# The Kalām Cosmological Argument

The kalām cosmological argument (hereafter, KCA) is a piece of natural theology that aims to show that there exists a beginningless, spaceless, timeless, immaterial, enormously powerful, personal, Creator of the universe. The most notable—and indeed, the most capable—defender of the argument today is the eminent Dr. William Lane Craig . Atheist philosopher Quentin Smith has noted,

The fact that theists and atheists alike ‘cannot leave [the] Kalam argument alone’ suggests that it may be an argument of unusual philosophical interest or else has an attractive core of plausibility that keeps philosophers turning back to it and examining it once again.

The argument can be formulated in three simple steps:

1.0 Everything that begins to exist has a cause.
2.0 The universe began to exist.
3.0 Therefore, the universe has a cause.

Stated formally in the notation of predicate calculus:

$\forall x(P(x) \to Q(x))$
$P(u)$
$\overline{\therefore Q(u)}$

$P(x): x$ begins to exist, $Q(x): x$ has a cause, $x =$all things, $u =$the spatiotemporal universe.

Clearly this is a logically valid argument; that is to say, if the premises are true, then the conclusion follows necessarily (i.e., it is impossible for the conclusion to be false). So the only question is, Is this a logically sound argument (i.e., are the premises true)?

# 1 Everything that begins to exist has a cause

In defense of premise (1.0), I’ll be brief and simply point out that this premise is constantly verified, and never falsified. That is to say that everything that we have ever known to begin to exist—e.g., The Superdome, The Star-Spangled Banner, the film Iron Man, Albert Einstein—did so from the actions of a cause of some sort; we know of nothing that has ever came into being without a cause. That fact alone is more than enough to make premise (1) much more plausibly true than its negation; a sufficient condition for a good argument.

# 2.0 The universe began to exist

With respect to premise (2.0), I will sketch briefly four reasons—two philosophical, two scientific—for believing it to be more plausibly true than its contradictory.

## 2.1 The impossibility of an infinite regress of events

The first philosophical argument can be formulated as follows:

2.1.1 An actual infinite cannot exist.
2.1.2 An infinite temporal regress of events is an actual infinite.
2.1.3 Therefore, an infinite temporal regress of events cannot exist.

When I use the word “exist,” I mean, “be instantiated in the mind-independent world.” By an “event,” I mean any change. The question is whether the series of past events in the history of the universe comprises an actually inﬁnite number of events, or not. If not, then since the universe cannot ever have existed in an absolutely quiescent state, the universe must have had a beginning.

To understand this argument, one needs to be aware of the difference between an actual infinite, and a potential infinite. When I speak of a potential infinite, I am referring to that—the lemniscate—which Cantor called the “variable finite” and denoted with the sign $\infty$. The role of this infinite is to serve as an ideal limit.

Now contrast that infinite with an actual infinite. This infinite, pronounced by Cantor to be the “true infinite,” is denoted by the symbol $\aleph_{0}$ (aleph zero). This infinite represents the value that indicates the number of all the numbers in the series 1, 2, 3, . . ., . This is the infinite of relevance to the above argument.

Just to be clear, an actual infinite is a collection of definite, distinct objects, and whose size is the same as the set of natural numbers. A potential infinite contains a number of members whose membership is not definite, but can be increased without limit. Thus, a potential infinite is more appropriately described as indefinite. The most crucial distinction that I am attempting to convey is that an actual infinite is a collection comprising a determinate whole that actually possesses an infinite number of members; while a potential infinite never actually attains an infinite number of members, but does perpetually increase. With that distinction in mind, let us turn to an examination of the premises of the argument.

### 2.1.1 Impossibility of the existence of an actual infinite

What premise (2.11) proposes is that actual infinites cannot exist in the real (i.e., mind-independent) world. If we imagine that they could, what follows is nothing short of absurdity. For example, suppose that I have an infinite number of coins, each individually numbered, 1, 2, 3, . . ., $\infty$. Now, suppose I were to give you all of the even numbered coins: How many coins would that be? Well, there are an infinite number of even numbers, so you would have an infinite number of coins. Since I kept all of the odd numbered coins for myself—and given the fact that the number of odd numbers is also infinite—, it follows that I, too, have an infinite number of coins left. Considering how I began with an infinite number of coins before giving any to you, the equation representing this transaction would be $$infinity – infinity = infinity.$$

