Today, Christians share the Kalam Cosmological Argument through social media posts, books, etc. Kalam is an Arabic word, and Muslims, al-Ghazali in particular, deserve much of the credit for the argument. The precursor to the argument, however, was a Christian named John Philoponus.
According to Philoponus, “The eternity of the universe would imply an infinite number of past motions that is continually being increased. But an infinite cannot be added to….” (quoted in William Lane Craig, The Cosmological Argument: From Plato to Leibniz, 1980; Eugene, Oregon: Wipf and Stock, 2001, 53). Philoponus was mistaken, however, in saying that the infinite number has been increased. Aleph-null, the actual infinite, the set, is as big as it can be (Paul Davies, The Edge of Infinity, 27 & 28). Aleph-null + 1 is simply aleph-null. Now, if you think that my rebuttal poses an even bigger problem, consider the following, more fundamental problem with Philoponus: the infinite that he wrote about is ultimately just a timeline. We have no problem with labeling points on a timeline with B.C. and A.D., negative and positive numbers. In what sense is the past being added to? A year passes, and that year, for example, 2020, is now in the past. Where does the collection exist? It doesn’t exist anywhere. Why then is there a problem?
There is a problem because everyday people think of the past in a particular way—they believe that the past is real in some sense. I am not saying that they are wrong, but there is a difference between real things and real events. There is disagreement over which events are real, and the debate isn’t just some digression. Rather, as we’ll see, the controversy about time among philosophers needs to be addressed before we make any judgements about the impossibility of the infinite.
According to the physicist Sean Carroll, there are, ultimately, two philosophical perspectives with respect to time: presentism and eternalism. Presentism is “the idea that what exists and what is real is the three-dimensional universe at some point in time…. The past and the future are not real.” Carroll called presentism “our everyday way of thinking about the world.” Perhaps, a less extreme form of presentism is even more descriptive of common sense notions. The milder form of presentism “treats the present and the past as real, but not the future.” Being a physicist, Carroll doesn’t embrace presentism. As he explained, physics suggests eternalism: “If we know the universe exactly right now, we can predict what the future will be and can reconstruct what the past was…. From that perspective, we begin to think that the past, present, and future are perhaps all equally real” (Sean Carroll, Mysteries of Modern Physics: Time, Chantilly, Virginia: The Teaching Company, 2012, 12 & 13).
What does the eternalism-presentism controversy have to do with real infinites? Quite a lot. Think about it. If the past is only fixed but not real, then the infinite collection doesn’t really exist. If it doesn’t exist, then nothing is being added to it. If, however, our natural way of thinking is correct, then the past is real, and it is intuitive to think that it is being increased. Of course, our intuition may simply be mistaken. The mathematicians can say that aleph-null + 1 = aleph-null, but can we really believe that? It’s possible that ordinary people already believe something just as strange.
Before continuing, I want to preface what follows by saying that I may not be as rigorous as I would be if I were writing for a scholarly audience. Like the science popularizer Isaac Asimov, I see no problem with using “infinity” instead of “aleph-null.” He argued, “What we usually consider as infinity, the endlessness of the integers, has been shown to be equal to aleph-null” (Isaac Asimov, Asimov on Numbers, 1977; New York: Pocket Books, 1978, 82). The Kalam defender may wish to make a distinction. As one proponent has noted, “An actual infinite is not a series but a set” (J. P. Moreland & Kai Nielsen, Does God Exist?, 1990; Prometheus, 1993, 229). Like Asimov, I don’t think it’s necessary to use aleph-null. I will therefore use the word or symbol that is more familiar to a popular audience.
