Boxing Pythagoras

Philosophy from the mind of a fighter

WLC doesn’t understand infinity, Part 2

In my previous article, we began to take aim at William Lane Craig’s misconceptions regarding the nature of infinity. We continue on that theme, today, by taking a look at the further arguments which Dr. Craig makes in Part 10 of his Excursus on Natural Theology. While most of the objections which Dr. Craig espouses in this episode fall prey to the same mistakes which he was making last time, I still thought it might be instructive to respond to each one, in turn. Suffice to say, the arguments which Dr. Craig levies this time around are absolutely no better than the ones which he raised previously.

In fact, I’d argue that– for the most part– they are far worse.

The first argument which is brought out is the question of constructing an actual infinite by means of iterative addition.

Now, the series of past events, Ghazali observes, has been formed by adding one event after another. The series of events, in the past, is like a sequence of dominoes, falling one after another, until the last domino, today, is finally reached. But, he argues, no series which is formed by adding one member after another can be actually infinite. For you cannot pass through an infinite number of elements, one element at a time.

Before I even address the argument, itself, I would like to know just where it is that “Ghazali observes” and “[Ghazali] argues” these things, as Dr. Craig claims. It does not appear in Ghazali’s Incoherence of the Philosophers, which is where the Iranian theologian makes his case for the world not being eternal. I am, in fact, not aware of Ghazali ever making this argument. This, of course, does not invalidate the argument, itself– to suggest otherwise would be a fairly flagrant Genetic Fallacy. However, I am curious as to why Dr. Craig would claim that Ghazali made this argument.

That said, the argument, itself, is not a very good one. Craig attempts to support the position by saying,

Now, I think this is easy to see in the case of trying to count to infinity. No matter how high you count, there is always an infinity of numbers left to count.

Once again, Dr. Craig is conflating the ideas of potential infinity and actual infinity, despite the fact that he made such a clear attempt to delineate the two at the onset of his discussion about the subject. In mathematics, when we say, “count to infinity,” we do not mean to imply that we are counting toward a specific number. Remember, once again, that infinity is not a number. “Count to infinity” is not analagous, for example, to the phrase, “count to 10,” where the implication is to stop counting once an upper bound has been counted– in this case, the number 10. Indeed, the phrase, “count to infinity” is the complete opposite concept, implying that one is to count indefinitely.

It is absolutely correct to note that no single iteration during this process will have an infinite value. When Dr. Craig says “count,” he means to list the Natural numbers sequentially, completely, and in order. So, “count to 10” would mean, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10,” and would not mean any other pattern– for example, “1, -3, 17, 10.” Since no Natural number is an infinite number, it is rather obvious that no iteration of this process will be an infinite number.

However, the absurdity of Dr. Craig’s position can be illustrated quite easily. He would almost certainly not deny that there are actual Rationals in the real world. For example, I am rather sure that Dr. Craig would agree that \frac {1}{2} is a number which is applicable to the real world. However, it is just as impossible to “count to \frac {1}{2}” in the manner Dr. Craig implies as it is to count to an infinite number. So, presumably, the ability to count to a number in this manner tells us nothing at all about that number’s applicability to the real world.

Dr. Craig continues in this line of thought by asking,

But, if you can’t count to infinity, how can you count down from infinity?

In addition to falling prey to the same error as above, Dr. Craig seems to have (once again) forgotten that infinity is not a number. We stressed this point in the previous article– a point which Craig, himself, noted before neglecting it several times. The phrase “count down from infinity” makes no more sense than would phrases like “count down from evenness,” “count down from primeness,” or “count down from Rationality.” You can’t count down from a property of a number, and being infinite is every bit as much a property of a number as is its being even, or being prime, or being Rational.

Dr. Craig is attempting to say that it is impossible for a completed set to be infinite. So, perhaps he is not referring to any single finite step in the iterative algorithm, but is rather referring to whole process. If that is the case, however, his argument seems to simply be an example of egregious Question Begging. He gives no real reason to reject that such a process can have an actually complete infinite number of steps. He simply asserts that it cannot.

Skipping ahead, Dr. Craig starts discussing the idea of supertasks, which he claims are metaphysically impossible. In defense of that claim, he says,

Let’s use the letter omega (ω) to symbolize that process [of the supertask], okay? Omega is an ordinal number of infinity. Now, you say, “Wait a minute! I thought that the number of infinity was \aleph_0— the Hebrew letter, uh, Aleph. That was the number of infinity.” Well, to be precise, that’s the cardinal number of infinity.

