporal cosmological argument which is predicated upon the universe’s being temporally finite. That is to say, the argument is based upon the assertion that the universe had a first temporal moment at some point in the finite past. Here is the argument in syllogistic form once more:

1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.2
Because things which begin to exist have causes, it follows that if the universe began to exist, then it has a cause. The pertinent question for this post, then, is “Did the universe begin to exist?” But before we get there, we need to fit ourselves with a helpful conceptual tool and make a very important distinction between two terms.

The Reductio ad Absurdum

    In philosophical discourse, philosophers will sometimes employ what is called the reductio ad absurdum (Latin for “Reduction to the absurd”) in their attempts to show that a certain proposition is false. Before explaining further, take a look at this example of a reductio:
1. If God does not exist, then objective moral values do not exist.
2. Objective moral values do exist.
3. Therefore, God exists.3    
This argument, which is one version of the moral argument for God’s existence, attempts to show that the proposition “God does not exist” is false by demonstrating that the amoral implications of God’s nonexistence are untenable or absurd. In its symbolic form, the reductio often looks like this:
1. P → Q
2. ¬ Q
3. ∴ ¬ P4
But in the case we are now considering, it looks like this:
1. ¬ P → ¬ Q
2. Q
3. ∴ ¬ ¬ P, or P5
    In the word of philosophers Steven Cowan and James Spiegel, “The basic idea behind the reductio argument is to show an opponent’s view to be false by arguing that it leads to a contradiction or an otherwise obvious falsehood.”6 Now, whether the moral argument is sound or not is another question for another post, but it provides for us a good example of a reductio that is used widely in the world of natural theology today.
    The relevancy of the reductio to the second premise of the KCA will be shown shortly, but before that, as I stated above, we need to make an important distinction between two terms.
    To deny that the universe began to exist is to affirm that the universe has existed for a limitless amount of time. In other words, if the universe had no beginning, then there have been an actually infinite number of temporal moments in the universe’s history. But what exactly does it mean for something to be “actually infinite,” anyway? Is there another type of infinite that we should be aware of? If so, how do we differentiate between the two? Let me expound a little.

Distinguishing between Actual and Potential Infinities

    Peter S. Williams defines “actual infinite” thus: “An ‘actual infinite’ is any collection of things which at any given time has a number of members greater than any natural number {0, 1, 2, 3, etc.}.”7 An actual infinite stands in contrast with a merely potential infinite, which is, in the words of William Lane Craig, “a collection that is increasing towards infinity as a limit but never gets there.”8 To offer a helpful picture which distinguishes the actual from the potential infinite, consider the Christian conception of eternal life.
    The free gift of God received through faith in Christ is eternal life with him, but what that gift amounts to is an unending life; a life that will progress towards infinity as an ideal limit, but never get there. In heaven, one will never reach their infinitieth year (the year with a greater value than any natural number) of being there, but will simply add the years indefinitely as they go along. This is expressed well by John Newton in his classic hymn Amazing Grace, where he pens:

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

One’s time spent in heaven will never comprise an actual infinite, but a potential infinite with one experiencing unending, ever-lengthening life in God’s presence.

    Now, why do we have to make this distinction? This distinction is important because actual infinities, in contrast with potential infinities, are metaphysically impossible.9 This has some serious implications for the second premise of the KCA, for if it is true that actual infinities are metaphysically impossible, then it follows that the universe’s history is not comprised of an actually infinite number of temporal moments, but had a first temporal moment a finite time ago.

