In Response to Steven Den Beste

As I mentioned earlier, Steven Den Beste at USS Clueless fisked my summary of the kalam cosmological argument--and boy was he peeved. Entitled "Abusing Math for God" (sounds like an allusion to Philip Kitcher's Abusing Science: The Case Against Creationism) his scathing blog calls me an idiot and a tyro, labels my argument as "an almost ideal example of a specious argument," and then builds up to the climactic ad hominem that I'm worse than--horror of horrors--a creation scientist. In the blog's denouement, Den Beste opines "It's a sorry performance indeed." Gee, I'm surprised he didn't liken my summary to a Jack Chick tract.

No doubt Den Beste would say he's just telling it like it is. That's one interpretation. Another interpretation is that his sneering contempt is completely unwarranted. Not surprisingly, I endorse the latter interpretation, for reasons that will become clear throughout this blog. Shall we begin?

Den Beste begins with a brilliant rhetorical bang:

Good grief; where did these idiots study mathematics?

Why is this brilliant? Because he manages to poison the well, make an ad hominem attack, and ask a loaded question in one line. For the record, I'm a fourth-year grad student in the philosophy department at the University of Missouri. That certainly doesn't prove that I'm not an idiot or not a mathematical tyro, but I think it begins to show that Den Beste's smarter-than-thou attitude rings a bit hollow. To wit: philosophers, not just mathematicians, are concerned with set theory and infinity. Of course, I could be a bad philosopher, so let's get to the substance of Den Beste's criticism.

Den Beste takes issue with the following argument in my summary:

1. An actual infinite cannot exist.
   
2. A beginningless series of events in time is an actual infinite.
   
3. Therefore, a beginningless series of events in time cannot exist.

Regarding (1), he asks:

So they claim that an "actual infinite" can't exist so that they can argue that time is bounded in the past. But that doesn't mean it's true. Who says?

Well, they've got a snappy appeal to authority in the form of a quote from David Hilbert, an eminent mathematician who died in 1943.

In the actual blog, "appeal to authority" is hyperlinked to an entry in an online collection of logical fallacies. So we can infer that he's accusing me of committing the fallacy of appealing to authority. The problem is that this fallacy is an informal fallacy, not a formal one. That means that an argument taking the form of an appeal to authority is not inherently fallacious, as is an argument taking the form of, say, affirming the consequent. In other words, there can be good appeals to authority and bad ones. Mine is a good one. Even Den Beste accepts Hilbert's point; he just (wrongly, as we will see) thinks that there is a larger issue not addressed by it. I'll return to Hilbert later.

Den Beste quotes my paragraph discussing the nature of actual infinity and takes issue with several its points. First, I write: "One of the unique traits of an actual infinite is that part of an actually infinite set is equal to whole set." Den Beste responds:

It's true that in some cases it can be demonstrated that a part of an infinite set can be shown to be the same size as the whole set. But, for instance, I can take the infinite set of natural numbers and create a subset consisting of the numbers 5 and 6, and that subset is not infinite.

It's true, albeit irrelevant, that finite sets created from an AI are not equal to the AI, but's it clear from my examples that I'm talking about infinite subsets of the AI. So to be more precise: an AI is a set containing subsets that are equal to the AI set. It is not the case that all subsets of an AI are equal to the AI set.

But even if the subset is infinite in size, that doesn't mean it's equal to the parent set. There's more to equality than size and the set of even numbers is not equal to the set of natural numbers, even though they're both infinite sets.

On the contrary, an infinite subset is equal to the infinite set of which it is a subset. Why? Because the subset's members can be put into a one-to-one correspondence with the members of the AI. Take the number of even numbers and the number of natural numbers. Every even number can correspond to a natural number ad infinitum:

   Even numbers:   2  4  6  8  ...
Natural numbers:   1  2  3  4  ...

Two sets are equal when their members can be put into such a correspondence. So these two sets are in fact equal. One will never have a member that the other will not. The upshot is that the whole is not greater than the part when it comes to AIs and their infinite subsets. That property is what precludes their existence. Why? If one doesn't care for hotels or museums or libraries, let's take Bonaventura's example of celestial revolutions.

Imagine a moon in our universe that revolves around a planet three times to every one time the planet revolves around a sun. Which object has made the most revolutions? Obviously the moon, at a 3-to-1 ratio. Now imagine the moon's revolutions and the planet's revolutions to be AI sets. The number of revolutions made by the moon can be put into a one-to-one correspondence with the numbers of revolutions made by the planet. At no point will the set of the moon's revolutions have more revolutions than the set of the planet's revolutions and vice versa. Yet if this moon and planet were actually to exist, we would expect the revolutions of the moon to be three times the number of the revolutions of the planet. We would not expect them to be equal.

Second, I write: "Another trait of the actual infinite is that nothing can be added to it." Den Beste makes a good deal of hay out of this:

It is not true that nothing can be added to a set which is "actually infinite" in size. What they think they're saying is that you can't add anything because it's already in there.
. . . .
It is wrong to even state that an infinite set must necessarily already contain everything there is.
. . . .
Why can't I add books to the infinite museum?
. . . .
Once they've formed their "actually infinite" set of natural numbers, I can add "blue" or "popcorn" or "Steven Den Beste" to it without duplication.