Now suppose instead that I were to give to you all of the coins numbered greater than 3: How many coins would that be? Well, there are an infinite number of numbers greater than 3, so you would have an infinite number of coins. But here is the kicker: How many coins would I have left? Obviously, 3, since I gave you all of the coins numbered greater than 3. Considering that I started with an infinite number of coins, and I gave you an infinite number of coins, the equation would be $$infinity – infinity = 3.$$

How can this be?! In both cases I subtracted an identical (infinite) number of coins, from an identical (infinite) number of coins, but came up with non-identical (infinite; 3) answers! Clearly this is a contradiction. What this absurdity therefore shows is that the existence of an actually infinite number of mind-independent things is impossible. As the great German mathematician David Hilbert once put it,

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.

### 2.1.2 An infinite regress of events is an actual infinite

Premise (2.12) states that an inﬁnite temporal regress of events is an actual inﬁnite. This seems beyond dispute: if the universe has existed eternally, then there would be a sequence composed of an inﬁnite number of events stretching back into the past; and the set of all events in the series would be an actually inﬁnite set.

### 2.1.3 Impossibility of the existence of an infinite regress of events

Given the truth of both premises—that an actual inﬁnite cannot exist and that an inﬁnite temporal regress of events is an actual inﬁnite—, it follows necessarily that an inﬁnite temporal regress of events cannot exist. Thus, since the temporal regress of events is ﬁnite, the universe began to exist.

## 2.2 The impossibility of an actually infinite temporal series of events

That brings us to the second philosophical argument in support of the finitude of the universe—the argument from the impossibility of the formation of an actual inﬁnite by successive addition. It too can be formulated in three simple steps:

2.2.1 A collection formed by successive addition cannot be an actual inﬁnite.
2.2.2 The temporal series of events is a collection formed by successive addition.
2.2.3 Therefore, the temporal series of events cannot be an actual inﬁnite.

This argument is independent of the one argued above, for it does not dispute the possibility of the existence of an actual infinite; rather, it contends that a collection which contains an actually infinite number of members cannot be formed by the adding one member after another in succession. But if an actual inﬁnite cannot be formed by successive addition, then the series of past events must be ﬁnite since that series is formed by successive addition of one event after another in time.

### 2.2.1 Impossibility of an actually infinite successively-formed collection

Leaving aside for the moment all of the absurdities that would arise from the mere existence of an actually inﬁnite number of things, one is still left with the serious difﬁculties that arise as a result of attempting the temporal formation of such a collection through a process of successive addition. By “successive addition,” I mean “the accrual of one new element at a (later) time.” Thus, the impossibility of the formation of an actual inﬁnite by successive addition should manifest in the following example: Consider someone who begins at some arbitrary number, $n$, and then tries to count to inﬁnity. For given any ﬁnite number $n, n + 1$ equals a ﬁnite number. Therefore, $\aleph_{0}$ has no immediate predecessor, and thus the insurmountable obstacle should be obvious: If one cannot possibly count to the number just prior to infinity, how could one ever count to infinity? They can’t, because $\aleph_{0}$ is not the terminus of the natural number series; it is outside it and is the number of all the members in the series.

Consider the following illustration involving two beginningless series of coordinated events. Imagine there is a married couple who really enjoys reading books together. Suppose each day the husband reads one book, and the wife reads two. Thus, for every day that passes, the number indicating the total number of books read by the wife should get greater and greater than the number representing the husband’s total (in fact, her number should always be twice his). Now suppose that this (apparently immortal) couple has been reading at this rate every day from eternity past: How many books has the husband read? Since there has been an infinite number of days, and he reads at a rate of 1 book per day, it follows that he has read an infinite number of books. But how many has the wife read? She too has read an infinite number of books. How can this be?! For each day that passed, the disparity between them got greater and greater. But now, magically, the disparity vanishes, and their totals are equal. Surely, this is absurd.

### 2.2.2 A temporal series of events is a successively-formed collection

Granted the truth of the common sense notion of time—the A-Theory—, premise (2.2.2) is obvious: the past did not spring into being whole and entire but was formed sequentially, one event occurring after another.