Perhaps the evidence is only anecdotal, but I think that there is an asymmetry in the thinking of ordinary people when it comes to infinites, and it’s significant. Popularizers of science and public intellectuals may tell us that infinity – n = infinity and that infinity + n = infinity, but can we believe both claims? People believe the first claim, infinity – n = infinity. The philosopher Wes Morriston pointed out that the lyrics to “Amazing Grace” include the following verses:
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 first begun
Here, the hymn acknowledges that infinity – 10,000 = infinity. If these days are determined to happen, then, according to the eternalist view, they are part of reality. The hymn implies that God has willed that Heaven shall be never-ending. If we accept eternalism, then the duration of the afterlife is an actual infinite. The actually infinite set can lose a member without decreasing. If we can believe that infinity – n = infinity, then why can’t we believe that infinity + n = infinity? Philosophers and everyday people may predictably object that the past is real while the future really isn’t because the past happened, and the future, although determined, has not. The paradoxes caused by actual infinites can then be eliminated. The actual infinites cease to be parts of reality, and the paradoxes go away with them. Just to be clear, one would have to deny that pre-determined events are real things, parts of the real world. Such a view may be correct, but it is still just one way of viewing reality. (Considering how economical it is to be forward-looking, presentism doesn’t even seem pragmatic. Whether it’s “the truth” is a separate issue.) Furthermore, one could eliminate the paradoxes by adopting the most extreme form of presentism. One could acknowledge that the past is fixed. Who could argue otherwise? Saying that the past is fixed, however, is not the same thing as saying that the past is real. The past does not exist; so how is the past real?
Perhaps it was the translator and not Philoponus who was responsible, but something must be said about the phrase “infinite number.” Strictly speaking, there are only finite numbers. Infinity, according to Asimov, “is not a large number or any kind of number at all…. Infinity is not an integer or any number of a kind with which we are familiar. It is a quality; a quality of endlessness. And any set of objects (numbers or otherwise) that is endless can be spoken of as an ‘infinite series’ or an ‘infinite set’” (Asimov on Numbers, 69 & 72). If Philoponus had written, “The eternity of the universe would imply an infinite set that is continually being increased,” we could spot the fallacy more easily. Infinite sets can’t be increased. Paradoxically, however, they can be added to. All of the numbers from 3 and up is an infinite set. We can easily add to this set. By adding 2 & 1 to the end, we now have a set of all of the integers. For the above reasons, the eternity of the past simply implies a set that is always infinite. Perhaps, there is a problem with that, but Philoponus wasn’t making that point.
As mentioned, al-Ghazali, a Muslim, is truly the author of the Kalam argument. When people recite the argument, they are almost quoting al-Ghazali. He wrote, “It is an axiom of reason that all that comes to be must have a cause to bring it about. The world has come to be. Ergo the world must have a cause to bring it about” (quoted in William Lane Craig, The Cosmological Argument: From Plato to Leibniz, 58). Instead of critiquing one argument or one man, I’ll begin by covering an earlier Islamic philosopher, al-Kindi. Ghazali wrote over a century after al-Kindi. After we grasp al-Kindi’s teachings, we’ll know what al-Ghazali and his successors really mean when they talk about “the world” and “the universe.” A beginning to the universe, all of physical existence, would help believers make their case without any apparent drawbacks. Their arguments, however, prove too much. They show that the universe had a beginning. By universe, they mean “not just … the space, time, and matter of our universe, but … the space, time, and matter of any universe that might ever have existed” (Wes Morriston, “Doubts about the Kalam Cosmological Argument”). How then does God have a referent in reality? According to Thomas Hobbes, “because the universe is all, that which is no part of it is nothing, and consequently nowhere” (Leviathan). Even the orthodox philosopher St. Augustine argued, “Since … the motion of a body is one thing, and the norm by which we measure how long it takes is another thing, we cannot see which of these two is to be called time. For, although a body is sometimes moved and sometimes stands still, we measure not only its motion but also its rest as well; and both by time!… Therefore, time is not the motion of the body” (St. Augustine, Confessions, Book Eleven, Chapter XXIV, 31). If being unchanging doesn’t make God timeless, what could it mean to be timeless? Like Philoponus before them, the Arabic philosophers we’ll cover here argued directly for a beginning to time. A beginning to time raises concerns that a beginning to “the universe” might not necessarily raise. At least, the potential problems aren’t as obvious.
