WLC doesn’t understand Infinity, Part 1
One of the topics which William Lane Craig often discusses is a question which has been argued in the Philosophy of Mathematics for at least 2300 years. Can an infinite number of things actually exist? Dr. Craig asserts that such actual infinites cannot exist. This is actually a topic which I have discussed before, on this blog, but Dr. Craig attempts to tackle the question quite differently than does Dr. Wildberger. Interestingly, Dr. Wildberger is a mathematician, and most of my objections to his argument pointed out his unfamiliarity with philosophy; while Dr. Craig, on the other hand, is a philosopher, and most of my objections to his argument will point out his unfamiliarity with mathematics.
Dr. Craig has discussed the topic of actual infinities in a number of different places, but I will be referring to his Excursus on Natural Theology, Part 9, for our discussion today. These are the same arguments which I have generally seen Dr. Craig present in his other work, but this happens to be the most recent exploration of the topic from WLC which is available to us.
Early on in the Excursus, Dr. Craig is attempting to explain the difference between an actual infinite and a potential infinite. These are concepts drawn largely from Aristotle’s work, Physics, which were quite central to the discussion of infinite quantities in centuries past. An actual infinity refers to a complete set of entities which has a quantity greater than any finite number. A potential infinity refers to a process or algorithm which can be repeatedly iterated without end. It is at this point in his discussion that Dr. Craig makes the first of many claims about mathematics which are simply not true. He says:
This [potential infinity] is the type of infinity which is used in Calculus, in mathematics, where you have infinite limits.
Yes, the concept of potential infinity is utilized in Calculus, but Dr. Craig seems to be implying that this is done to the exclusion of the actual infinite. This is absolutely not the case. Quite the contrary, the entire purpose of Calculus is to perform calculations over actually infinite sets of entities.
Let’s take a very, very basic example. Let’s say that I am presented with the integral , and asked to evaluate the expression. This is a very simple problem which most students will encounter when they are first introduced to Integral Calculus. Now, let’s take a look at what this expression represents. In Figure 1, below, I have graphed the function , for , as the dark purple curve. The shaded region underneath the curve is the value which we are attempting to calculate with the integral . We are trying to determine the area of this region.
To perform this calculation, we absolutely do make use of a potential infinite, as Dr. Craig alleges. We take the integral and evaluate it as follows:
We can iterate the limit repeatedly, over and over, in order to notice that it trends towards a particular value. There is no boundary to the number b which we can utilize in this iteration– it can become bigger and bigger and bigger, with no greatest possible value. This is what Dr. Craig means by a potential infinity.
However, the reason he is wrong (or, at the very least, misleading) to claim that this is “the type of infinity which is used in Calculus” is because the whole purpose of this type of calculation is to allow us to evaluate an actually infinite quantity. In the case of this particular integral, we are adding up an actually infinite collection of infinitesimally narrow areas underneath the curve in order to determine the whole area underneath the curve.
It is precisely the fact that Calculus makes use of actual infinites which made it (and the 17th Century advancements upon which it was built) controversial amongst mathematicians and philosophers when it was originally developed. There’s a wonderful book on precisely this subject, written by Amir Alexander, called Infinitesimal: How a dangerous mathematical theory shaped the modern world. I highly recommend it to anyone interested in Math History.
Moving on, Dr. Craig then begins to discuss the concept of actual infinity. Here, again, he makes a claim which demonstrates that he really doesn’t understand the mathematics which he is attempting to discuss.
This type of infinity [actual infinity] is symbolized by the Hebrew letter, (Aleph)…
The Aleph symbol, in mathematics, certainly does represent something infinite. However, it is not a symbol which is meant to represent “actual infinity” as a general concept. Rather, the Aleph is a very specific symbol with a very specific meaning. An Aleph describes the cardinality of an actually infinite set. This is a very important distinction, and one which will illuminate a number of the errors which Dr. Craig will make during his discussion of Infinity.
So, Aleph is a number. If you were to ask, “What is the number of elements in the set of Natural numbers?” the answer would be . That is the number of members in the set of Natural numbers.
This is one such error. As I just mentioned, Aleph is the cardinality of a set. It does not refer, necessarily, to the quantity of elements in that set. So, then, what does “cardinality” mean? Cardinality is a description of the algorithm which one might utilize in order to count or list the elements of a particular set. A set is said to have a cardinality of if the elements of that set can be listed in a one-to-one correspondence with the elements of the set of Natural numbers. While this is sometimes referred to as the “size” of the set, this is not at all the same thing as a “quantity,” in the usual sense of the word.
Soon after this, Dr. Craig makes mention of something which is very important, and very true, but something which he will (unfortunately) either forget or ignore in his subsequent points. He says,
The lemniscate (), or the potential infinity, isn’t a number. It’s not a number.