### 2.2.3 Impossibility of an actually infinite temporal series of events

It follows, therefore, that the temporal series of events cannot be actually inﬁnite.

## 2.3 The expansion of the universe

We will now discuss the first of two scientific discoveries that serve as confirmations of premise (2.0): the discovery that the universe is (and has been) in a state of cosmic expansion. The evidence for this discovery is found in the field of astronomy and astrophysics. Before the early twentieth century, the standard view was that the universe was static and eternal. But all of that changed in 1917 when Albert Einstein decided to make a cosmological application of his new gravitational theory—the General Theory of Relativity (hereafter, GTR). In so doing he made the assumption that the universe was in a steady state, with a constant mean mass density and a constant curvature of space. Much to his dismay, however, he found that GTR would not permit such a model of the universe unless he introduced into his gravitational ﬁeld equations a certain “fudge factor,” $\Lambda$, in order to counterbalance the gravitational effect of matter and so ensure a static universe. He would later go on to call this is “greatest blunder ever.”

### 2.3.1 The Friedmann–Lemaître model

On the other hand, Alexander Friedmann and Georges Lemaître were able to formulate (independently) solutions to the ﬁeld equations which predicted an expanding universe. Friedmann’s first equation reads

$$H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G \rho}{3} – \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$$

where $H$ = Hubble parameter, $a$ = scale factor, $G$ = gravitational constant, $\rho$ = mass density of universe, $\Lambda$ = cosmological constant, $k$ = curvature parameter, $c$ = speed of light. Friedmann’s ﬁrst equation tells us how the scale factor changes as time elapses (i.e., how the universe expands over time).

Friedmann’s second equation gives the rate of change of the expansion rate: $$\frac{\ddot a}{a} =-\frac{4\pi G}{3} \left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$$

It indicates whether the expansion is slowing or accelerating. A new term “$p$” appears in the equation—it represents pressure.

What was of monumental significance with respect to the Friedman-Lemaître model was its (at the time) radical postulation of an expanding cosmos. As I previously noted, throughout all of human history the universe was regarded as fixed and immutable; the idea that it might actually be changing was inconceivable. But if the Friedman-Lemaître model were correct, then the universe could no longer be adequately treated as a static entity existing in a steady state for eternity past. This drastically changed the field of cosmology.

I think it wise here to take the time to offer some clarity so as to avoid a very common misconception. The Big Bang\Friedman-Lemaître cosmogonic model does not posit some sort of an initial explosion, after which all of the matter in the universe goes flying out into some preexisting space, similar to shrapnel from a hand grenade explosion. It’s critical to understand that, as a GTR-based theory, the model does not describe the expansion of the material content of the universe into a preexisting, empty space, but rather, the expansion of space itself. The galaxies are conceived to be at rest with respect to space but to recede progressively from one another as space itself expands or stretches.

Imagine buttons glued to the surface of a balloon: they will recede from one another further and further the more the balloon inflates. As the universe expands, it becomes less and less dense. This has the astonishing implication that as one reverses the expansion and extrapolates back in time, the universe becomes progressively denser until one arrives at a state of infinite density at some point in the finite past. This state represents a singularity, at which spacetime curvature, along with temperature, pressure, and density, becomes infinite. It therefore constitutes an edge or boundary to spacetime itself. As world renowned physicist P. C. W. Davies explains,

If we extrapolate this prediction to its extreme, we reach a point when all distances in the universe have shrunk to zero. An initial cosmological singularity therefore forms a past temporal extremity to the universe. We cannot continue physical reasoning, or even the concept of spacetime, through such an extremity. For this reason most cosmologists think of the initial singularity as the beginning of the universe. On this view the big bang represents the creation event; the creation not only of all the matter and energy in the universe, but also of spacetime itself.

What is also noteworthy is the fact that the origin it posits is an absolute origin out of nothing. For not only all matter and energy, but space and time themselves come into being at the initial cosmological singularity. As physicists John Barrow and Frank Tipler emphasize,

At this singularity, space and time came into existence; literally nothing existed before the singularity, so, if the Universe originated at such a singularity, we would truly have a creation ex nihilo.