Last time, I explained that Dr. Craig was wrong to claim that the Aleph symbol is the one used by mathematicians to represent actual infinites because there are a whole host of different symbols which are used to represent infinite numbers. Yes, \aleph_0 is an infinite number, but it is by no means the only infinite number; nor are all infinite numbers equivalent to one another. This time around, WLC trips into that hole which he dug for himself as he is forced to acknowledge the existence of a different infinite number, ω.

The ordinal number for infinity is ω.

This is a gross misunderstanding. There is no single ordinal number for infinity. Rather, ω is the smallest infinite ordinal. After ω comes ω+1, ω+2, ω+3, et cetera, all of which are also infinite ordinals. This isn’t some sort of occult, esoteric bit of mathematics, stuffed in the back pages of a lost textbook. This is information that Dr. Craig could have learned if he had simply taken a few minutes to read the Wikipedia article on Ordinal numbers. It is absolutely ludicrous that a man so ignorant of these concepts would think himself qualified to confidently pronounce on the metaphysical impossibility of actual infinites.

What that means is, the state that exists at ω+1 is completely indeterminate, with respect to the ω-series… One philosopher who discussed this used the example of a light that is turned on and off, faster and faster and faster and faster, and his question was, “At ω+1, is the light on, or is it off?” And, the answer is, there isn’t any answer, because the state of the lamp at ω+1 is completely unconnected to its state during the ω-series.

No, Dr. Craig, the reason that there isn’t any answer is because the state of the lamp is indeterminate. If we cannot determine the state of the lamp, we cannot answer questions about the state of the lamp. That seems perfectly obvious, but here’s an analogy, in case you are confused. Let’s say that we have some Integer, x, such that x is an even number. Is x divisible by 4? There is no answer to this question, because x‘s “divisibility by 4” is indeterminate. We do not have enough information to determine the answer to the question. This, however, does not imply that it would remain impossible to determine an answer if we were given more knowledge.

Our ignorance of the state of the lamp at ω+1 does not imply a “hole in causality or nature,” as Dr. Craig claims.

So, [Ghazali] says, let’s imagine our solar system, and here is Saturn. And let’s imagine that for every one orbit that Saturn completes, around the sun, Jupiter– which is closer in– completes two… Now, notice that the longer they orbit, the further Saturn falls behind. If Jupiter has done ten-trillion orbits, Saturn has only done five-trillion, and the longer they orbit, the farther and farther Saturn falls behind. If they continue to orbit forever, they will approach a limit at which Saturn is infinitely far behind Jupiter…

Now, let’s turn the story around, says Al Ghazali. Suppose that they have been orbiting the sun, from eternity past, now which one has completed the most orbits? Well, the answer mathematically is that the number of orbits completed is exactly the same.

…As I say, this is his argument, in the 12th Century. It’s just amazing to read this stuff.

Now, I’m really starting to question Dr. Craig’s integrity, because contrary to his assertion, this is not Al Ghazali’s argument, in the 12th Century. Though he makes a very similar argument, Ghazali formulates the situation in a very different way– and for good reason. You see, Al Ghazali lived from the years 1058 to 1111 CE. He died almost five hundred years before Johannes Kepler showed that the planets orbit around the sun. Al Ghazali had no concept of heliocentrism. He had no idea that Saturn and Jupiter orbit the sun. Instead, Ghazali believed that the Earth stood at the center of the universe and that the Heavens were literal spheres surrounding Earth. To Ghazali, the reason that Saturn and Jupiter moved at different rates was because each was attached to a different Heaven which rotated about the Earth at different speeds. The sun, he believed, was also attached to such a Heaven, revolving around the Earth.

The reason that Al Ghazali supposed these revolutions to be occurring from eternity past wasn’t just a simple thought experiment, as Dr. Craig seems to be framing it. Ghazali addressed these revolutions as being past eternal because there were actually people claiming that these revolutions were past eternal, due to their Heavenly nature. Today, no one claims that there are actual spheres surrounding the Earth to which the planets, sun, moon, and stars are attached and which have been rotating in the exact same manner for an infinite amount of time. So, obviously, Dr. Craig decided to update Ghazali’s argument somewhat. Instead of addressing a laughably antiquated cosmology, Dr. Craig decided to talk about our current model of the Solar System. However, since no one in modern times posits that Saturn and Jupiter have been orbiting the sun for an infinite period of time, Dr. Craig still needed to shoehorn modern astrophysics into ancient cosmology in order for the analogy to be, at all, useful.