This is an extremely important point, because it underlies the vast majority of the problems and misunderstandings which most people have when they discuss infinities. Arithmetic operations can only be performed on numbers. You cannot, for example, answer the questions, “What is ?” or “How much is ?” or “What do you get when you divide Astronomy by John Wilkes Booth?” These are entirely nonsensical, because arithmetic operations can only be performed on numbers. In exactly the same way, one cannot perform arithmetic operations on , because is not a number. Keep this in mind, for later on.
Dr. Craig is then asked about the Aleph numbers, by someone in his class. The questioner wants to know in what way the Alephs can actually be considered numbers, presuming that arithmetic operations cannot be performed upon them (similar to the manner in which we cannot perform arithmetic with the lemniscate). However, WLC corrects this questioner by saying,
You can do transfinite arithmetic using these numbers. You can do multiplication, and you can do addition, and you can do exponentiation (like , for example). So, this is a number that can be manipulated in arithmetic, in this way. What is interesting (and this will become significant when we talk about whether actual infinites can really exist) is that you can’t do inverse operations, like subtraction and division, with them.
Now, Dr. Craig is certainly correct to point out that can be utilized in mathematical operations. However, he introduces a bit of an equivocation, at this point, which he never corrects throughout the whole of the discussion. The operations of transfinite arithmetic are not the same operations as utilized in basic arithmetic. Transfinite Addition, Multiplication, and Exponentiation are very different from the Addition, Multiplication, and Exponentiation which children learn in school. It is absolutely fallacious to conflate the two, and yet Dr. Craig will do so many times during his discussion.
After this, Dr. Craig goes on to acknowledge that modern mathematics utilizes actually infinite sets all the time, and discusses the topic without issue. He explicitly notes that mathematicians have developed logically consistent and sound systems for discussing such actual infinites. Dr. Craig raises these points because one common objection to the claim that “actual infinites cannot exist” is that modern mathematics invalidates the sorts of arguments classically used in support of the claim. He responds to this objection by saying,
But is that really the case? Modern Set Theory shows that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way, without contradicting yourself (as I said in response to John’s question). Now, all this does is succeed in setting up a certain universe of discourse for talking consistently about actual infinites, but it does absolutely nothing to show that such mathematical entities really exist or that an actually infinite number of things can really exist. If Ghazali is right [about the impossibility of the actual infinite], this universe of discourse may be regarded simply as a fictional realm, rather like the world of Sherlock Holmes, in the Arthur Conan Doyle novels; not something that exists in the real world.
Of course, Dr. Craig neglects to mention that if Ghazali is wrong, then this universe of discourse might be perfectly applicable to the real world. So that leaves us with a rather important question: why should I think that this discourse does not apply to the real world? Well, one way (and, likely, the best way) to show that Modern Set Theory is inapplicable to the real world would be to attack the axioms upon which it has been built. This is the strategy employed by those very few mathematicians in the world (like the aforementioned Dr. Norman Wildberger) who reject the idea of actual infinites. However, this is not the tack taken by Dr. Craig. He doesn’t make any attempt to refute the axioms of Set Theory. In fact, Dr. Craig never even mentions any of the specific axioms of Set Theory.
The only discussion which Dr. Craig proffers in order to support his claim that actual infinites cannot exist is the example of Hilbert’s Hotel. Unfortunately, while he does an adequate job of explaining just what Hilbert’s Hotel is, and the different situations which can arrive in the thought experiment, he doesn’t actually give us very good reasons to think Hilbert’s Hotel refutes the idea of actually existing infinite sets. He simply shows that the thought experiment illustrates counter-intuitive properties of infinite sets, as if that acts as some sort of refutation.
Furthermore, there actually does exist a rather glaring problem with the manner in which Dr. Craig describes the entire thought experiment. Remember that earlier in the article, we explicitly noted that is not a number, and Dr. Craig acknowledged in his discussion that there are some infinite numbers which are larger than other infinite numbers. However, he seems to have completely forgotten this fact as he starts to discuss the Hotel. To simply tell us that the hotel contains “an infinite number” of rooms gives us very little information. What infinite number are we talking about? There are an infinite number of infinite numbers, and some have very different properties from others!
Dr. Craig does not clarify, and seems content with his description of the Hotel as it stands. However, a few small clarifications can alter the whole thought experiment significantly. For example, suppose the Hotel has a countably infinite number of vacant rooms, and an uncountably infinite number of guests arrive. Well, despite the fact that the Hotel currently has an infinite number of empty rooms, it cannot possibly accommodate all of the guests which are arriving (yet another counter-intuitive property)!
This, combined with Dr. Craig’s confusion about cardinality as opposed to quantity, is what leads to his next misunderstanding.