### 2.3.2 Galactic redshift

In 1929 the American astronomer Edwin Hubble showed that the light from distant galaxies is systematically shifted toward the red end of the spectrum. This redshift was taken to be a Doppler effect, indicative of the recessional motion of the light source in the line of sight. In actuality, what Hubble had discovered was the isotropic expansion of the universe predicted by Friedmann and Lemaître on the basis of Einstein’s GTR.

### 2.3.3 Singularity/kinematic-incompleteness theorems

I will only mention one—the Borde-Guth-Vilenkin theorem which was proven in 2003. As I previously explained, it proves (in very non-technical terms) that any universe that has been, on average, expanding throughout its history, cannot be eternal to the past, but must have a past-spacetime boundary (i.e., a beginning).

## 2.4 The second law of thermodynamics

The final piece of evidence in support of premise (2.0) comes from one of the most well-established areas in all of physics—thermodynamics. According to the second law of thermodynamics, entropy in a closed system almost never decreases. According to the second law of thermodynamics, processes taking place in a closed system always tend toward a state of equilibrium. In other words, unless energy is constantly being fed into a system, the processes in the system will tend to run down and quit. For example, if I had a bottle that was a sealed vacuum inside, and I introduced into it some molecules of gas, the gas would spread itself out evenly inside the bottle. It is virtually impossible for the molecules to retreat, for example, into one corner of the bottle. This is why when you walk into a room, the air in the room never separates suddenly into oxygen at one end and nitrogen at the other. It’s also why when you step into your bath you may be confident that it will be an even temperature instead of frozen solid at one end and boiling at the other. It’s clear that life would not be possible in a world in which the second law of thermodynamics did not hold. Given the truth of naturalism, the universe is a closed system. The second law therefore implies that, given enough time, the universe will come to a state of thermodynamic heat death. However, it is possible (rather, likely) that the universe will expand forever; so it may never reach a state of equilibrium. But it will in fact grow increasingly cold, dark, dilute, and dead. So the obvious question arises: Why, if the universe has existed forever, is it not now in a cold, dark, dilute, and lifeless state? P. C. W. Davies provides the painfully obvious answer: “The universe can’t have existed forever. We know there must have been an absolute beginning a finite time ago” . The universe’s energy, says Davies, was simply “put in” at the creation as an initial condition.

# 3 Therefore, the universe has a cause

It follows logically from the truth of the two premises, therefore, the universe has a cause. The final step of the argument is to perform a conceptual analysis to see what properties a cause of the universe must possess.

First, the cause must be uncaused, since, as we have seen from sub-argument (2.0), an infinite regress of causes is impossible. This First Cause must also be beginningless, since by contraposition of Premise (1.0), whatever is uncaused does not begin to exist. Moreover, this cause must be changeless, at least insofar as it exists sans the universe since, once more, an infinite temporal regress of changes cannot exist. From the changelessness of the First Cause, its immateriality follows. For whatever is material involves incessant change on at least the molecular and atomic levels, but the uncaused First Cause exists in a state of absolute changelessness. Given some relational theory of time, the Uncaused Cause must therefore also be timeless, at least sans the universe, since in the utter absence of events time would not exist. It follows that this Cause must also be spaceless, since it is both immaterial and timeless, and no spatial entity can be both immaterial and timeless. If an entity is immaterial, it could exist in space only in virtue of being related to material things in space; but then it could not be timeless, since it undergoes extrinsic change in its relations to material things. Hence, the uncaused First Cause must transcend both time and space and be the cause of their origination. Such a being must be, moreover, enormously powerful, since it brought the entirety of physical reality, including all matter and energy and space-time itself, into being without any material cause.

Finally, and most remarkably, such a transcendent cause is plausibly taken to be personal. This conclusion is implied by the fact that only personal, free agency can account for the origin of a first temporal effect from a changeless cause. We have concluded that the beginning of the universe was the effect of a First Cause. By the nature of the case, that cause cannot have any beginning of its existence nor any prior cause. Nor can there have been any changes in this cause, either in its nature or operations, prior to the beginning of the universe. It just exists changelessly without beginning, and a finite time ago it brought the universe into existence. Now this is exceedingly odd. The cause is in some sense eternal, and yet the effect which it produced is not eternal but began to exist a finite time ago. How can this be? If the necessary and sufficient conditions for the production of the effect are eternal, then why is not the effect eternal? How can all the causal conditions sufficient for the production of the effect be changelessly existent and yet the effect not also be existent along with the cause? How can the cause exist without the effect?