Still, that shouldn’t cause us to dismiss the thought experiment out of hand. If actual infinites are metaphysically possible, then we can certainly conceive of two cyclical processes which have differing periods of completion existing over the same infinite set of time. Unfortunately, the same pitfall which befell Dr. Craig in the last article rears its ugly head, yet again. He continues to presume that the fact that two numbers are both infinite implies that those numbers are equal. This continues to be just as wrong as it was before. Reformulating Dr. Craig’s argument, let’s say Saturn completes an orbit in exactly one Saturn-year, while Jupiter completes two orbits in exactly one Saturn-year. Now, let’s assume that both planets have been doing so for a whole, completed infinite set of time. We’ll call that number Saturn-years, such that S is a positive, infinite Hyperreal number. If this is the case, then Saturn has completed S orbits while Jupiter has completed 2S orbits. Contrary to Dr. Craig’s assertion, S is not “exactly the same” number of orbits as 2S. It is, in fact, half as many.

They have both completed just infinity. An infinite number of orbits. You can’t get out of this argument by saying that infinity isn’t a number, because it is a number in this case. We’re dealing with an actually infinite number of orbits.

Dr. Craig, please try to remember: infinity is not a number. It is not a number in this case, nor is it a number in any other case. Infinity is a property of numbers. Numbers can be finite, or they can be infinite. The fact that one number is infinite does not imply that it is therefore equivalent to all other infinite numbers, any more than the fact that one number is prime implies it is equivalent to all other prime numbers.

There’s just one more juicy little tidbit about this illustration: Al Ghazali asks, “Is the number of orbits completed odd or even?” And, you know, what the answer is, mathematically– it’s both. It is! It’s both odd and even. So, that again just shows, I think, the absurdity of trying to form an actually infinite number of things by successive addition.

Until such time as Dr. Craig actually understands the mathematics he’s trying to describe, he would do well to avoid claims about what something “is, mathematically.” Mathematically, the definition of an “even number” is a number, n, which can be expressed in the form n=2k, where k is an Integer. Invoking the Transfer Principle of the Hyperreals allows us to extend this definition for infinite numbers. In this case, an even Hyperreal number, N, is any number which can be expressed in the form N=2K, such that K is a Hyperinteger; similarly, an odd Hyperreal number would be of the form N=2K+1. It is important to note that by these definitions, numbers which are not Integers (or Hyperintegers) are neither even nor odd. Though Ghazali claims otherwise in the Incoherence, there is absolutely nothing absurd about saying a number is neither even nor odd– for example, \frac {1}{2} is neither even nor odd.

So, if we have adequate information about the number S, in our above example, we can quite easily determine whether the number of orbits is even, odd, or neither.

Here’s another illustration: suppose we meet a man who claims to have been counting down from eternity past, and he is now finishing. “Negative three. Negative two. Negative one. Zero!” At last! Why, we may ask, is he just now finishing his countdown, today? Why didn’t he finish it yesterday, or the day before that, or the year before that? After all, by then, an infinite amount of time had already elapsed. So, if the man were counting at the rate of, say, one number per second, he’s already had an infinite number of seconds to finish his countdown. He should already be done! In fact, at any point in the infinite past you pick, the man will already be finished with his countdown; which implies that no matter how far back in time you go, you’ll never find the man counting. And that contradicts the hypothesis that he has been counting from eternity!

By now, dear reader, I hope you can anticipate my response before you even read it. Yet again, Dr. Craig declares that actual infinites must be absurd based on his misconception that all infinite numbers are the same number. We can, again, reformulate Dr. Craig’s argument very easily in a manner which completely eliminates the supposed absurdity. Let’s say that our man has been counting one number per second for N seconds, where N is a positive infinite Hyperinteger. At precisely time t, the man counts “zero,” and declares himself finished. Why is he just finishing his countdown at time t rather than at t-1? Because N seconds have elapsed at time t, whereas only N-1 seconds have elapsed at t-1. Yes, N-1 is still an infinite number, but it is not the same number as N. It is, in fact, exactly 1 less than N.

The final argument which Dr. Craig levies against actual infinites, in this lecture, is a quick exposition of the Grim Reaper paradox, but since I have discussed this problem in a previous post, I won’t rehash it here. In summary, I do not find the Grim Reaper paradox convincing as it is predicated upon the idea of a Tensed Theory of Time, and I thoroughly reject that model.

Every single argument which William Lane Craig raises in opposition to the existence of actual infinites is tainted by his ignorance of mathematics. Every absurdity which he proposes can be easily resolved using tools developed and proven more than half a century ago. It’s been more than a century since the idea of actual infinites were controversial, in mathematics. It’s been more than three centuries since we’ve been routinely utilizing completed infinite sets in Calculus, and Archimedes was toying with the concept 2000 years before that.

William Lane Craig does not understand infinity. Worse than that: he thoroughly misunderstands the concept, and frames arguments based upon these misconceptions. These bad arguments are then propagated to his rather sizable audience. Many of those who hear Dr. Craig lay out these positions then regurgitate his Straw Men, never realizing that William Lane Craig is not at all qualified to pronounce on age-old questions of mathematics.

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