Al-Kindi, like other Muslims, was eager to prove that the universe was created. He believed that Philoponus’s arguments would be of assistance. According to William Lane Craig, Al-Kindi argued from a beginning to time to a beginning of the universe in the following way:
Time must have had a beginning. Now time is not an independent existent, but is the duration of the body of the universe. Because time is finite, so is the being of the universe. Or, to put it in another way: time is the measure of motion; it is a duration counted by motion. Now motion cannot exist without a body—this is obvious, for change is always the change of some thing. But it is equally true that a body cannot exist without motion. (The Cosmological Argument, 62)
According to Al-Kindi, if time began, then the universe couldn’t have existed before time began (if it even makes sense to talk about “before time”). If the universe did exist, then time would have existed then. The beginning of time would have been before the beginning of time—a contradiction. Why? Because if there was a universe, then there is change, and if there is change, there is time. From the above argument, he concluded that the universe began. Denying eternal time in order to deny an eternal universe only works for the believer because God somehow can exist without time; the universe cannot. If either of the two assertions is rejected, then the Kalam fails.
Let’s consider first the assumption that some being can exist timelessly. There is no consensus on this topic. For example, Frederick Engels attacked an author who agreed with Al-Kindi on the nature of time. Engels wrote,
[E]xistence out of time is just as gross an absurdity as existence out of space…. According to [my opponent] time exists only through change, not change in and through time. Just because time is different from change, is independent of it, is it possible to measure it by change, for measuring always implies something different from the thing to be measured. And time in which no recognizable changes occur is very far removed from not being time; it is rather pure time, unaffected by any foreign admixtures, that is, real time, time as such. (Anti-Dühring, trans. Emile Burns, New York: International Publishers, 1939, 60)
Engels was not alone in being suspicious about timeless existence. Consider the following comments, which evidently addressed the Kalam argument:
[S]ome philosophers have thought that the idea of infinite time involves contradictions. But the point to be made is that the idea of God as a first cause presents exactly the same difficulties and contradictions and offers no solutions of them. For the existence of God, on the traditional view, runs back into an infinite past in exactly the same way as the suggested chain of causes. It is true that some theologians, seeing this, have said that God’s eternity is not an infinite extension of time, and that God created time along with the temporal world. (W. T. Stace, Religion and the Modern Mind, 1952; Philadelphia: Keystone-J. B. Lippincott Company, 1960, 245)
God’s eternity can be special and qualitatively different only because God can exist motionlessly. God can do so because He is a spiritual being. Spiritual, in this case, means completely immaterial. To embrace the Kalam, we must not only accept timeless existence; we must accept immaterial existence. It’s even worse. If God acts in a purely mental way, does that count as change? To eliminate time, would we have to believe that God simply did nothing for an eternity? We won’t explore all or even some of these concerns. I just mention them to reinforce what I will postulate later—God as an explanation should be a last resort, even if all other possibilities are really weird.
Al-Ghazali, as mentioned, condensed the Kalam philosophy into an argument that is easy to remember. Starting with his comments, it is a small step to get to the now-famous Kalam Cosmological Argument:
1. Everything that begins to exist requires a cause for its origin.
2. The world began to exist.
3. Therefore, the world has a cause for its origin…
According to al-Ghazali, “no time existed before the universe.” We’ve seen how his position is not held unanimously. At most, al-Ghazali proved that the “series of temporal phenomena must have a beginning” (The Cosmological Argument, 102 – 104). At least one thinker has denied Premise 2 while accepting the finitude of temporal phenomena. As Engels summarized the thinker’s views, “So time had a beginning. What was there before this beginning? The universe, which was then in an identical, unchanging state” (Anti-Dühring, 59). Engels objected, “[S]o long as present-day mechanics holds good … it cannot be explained how it is possible to pass from immobility to motion” (Anti-Dühring, 63). Engels was correct, but present-day mechanical laws have no bearing on what was possible. The believers wouldn’t accept it if you used the Law of Conservation of Mass-Energy to reject their explanation: a supernatural disembodied mind. After all, such a being is logically possible. Unfortunately for them, a universe that enters time after being eternally immobile is also logically possible.
Furthermore, no one can explain how a disembodied mind created a universe from nothing. It’s not necessary for someone to explain how a universe can go from non-motion to motion. Reportedly, Immanuel Kant argued that “even if an infinite series of causes were impossible, it does not follow that there must be a First Cause, just because this is the only way we can conceive of having a start to a series” (Donald A. Wells, God, Man, and the Thinker, 1962; New York: Delta-Dell, 1967, 103). There is nothing, however, inconceivable about a universe that is determined from eternity to become temporal. We may find such a conception to be ridiculous, but we would not, consequently, be justified in accepting the conclusions of the Kalam Cosmological Argument. We would instead be obliged to reconsider whether an infinite series of causes is indeed impossible.