One might say that the cause came to exist or changed in some way just prior to the first event. But then the cause’s beginning or changing would be the first event, and we must ask all over again for its cause. And this cannot go on forever, for we know that a beginningless series of events cannot exist. There must be an absolutely first event, before which there was no change, no previous event. We know that this first event must have been caused. The question is: How can a first event come to exist if the cause of that event exists changelessly and eternally? Why is the effect not co-eternal with its cause?

It’s easy to see this fact with a simple illustration: The necessary and sufficient conditions to account for water’s freezing is sub-zero temperature; if the temperature is sub-zero, then any water around will necessarily be frozen. Now think about this: If the temperature were sub-zero from eternity past, wouldn’t any water that was around be eternally frozen? Would it not be impossible for the water to begin to freeze merely a finite time ago? Indeed, how could causally sufficient mechanical conditions (sub-zero temp.) for the production of an effect (water’s freezing) be eternally in place, and yet, the effect not be co-eternal with the cause? How can the cause exist without its effect?

Another way of seeing the severity of this dilemma is by reflecting on the different types of causation. For instance, there is what philosophers call “state/state causation”: the effect is some state of affairs (e.g., a ceiling fan rotating with constant angular velocity) produced by some other state of affairs (e.g., the switch being in the “on” position). In contrast, we have what’s known as “event/event causation”: the effect comes in the form of some event (e.g., the rotational motion of the fan undergoes a constant rate of deceleration until its angular velocity reaches zero. In other, simpler words: it stops.) which is caused by some other event (e.g., my Wife’s exercising her causal powers to alter the position of the switch from the “on” to the “off” position). The significance here is that in the former type, the cause/effect relationship between the two states could exist eternally; if the switch is eternally in the “on” position, then the fan will eternally rotate at a constant rate.

However, in the case of the origin of the universe we have a peculiar case of what appears to be “state/event causation.” Namely, the effect that we are trying to explain is the origin of the universe (an event); but given the fact that nothing in the spatiotemporal realm existed prior to that first moment, what follows is that it’s logically impossible for whatever ultimately produced our universe to be, itself, also an event. How so? Because events, by their very nature, must have a temporal connotation—an “event” is an occurrence; an instance of something. Thus, if nothing existed—no space, time, matter, or energy—then there could not have been any events. Moreover, since the universe’s coming into being at $t_{0}$ simply is the first spatiotemporal event, it follows logically that there can be no time $t’$ prior to $t_{0}$ at which an event occurs. Therefore, the dilemma confronted is the need to provide a plausible account of how a past-eternal state of affairs, could give rise to a first, temporal event; in what intelligible way can naturalism account for state/event causation?

The best (and perhaps the only) way out of this dilemma is agent causation, whereby the agent freely brings about some event in the absence of prior determining conditions. Because the agent is free, he can initiate new effects by freely bringing about conditions which were not previously present. For example, a man sitting changelessly from eternity could freely will to stand up; thus, a temporal effect arises from an eternally existing agent. Similarly, a finite time ago a Creator endowed with free will could have freely brought the world into being at that moment. In this way, the Creator could exist changelessly and eternally but choose to create the world in time. By “choose” one need not mean that the Creator changes his mind about the decision to create but that he freely and eternally intends to create a world with a beginning. By exercising his causal power, he therefore brings it about that a world with a beginning comes to exist. So the cause is eternal, but the effect is not. In this way, then, it is possible for the temporal universe to have come to exist from an eternal cause: through the free will of a personal Creator. The contrast is, the only way in which a temporal effect could originate from an eternal, changeless cause, would seem to be if the cause is a personal agent who eternally chooses to create an effect in time. A changeless, mechanically operating cause would produce either an immemorial effect or none at all; but an agent endowed with free will can have an eternal determination to operate causally at a (first) moment of time and thereby to produce a temporally first effect. Therefore, the universe is plausibly regarded to be the product of a Personal Creator.

# Conclusion

Thus, the kalām cosmological argument provides sufficient warrant for believing in the existence of an uncaused, beginningless, changeless, immaterial, timeless, spaceless, enormously powerful, personal, Creator of the universe, who I happen to call “God.”

## References Cited

Posted in Natural Theology.