What would happen if people started checking out of the Hotel? Let’s suppose that the people in the odd numbered rooms check out: 1, 2, 3– uh, I mean 1, 3, 5, 7, and so forth. All the odd numbered guests check out. How many guests are left? Well, all the even numbered guests. So, an infinite number of guests are still in the hotel, even though an equal number has already checked out and left the hotel.
Once again, Dr. Craig is ambiguous in his description of these infinite numbers, and does not seem to realize that the simple fact that two numbers are both infinite does not imply that they are equal. Let me reformulate Dr. Craig’s statement, for a moment, to illustrate the problem. In this example, we won’t even talk about an infinite hotel. Let’s say our hotel is finite, with an even number of rooms which are all filled up. Now, let’s say an even number of guests check out of the hotel, leaving an even number of rooms still occupied. So, an even number of guests are still in the hotel, even though an even number of guests has already checked out of the hotel. Would you think that there is any absurdity in that statement? Of course not, because not all even numbers are the same number.
Now, it’s worth noting that all of the mathematics which Dr. Craig has mentioned in his discussion, thus far, is around 100 years old or older. Cantorian Set Theory, Hilbert’s Hotel, transfinite arithmetic– these are things which were developed in the late 19th and early 20th Centuries. And yet, Dr. Craig refers to them as if they are the boundary of modern mathematics with infinities. He does not show any indicator that he is even aware of the fact that infinite mathematics has progressed quite a bit farther since the time of Cantor and Hilbert. Particularly, there is a field of study (now more than 50 years old) which greatly illuminates the problems in WLC’s understanding. Our eminent philosopher seems entirely ignorant of the last half-century’s work with Hyperreal numbers.
You see, despite Dr. Craig’s earlier protestations that inverse operations like Subtraction and Division cannot be performed on infinite numbers, there actually do exist systems in which those operators are defined. The Hyperreal number system is one example. And, unlike the Addition, Multiplication, and Exponentiation operators of transfinite arithmetic, the Hyperreal operators do exactly what you would expect those operators to do based on your childhood math classes.
It’s very easy to now reformulate Dr. Craig’s discussion of people checking out of Hilbert’s Hotel using Hyperreal numbers. Let’s say that the Hotel is currently filled to capacity, and that the Hotel has a positive, countably infinite, Hyperreal number of rooms. Let’s call that number . We can even define exactly– let’s say that is the limit of the sequence . Now, let’s say that all of the occupants in odd numbered rooms check out of the Hotel. Since this amounts to half of the Hotel’s occupants, we can quite easily see that guests have checked out, and that another guests yet remain. So the original number of guests is infinite, and the number of guests which check out is also infinite, but these are not equal infinite numbers. Meanwhile, the number of guests which have checked out and the number of guests which remain are equal infinite numbers.
There is absolutely no absurdity in this. There is no inconsistency. The mathematics is every bit as sound as it would be if had been a finite number.
But now let’s suppose instead that all of the guests in the rooms 3, 4, 5, 6, 7, out to infinity, check out. How many guests are left now? Well, if there’s a room 0, just three are left. And yet, the same number of guests checked out this time as when all of the odd numbered guests left. So you subtract identical quantities from identical quantities and you get non-identical results, which is absurd.
Dr. Craig’s errors persist into this quote. Again, the cardinality of the sets is equal, but the quantity of elements is not. So, again putting this in terms of Hyperreal numbers, we started with guests, and in Scenario 1, we saw that guests checked out of the Hotel. Now, in Scenario 2, all but three of the guests check out of the hotel. Despite Dr. Craig’s claims, this is not an identical quantity to that which left in Scenario 1. Instead, the quantity of guests leaving in Scenario 2 is . Again, this is still an infinite number, but it is a different infinite number than the others we have mentioned. Contrary to WLC’s assertion, , and no absurdity is shown, at all.
Keep in mind that the Hyperreal number system is not just some crazy, esoteric, and untested construction. Quite the contrary, it has been extensively explored by mathematicians for a very long time. In fact, it has been more than half a century since Abraham Robinson proved that the Hyperreal number system is logically consistent if and only if the Real number system is, as well. As a consequence, if you believe that numbers like and are applicable to the real world, you’ll be very hard-pressed, indeed, to give a convincing reason to think that Hyperreal numbers are not.
William Lane Craig is woefully ill-equipped to pass judgment upon the metaphysical possibility of an actual infinite. He seems entirely ignorant of nearly a century’s worth of work, on the subject, and he greatly misunderstands those bits of infinite mathematics with which he is aware. The mathematics which deals with the infinite is logically consistent, and does not produce the absurdities which Dr. Craig claims are produced. He does absolutely nothing to discuss, let alone refute, the axioms upon which infinite mathematics are based. He gives us absolutely no good reason to reject the idea that infinite mathematics might be applicable to the real world.
William Lane Craig simply doesn’t know what he’s talking about, when it comes to Infinity.
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