The next Kalam defender we’ll cover is significant for two reasons. First, he’s Jewish. Second, he published an argument that has since become perhaps one of the most famous in the history of philosophy. Saadia ben Joseph was born before al-Ghazali, and, according to one scholar, he was the first to argue that the universe has not always existed (Arnold A. Smith II, “Time and the Medieval World,” Philosophy Now: The Ultimate Guide to Metaphysics, n.d., 12). Given that Philoponus wrote well before Saadia, it’s a reach to say that he was really the first. Perhaps, he was the first to think of the particular argument we’ll examine shortly. Those who are well-read in philosophy will be familiar with Saadia’s argument since it is “nearly identical” to an argument which was published in Immanuel Kant’s Critique of Pure Reason (Craig, The Cosmological Argument, 130).
Saadia “claimed that the past cannot be eternal, for if it were, an infinite amount of time must have already passed prior to the present point in time—so if time were eternal, we could not be at the point in time we are now. If there were no starting point in eternity we can never go half the length of time, since half of infinity is still infinity, which by definition is non-traversable” (Arnold A. Smith II, “Time and the Medieval World,” 12). According to Wallace Matson, however, “it is only impossible to run through an infinite series in a finite time” (The Existence of God, 1965, 60 & 61). Keep in mind that, in Saadia’s argument, time is ex hypothesi, infinite. It’s therefore baffling when a Kalam defender argues, “The past cannot be actually infinite, one might argue, because an infinite number of equal time segments, say, hours, could not successively elapse. It would be foolish to say that they could elapse given infinite time, for the argument is precisely about time itself, and the objector fallaciously posits a time ‘above’ time” (The Cosmological Argument, 290, emphasis in original; see also J. P. Moreland in Does God Exist?, 229). As another Kalam defender put it, “[my opponent] postulat[ed] an infinite series of temporal moments to explain how the infinite series of tasks can be completed” (Does God Exist?, 229). I share the confusion of the late Quentin Smith, perhaps the most articulate critic of the Kalam argument:
As for explaining how an infinite amount of time (e.g., an infinite number of hours) can elapse, it is hard to see why or in what sense this needs to be “explained.” The point is that it is consistent to suppose that this amount of time elapses and that arguments against this supposition are unsound. Stated in terms of hours, I would say that it is a contradiction to suppose that an infinite number of hours has elapsed during a time period consisting of a finite number of hours, but it is consistent to suppose that an infinite number of hours has elapsed during a time period composed of an infinite number of hours. Indeed, it is an analytic truth that “an infinite number of hours take an infinite number of hours to elapse.” (“Reply to Craig: The Possible Infinitude of the Past,” International Philosophical Quarterly, 33.1, 1993, 133)
Let’s be clear. Saadia postulated that “time is infinite” (quoted in Craig, The Cosmological Argument, 128). From the above premise, a Kalam defender concluded, “if there was no beginning, the past could have never been exhaustively traversed to reach the present” (J. P. Moreland, Scaling the Secular City, 1987, 29). Saadia, though, merely asserted that “what is infinite cannot be completely traversed mentally in a fashion ascending [backward to the beginning]” (quoted in The Cosmological Argument, 128). We see here that Kant was possibly influenced by Saadia since Kant also used the counter-intuitive labels (“ascending,” “descending”) in his own work. As he explained, “The series ascends from the conditioned n to m (l, k, i, etc.), and also descends from the condition n to the conditioned o (p, q, r, etc.)” (Critique of Pure Reason, trans. Norman Kemp Smith, New York: St Martin's, 1965, 387). Regardless, what Saadia was conveying here wasn’t very controversial. Of course, the infinite cannot be completely traversed mentally, and it doesn’t seem to matter which direction we’re considering. What we can do mentally and what existence can do are two very different questions.