The Impossibility of an Actually Infinite Number of Things

Let us put the above argument in syllogistic form:
​

1. A beginningless series of events in time entails an actually infinite number of things.
2. An actually infinite number of things cannot exist.
3. Therefore, a beginningless series of events in time cannot exist.10

    This, as you might recognize, is a reductio with premise two being the negation of the consequent of premise one.11 So, why should we think that an actually infinite number of things cannot exist? Well, because it leads to intolerable absurdities. Consider the following thought experiment named “Hilbert’s paradox of the Grand Hotel” after the famous German mathematician David Hilbert:

[S]uppose that there existed…a hotel with an actually infinite number of rooms and an actually infinite number of guests. Every room in Hilbert’s Hotel is booked out to a guest…When a new guest arrives, the manager asks the person in room #1 to move into room #2, and so one, in order to free up room #1 for the new arrival. Before this new guest checked into the hotel there were no vacancies, because an actually infinite number of guests had booked out each and every one of the hotel’s actually infinite number of rooms. After the new guest has been added to the guest list, that guest list is no longer than before! The hotel contains the same indefinite number of guests before and after the new guest checks in, namely, ‘an actual infinity’! This story about Hilbert’s Hotel is consistent with the theoretical definition of an actual infinity. However, is it plausible to believe that Hilbert’s Hotel could actually exist in the real world?12


    The answer to that question is clearly “no.” Such a hotel is patently absurd, and yet it is what one would expect if it were the case that actual infinities could exist. Amazingly, however, Hilbert’s Hotel gets even more preposterous! Craig asks us to take the thought experiment even further, writing,

[T]he situation becomes even stranger. For suppose an infinity of new guests shows up at the desk, each asking for a room. ‘Of course! Of course!’ says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room with a number twice his own. Because any natural number multiplied by two always equals an even number, all the guests wind up in even-numbered rooms. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were already occupied!13

    It isn’t over yet, however, for what would happen if guests at Hilbert’s Hotel started to check out? So far we’ve only seen the absurdities that result from new guests checking into the hotel, but things get even crazier whenever those guests have had their stay! Craig continues, writing,

Suppose the guests in rooms #1, #3, #5…check out. In this case an infinite number of people has left the hotel, and half the rooms are now empty. Now suppose the proprietor doesn’t like having a half-empty hotel (it looks bad for business). No matter! By shifting occupants as before, but in reverse order, he transforms his half-vacant hotel into one that is jammed at the gills!…Can anyone believe that such a hotel could exist in reality? Hilbert’s Hotel is absurd. Since nothing hangs on the illustration’s involving a hotel, the argument, if successful, would show in general that it is impossible for an actually infinite number of things to exist.14

    Hilbert’s Hotel serves as a great example for elucidating the ludicrous consequences of the existence of an actual infinite, but it isn’t the only thought experiment that has been proposed. Consider Richard Creel’s example of a clock that has been running for an actually infinite amount of time:

If an analog clock with an hour hand, minute hand, and second hand has existed and run for an infinite amount of time, with the hour hand circling the face of the clock 1 time every 12 hours, the minute hand circling 12 times every 12 hours, and the second hand circling 720 times every 12 hours, then if the clock had been running for an actually infinite amount of time, then the hour hand would have circled the face of the clock exactly the same number of times as the second hand because each would have circled the face of the clock an infinite number of times, and one infinite is no larger than another infinite. But that conclusion is obviously absurd; therefore the concept of an actual infinite must be flawed.15

    Or, consider still another example. Imagine that there is a circle which possesses a circumference that is actually infinite in size. What would the diameter of the circle be? Well, the diameter size would be comprised of an actual infinite. And what about the radius of the circle? It, too, like the diameter and circumference of the circle, would be infinitely large. So, the circumference, diameter, and radius of the circle would all be identical in size. If they were not all identical, then we would no longer have a circle of an actually infinite size. But if it’s the case that they are all identical, then we no longer have a circle, for a circle’s circumference is necessarily π times larger than its diameter (c = πd) and a circle’s diameter is necessarily twice as large as its radius (d = 2r). The existence of an infinitely large circle, then, is an impossibility. We could go on and on with examples like these which demonstrate the intolerable results of the existence of actual infinities.