Assuming that Kant knew Saadia’s argument, we turn to the question: Did Kant improve upon the argument? Does his version succeed? After reading what follows, you can decide for yourself. The argument is short; so a paraphrase wouldn’t be much shorter than the argument itself. Here it is:
If we assume that the world has no beginning in time, then up to every given moment an eternity has elapsed, and there has passed away in the world an infinite series of successive states of things. Now the infinity of a series consists in the fact that it can never be completed through successive synthesis. It thus follows that it is impossible for an infinite world-series to have passed away, and that a beginning of the world is therefore a necessary condition of the world’s existence. (Critique of Pure Reason, trans. Norman Kemp Smith, New York: St Martin's, 1965, 397)
Kant’s argument is still being discussed today (Raymond Tallis, “Did Time Begin with a Bang?” Philosophy Now: The Ultimate Guide to Metaphysics, 7-9); so it might be worthwhile to know how to respond to it. Another legend, Bertrand Russell, had the following to say about Kant’s argument:
[W]hen Kant says that an infinite series can “never” be completed by successive synthesis, all that he has even conceivably a right to say is that it cannot be completed in a finite time....
It is worthwhile, however, to consider how Kant came to make such an elementary blunder. What happened in his imagination was obviously something like this: Starting from the present and going backwards in time, we have, if the world had no beginning, an infinite series of events. As we see from the word “synthesis,” he imagined a mind trying to grasp these successively, in the reverse order to that in which they had occurred, i.e. going from the present backwards. This series is obviously one which has no end. But the series of events up to the present has an end, since it ends with the present. Owing to the inveterate subjectivism of his mental habits, he failed to notice that he had reversed the sense of the series by substituting backward synthesis for forward happening, and thus he supposed that it was necessary to identify the mental series, which had no end, with the physical series, which had an end but no beginning. It was this mistake, I think, which, operating unconsciously, led him to attribute validity to a singularly flimsy piece of fallacious reasoning. (Bertrand Russell, "The Problem of Infinity Considered Historically,” Our Knowledge of the External World, Lecture VI)
We now have reason to believe that Kant lifted the argument from Saadia, but Saadia is still, to this day, obscure; so Russell could not have been expected to know that.
Later in the lecture, Russell said, “The property of being unable to be counted is characteristic of infinite collections, and is a source of many of their paradoxical qualities. So paradoxical are these qualities that until our own day they were thought to constitute logical contradictions.” According to an important article by Max Black, “counting an infinite collection is self-contradictory” (“Achilles and the Tortoise,” Analysis, 11.5, 1951, 100). We’ll give the article the attention it deserves, but before we cover it, I’ll explain briefly why it matters. Some back story: I first learned of the Thomson Lamp from William Poundstone’s book Labyrinths of Reason. I never would have guessed that someone would have used something like the Thomson Lamp to prove that the past couldn’t be beginningless. Evidently, Kalam defenders have resorted to using any puzzle or paradox as long as it makes people uncomfortable with the infinite. For example, a Kalam defender has written that the article from Max Black “has shown that the difficulty with traversing an actual infinite is not related to having enough time” (Does God Exist?, 243). Recall what Saadia taught: half of infinity is still infinity, which by definition is non-traversable. Philosophers have a response:
[T]o say that an infinite series can never be completed simply begs the question. This statement rests on the assumption that however many members of an infinite series are taken, others will remain. Now it is certainly true that if any finite number is taken from an infinite series, other members will remain. However, a reason must be given why an infinite number of members cannot be taken from the series. (Keith Parsons, “Is There a Case For Christian Theism?,” Does God Exist?, 187)
A reason was given, but we have reasons to disagree with the conclusion. Before continuing, ask yourself: Is it really the case that infinite movements can only move a finite number of objects?
As mentioned, philosopher and mathematician Max Black wrote an interesting article about infinity. The part that is most relevant begins after the halfway point. He wrote, “Some writers … have said that the difficulty of counting an infinite collection is just a matter of lack of time” (“Achilles and the Tortoise,” 96). Perhaps, the obscure author Eugen Dühring was ahead of his time when he wrote about “the impermissible contradiction of an infinite series which has been counted” (quoted in Anti-Dühring, 55). According to Black, an infinite count is “impossible.” His argument involves a so-called “infinity machine.” He wrote,
[L]et us imagine a machine…. Let us suppose that upon our left a narrow tray stretches into the distance as far as the most powerful telescope can follow; and that this tray or slot is full of marbles. Here, at the middle, where the line of marbles begins, there stands a kind of mechanical scoop; and to the right, a second, but empty tray, stretching away into the distance beyond the furthest reach of vision. Now the machine is started. During the first minute of its operation, it seizes a marble and transfers it to the empty tray; then it rests a minute. In the next half-minute the machine seizes a second marble on the left, transfers it, and rests half-a-minute. The third marble is moved in a quarter of a minute, with a corresponding pause; the next in one eighth of a minute; and so on until the movements are so fast that all we can see is a grey blur. But at the end of exactly four minutes the machine comes to a halt, and we now see that the left-hand tray that was full seems to be empty, while the right-hand tray that was empty seems full of marbles. (“Achilles and the Tortoise,” 96 & 97, emphasis added)
So far, there doesn’t seem to be a problem with conceiving of a machine that can count infinitely many marbles. Mathematicians apparently do believe that infinitely many tasks could be completed if one could “speed up” (Rudy Rucker quoted in Parsons, “Is There a Case For Christian Theism?,” Does God Exist?, 187). Interestingly, Black argued that there was a problem with us merely conceiving a machine that counted infinitely many marbles:
An obvious difficulty in conceiving of an infinity machine is this. How are we supposed to know that there are infinitely many marbles in the left-hand tray at the outset? Or, for that matter, that there are infinitely many on the right when the machine has stopped? Everything we can observe of [the machine]’s operations … is consistent with there having been involved only a very large, though still finite, number of marbles. (“Achilles and the Tortoise,” 97)
Notably, there are differences between a scenario with infinite marbles and one with infinite cliffs. Cliffs aren’t moved; cliffs don’t fall out of a slot; etc. Black’s question, however, seems to be applicable to any task that involves an infinite number of tasks. Scholars call a task a “super-task” when there are an infinite number of tasks. It’s one thing to describe a super-task. It’s another thing to know whether we can really conceive such a task. If the man climbs the first cliff in one hour, the next cliff in half an hour, the one after that in a quarter of an hour, and, in general, the nth cliff in 1/2n-1 hours, it will look like he’s just climbing a lot of cliffs. Instead of a “blur,” the man will just become invisible.
In the case of the super-task involving cliffs, we would just have to assume that the man sped up enough and that he climbed infinitely many cliffs. What if, instead of cliffs, you had an escalator, and the escalator was trying to send you down, but you were trying to go up. If you sped up at the same rate that the escalator sped up, you wouldn’t go anywhere. Given certain assumptions, couldn’t you say that you took infinitely many steps? Let’s say the machine was programmed to keep speeding up, and we knew it. The problem with a marble though is that a marble has to be in a particular place, either on the left or the right. Keep that in mind when you read Black’s commentary:
Now there is a simple and instructive way of making certain that the machine shall have infinitely many marbles to count. Let there be only one marble in the left-hand tray to begin with, and let some device always return that same marble while the machine is resting. Let us give the name ‘Beta’ to a machine that works in this way. From the standpoint of the machine, as it were, the task has not changed. The difficulty of performance remains exactly the same whether the task … is to transfer an infinite series of qualitatively similar but different marbles; or whether the task … is constantly to transfer the same marble that is immediately returned to its original position….
I said, before, that “some device” always restored the marble to its original position in the left-hand tray. Now the most natural device to use for this purpose is another machine—Gamma, say—working like Beta but from right to left. Let it be arranged that no sooner does Beta move the marble from left to right than [sic] Gamma moves it back again. The successive working periods and pauses of Gamma are then equal in length to those of Beta, except that Gamma is working while Beta is resting, and vice versa. The task of Gamma, moreover, is exactly parallel to that of Beta, that is, to transfer the marble an infinite number of times from one side to the other. If the result of the whole four minutes’ operation by the first machine is to transfer the marble from left to right, the result of the whole four minutes’ operation by the second machine must be to transfer the marble from right to left. But there is only one marble and it must end somewhere! (“Achilles & the Tortoise,” 97 & 98)
In the scenario with Beta and Gamma, there are two machines doing super-tasks, and they cancel each other out. What if the same machine was doing both super-tasks? What if the same machine was moving the marble left to right, then right to left, and on and on? We would essentially have the Thomson Lamp that we’ll see later. It’s coherent to think that if only one super-task is being done, then we could know the outcome. Infinite movements should move infinite marbles just as ten movements should move ten marbles. Although I can’t spot a flaw in Black’s paper, I also can’t see an explanation for why infinite movements can’t move infinite objects. Likewise, it seems undeniable that infinite climbs would be enough to scale infinite cliffs.