Conclusion

    The above examples suffice, I think, to show that the existence of an actually infinite number of things is metaphysically impossible. If such is the case, then it follows necessarily that a beginningless series of events in time cannot exist. This conclusion leads us to the further conclusion that the series of events in time must have a beginning and since things which begin to exist have causes, consequently, the universe has a cause of its existence. Other arguments for the truth of premise two will be considered in the next post.


Notes
1. Thank you to my friend Kit Alcock for his helpful comments on my initial post on the KCA. That post can be read here: https://www.apologeticsdiscipleship.com/blog/premise-one-the-kalam-cosmological-argument. I plan on looking deeper at more issues regarding the first premise of the KCA in the near future.
2. William Lane Craig. Reasonable Faith: Christian Truth and Apologetics, 3rd ed. (Wheaton, IL: Crossway, 2008): 111.
3. For some contemporary defenses of the moral argument for God’s existence, see Paul Copan “A Moral Argument,” in Passionate Conviction: Modern Discourses on Christian Apologetics, ed. Paul Copan and William Lane Craig (Nashville, TN: B&H Academic, 2007); Douglas Groothuis. Christian Apologetics: A Comprehensive Case for Biblical Faith (Downers Grove, IL: IVP Academic, 2011): Chapter 15; William Lane Craig. On Guard: Defending Your Faith with Reason and Precision (East Sussex, England: David C. Cook, 2010): Chapter 6.
4. In symbolic logic, the symbols →, ¬, and ∴ stand for “Implies,” “Not,” and “Therefore,” respectively, while letters stand for individual propositions (most of the time). A proposition is simply a statement that can either be true or false (e.g. Water freezes at 32º, Los Angeles is in China, and Barack Obama was re-elected as president of the United States in 2012). Questions like “Where is the nearest gas station?” and “How long is this going to take?” are not propositions because they cannot be true or false.
5. Negating the negation of a proposition entails the truth of said proposition. So, for example, saying, “Zech is not not going to the football game tonight” is equivalent to saying that Zech is going to the football game tonight.
6. Steven B. Cowan and James S. Spiegel. The Love of Wisdom: A Christian Introduction to Philosophy (Nashville, TN: B&H Academic, 2009): 24.
7. Peter S. Williams. A Faithful Guide to Philosophy: A Christian Introduction to the Love of Wisdom (London, England: Paternoster, 2013): 90.
8. Craig, Reasonable Faith. 116.
9. It is important here to distinguish metaphysical possibility, or narrow logical possibility, from broad logical possibility. A proposition is broadly logically possible if and only if it does not violate any logical rule. On the other hand, a proposition is narrowly logically possible if and only if it can be actualized. So, for example, the proposition “The prime minister is a prime number” is broadly logically possible because it does not violate any of the laws of logic, but it is not narrowly logically possible because there is no possible world in which the proposition is true. For a deeper look at the notion of narrow and broad logical possibility, see J.P. Moreland and William Lane Craig. Philosophical Foundations for a Christian Worldview (Downers Grove, IL: InterVarsity Press, 2003): 49–50.
10. This argument is a slightly modified version of Craig’s in Craig, Reasonable Faith. 116.
11. The consequent is “the second part of a conditional proposition, whose truth is stated to be conditional upon that of the antecedent” (New Oxford American Dictionary). Here’s an example that the reader may find helpful: “If it is raining, then the ground will be wet” is a conditional proposition. The statement “If it is raining” is the antecedent, while the statement “then the ground will be wet” is the consequent. The ground’s being wet is conditional upon it raining, which means that if it were false that the ground was wet, then it follows that it is not in fact raining. It has been helpful for this author to remember that the “if” part of a conditional proposition is the antecedent and the “then” part is the consequent.
12. Williams, A Faithful Guide. 91–92.
13. Craig, Reasonable Faith. 118.
14. Ibid., 118–119.
15. Richard E. Creel. Philosophy of Religion: The Basics (Somerset, NJ: Wiley-Blackwell, 2014): 96.