If there is a flaw in Black’s argument, we can see it if we tell the story using math. Ironically, James Thomson, who gave us the Thomson Lamp puzzle, told his story by using numbers and mathematical operations (Paul Benacerraf, “Tasks, Super-Tasks, and Modern Eleatics,” The Journal of Philosophy, 59.24, 1962, 770 & 771). Let’s do the same for Black’s infinity machines. Let’s assume, for the sake of argument, that there are infinitely many marbles in the left-hand tray. The infinity machine moves the infinitely many marbles, one by one. The left-hand tray is represented by the equation ∞ -1 -1 -1 … = x. Black says that the left-hand tray is indeed empty, but he says that we can’t know how many were really in the tray; so his equation is x -1 -1 -1 … = 0. The scenario with Beta and Gamma is totally different no matter what you may think about infinity. Look at what the left-hand tray looks like now when we translate the story into math. As Black wrote, “Let there be only one marble in the left-hand tray to begin with.” The equation is then 1 -1 +1 -1 +1 … = x. Beta moves the marble from left to right (-1), but Gamma moves it right to left (+1). The process keeps repeating. When we translate the stories into math, they don't look identical at all. It is true that when we imagine the two scenarios in a “split-screen” in our mind, they look the same in the beginning—in the part that we can visualize before everything becomes a blur—especially if we imagine the marbles coming out of a slot. Mathematically, however, they don’t look the same.
Since x was used as the unknown variable earlier for multiple equations, there may be some confusion. I am not at all implying that x can be solved for if we only have the equations in front of us. To clarify, here is what Black is saying:
Scenario 1: n -1 -1 -1 … = 0
Scenario 2: 1 -1 +1 -1 +1 … = m
If I understand him, he argued that Scenario 2 and its implications should make us wary of saying that n = ∞. Perhaps, we can’t model the second scenario. Why? In math, the sum of +1 -1 +1 -1 … is ½, and the marble has to be either on the left or right. Scenario 2 is obviously problematic; so if one can tie it to Scenario 1, then Scenario 1 implicitly has a problem with it. I’ve given reasons to doubt that the two scenarios are perfectly analogous.
I think Black’s argument is plausible because if we narrow our focus to the left-hand tray, then in both scenarios, we could see a situation modeled by -1 +1 -1 +1 …. Notice that he wrote “tray or slot.” We can think of the remainder of the entire collection rolling down every time one of the marbles is transferred from left to right. In other words, if we only focus on one detail, the two scenarios are indistinguishable, at least in the early stages. If we focus, however, on the whole collection, then there is only subtraction, and the scenario can be modeled with ∞ -1 -1 -1 … = x. We do have a paradox here. Black is correct to focus on the infinite additions, and it is understandable that he would think of a machine (Gamma) that brings about endless additions (whenever it moves the marble from right to left). We must not forget that in the original scenario, every addition is also a subtraction—an addition to the tray is a subtraction from the collection. They are just different ways of describing the same event. Instead of 1 -1 +1 -1 +1 … = m, we have ∞ - ∞ = 0.
Before I get accused of being inconsistent for typing ∞ - ∞ = 0, I’ll briefly explain why I believe it’s permissible here. Isaac Asimov wrote about subtracting infinity from infinity. He included a disclaimer: “Even though ∞ is not a number, we can still put it through certain arithmetical operations” (Asimov on Numbers, 72 & 73). There is no solution for ∞ - ∞. I am not saying that the solution must be zero. I am only arguing that the solution could be zero.
I’ll briefly mention the Thomson Lamp. It was named after J. F. Thomson, a respected scholar. Originally, it was just a lamp that he discussed in a paper:
There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button, the lamp goes off. So if the lamp was originally off and you pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half minute, and so on…. After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off?… It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must either be on or off. This is a contradiction. (quoted in Benacerraf, “Tasks, Super-Tasks, and Modern Eleatics,” The Journal of Philosophy, 59.24, 1962, 767 & 768)
Why do I even bother mentioning this lamp? As I explained earlier, the Thomson Lamp and Beta-Gamma are just two versions of the same device. Translate the story into math and you’ll see:
Say that the reading lamp has either of two light values, 0 (“off”) and 1 (“on”). To switch the lamp on is then to add 1 to its value and to switch it off is to subtract 1 from its value. Then the question whether the lamp is on or off after the infinite number of switchings have been performed is a question about the value of the lamp after an infinite number of alternating additions and subtractions of 1 to and from its value, i.e. is the question: What is the sum of the infinite divergent sequence +1, -1, +1,…? (quoted in “Tasks, Super-Tasks, and Modern Eleatics,” 770 & 771)
When I first learned about the Thomson Lamp, I couldn’t fathom how someone could ever use it to defend the Kalam argument. It was clear that they both involved actual infinites, but other than that, I didn’t see how they could be related. If some philosopher used it, I figured that ordinary folks would view it as a reach. Using Beta-Gamma is just as ridiculous as using the Thomson Lamp, and to my knowledge, no one has ever used the Thomson Lamp in a philosophy of religion debate.
“The kalām argument for the beginning of the universe became a subject of heated debate, being opposed by Aquinas…” (The Cosmological Argument, 110). “[W]hen someone wants to support faith by unconvincing arguments, he becomes a laughing stock for the unbelievers” (Aquinas quoted in The Cosmological Argument, 159). The Kalam argument, the infinity machines, etc. didn’t reach the educated public. Aquinas rejected the ontological argument too (Antony Flew, An Introduction to Western Philosophy, Indianapolis: Bobbs-Merrill, 1971, 183, 187 & 188), but at least it was taught in schools. (Even a Chaplain could write, "The mistake in [the traditional ontological argument] has usually been supposed to be that while the idea of God may include the idea of existing the fact that people have ideas of a certain sort is, of itself, no evidence at all that anything corresponding to the ideas really exists.") The Kalam argument was obscure until fairly recently. A philosophy textbook from the nineteen-sixties could never be written today:
It is hard to grasp the idea of an infinitely old Universe. It is all very well to allow infinity in mathematics, but can there really be a physically existing infinity? One way to persuade oneself of the legitimacy of the idea is to ask oneself whether one thinks the Universe must suddenly come to an end. There seems to be no absolute necessity about this, and hence, turning our gaze backward in time instead of forward, there is surely no necessity for the Universe to have begun at any time. Hence it may be infinitely old. (Michael Scriven, Primary Philosophy, New York: McGraw-Hill, 1966, 118)
In the twenty-first century, philosophers have to grapple with Kalam. Unfortunately, they are playing defense. Prior to J. L. Mackie’s The Miracle of Theism, I am not aware of any mainstream philosopher who tried to adequately debunk Kalam arguments. As of 2021, there is at least one textbook that deals with the Kalam argument the way that textbooks from fifty years ago dealt with the ontological argument, etc. Definitely read A Thinker’s Guide to the Philosophy of Religion by Allen Stairs & Christopher Bernard (New York:Pearson-Longman, 2007) pp. 60-66 for a takedown of the modern Kalam argument.
I know it’s cliché, but the history of the Kalam argument is stranger than fiction. Its intellectual grandfather was Philoponus, a Christian, but the Muslims molded it into the “talking point” that we see today. Unlike the ontological argument, which also received “the scholastic kiss of death,” the Kalam argument doesn’t appear to have had any notable Christian defenders, with one exception in the thirteenth century, until very recently (Toni Vogel Carey, “The Ontological Argument and the Sin of Hubris,” Philosophy Now: The Ultimate Guide to Metaphysics, n.d., 81). Philosophers ignored it (The Cosmological Argument, 48). Bertrand Russell could say in the nineteen-twenties, “There is no reason to suppose that the world had a beginning at all.” Nearly a century later, people say confidently, “The universe began to exist.” Both claims can’t be correct.
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