Den Beste has badly misunderstood what's going on. I never state that an AI "must necessarily already contain everything there is." An AI is an infinite, completed set of particular things, not the infinite set of all things. Hence the examples: infinite sets of paintings, books, hotel rooms. I do deny that any new members can be added to such sets. So what about adding "popcorn" to the infinite set of, say, books? Why can't that be done? Because all that does is to create a new set: an infinite set of books and lonely popcorn. It does not add a new member to the AI set of books, because--to state the obvious--popcorn is not a book.

To put the matter differently, you certainly can carry out mathematical operations with an AI--e.g. add or subtract another set--but this has no impact on the number of members in the AI.

Third, I write: "Not one book can be added to an actually infinite library or one painting to an actually infinite museum." Den Beste agrees with this, but he thinks that this just means that I've demonstrated that a particular restricted type of set cannot exist:

It just means they've created one particular set with very restrictive limits on what may be added to it. Other sets with less restrictive limits may not be equally limited, thus their experience with this one set cannot be extrapolated to all conceivable sets. They have proved nothing. If all those things are prevented because of their set definition, that's fine. But it doesn't prove anything about any other set, or about all sets collectively.

This is another misunderstanding. Den Beste thinks that there are different kinds of AIs and that I've only shown that some of those cannot exist. But my argument is that no AI can exist due to properties that all AIs have in common: (1) an AI is a completed totality and (2) an AI can have a subset that is equivalent to the AI such that the members of the subset can be put into a one-to-one correspondence with the members of the AI. Much of the problem is that Den Beste thinks that an AI set of Xs and popcorn somehow escapes the stricture that nothing can be added to an AI. As I point out above, this isn't the case, but the even more salient point is that the AI set of Xs and popcorn cannot exist; the popcorn can, but the AI set of Xs cannot due to (1) and (2).

Just to drive the point home: an AI has certain properties that preclude it from existing. Those properties are common to all AIs. There are no AI sets that somehow escape from this. The AI set of Xs and some Y is not the same as the AI set of Xs itself. Finally, the Y can exist, but the AI set of Xs cannot exist.

That me brings me back to Hilbert. Den Beste thinks that Hilbert's argument is that no AI can exist because some particular AI cannot exist. So he thinks that Hilbert (and I) have made a hasty generalization from one case to all cases. But that's not Hilbert's argument (or mine) at all.

Hilbert's point is that any AI you please has the properties of (1) and (2). Take the AI set of occupied hotel rooms. Even Den Beste agrees that such a hotel is an impossibility; he just thinks that there are other AIs that are not impossible. But all AIs have the very properties that make Hilbert's Hotel impossible. There simply are no other AIs to be had. Den Beste does suggest the following:

It's possible to define a set consisting of all the points on a line between "0" and "1" and prove that this set is infinite (aleph-one) in size, even though it's based on a finite source (a bounded line segment). By the same token, there may well be sets in the universe which are infinite without requiring the universe to be infinite. For instance, there's no particular reason to believe that space is granular (i.e. that "position" is quantized) and if so then the set of "all possible positions between San Diego and Los Angeles" is infinite.

The problem here is that Den Beste is describing potential infinity, not actual infinity. The set of all positions between LA and San Diego is potentially infinite; à la Zeno's paradox, there is an infinite set of halfway points between the two cities--a potentially infinite set. If it were actually infinite, then travel between them would not be possible, just as Zeno argued that a runner could never reach the finish line.

So much for actual infinity. Let's move on to the next major point, i.e., that "a beginningless series of events in time is an actual infinite." Den Beste thinks that this, too, fails, because of my allegedly faulty arguments about actual infinity. I've addressed that issue above, so that leaves:

Their argument about dates is crap, too.
. . . .
... what these guys are trying to claim is that it isn't possible to actually calculate a difference between them, because each of them is actually infinite and you can't meaningfully subtract infinite numbers. What they're trying to claim is that I can't meaningfully manipulate those two numbers using subtraction unless I bound the line they're on to the left of them, and there's no such requirement I've ever heard of. Subtraction works perfectly well on the values of two points on an unbounded line. If the points are finite I can do subtraction on them even if the line they are on is infinite.

No, my claim is that an AI cannot be existentially instantiated. As I made clear in my original summary, I make no claim about the coherence of actual infinity as a mathematical concept. So one can carry out all the additions or subtractions wishes between two points on an unbounded line. My claim is that such a line cannot exist in the real world, just in the same exact way that Hilbert's Hotel cannot exist. Indeed, my argument about dates isn't "crap"; it's just a straightforward application of the above points about the properties of AI sets. To make it crystal clear--

An AI set has subsets that are equal to the set itself. On the assumption that the universe is actually infinite, that means that the set of events preceding the Battle of Hastings is equal to the set of events preceding the Declaration of Independence. Why? Because the members of each set can be put into a one-to-one correspondence with one another. Yet the set of events preceding the Declaration of Independence ought to--and in fact does--contain events not in the set of events preceding the Battle of Hastings. So the universe is not actually infinite.

Mathematics is the only realm of human intellectual activity where it is actually possible to arrive at absolute truths solely through the exercise of reason. But not if you go about it this way.

I think Den Beste is missing the philosophical nature of my argument. I am arguing that AIs have certain properties that preclude their existential instantiation. I say nothing about the mathematical coherence of the AI--indeed I assume its conceptual coherence--only the coherence of actually instantiating it.

This reads like one of those "proofs" that horses have an infinite number of legs.

It's amazing that Den Beste can say that with a straight face. I'm the one arguing that an actual infinity cannot exist. Guess what? That would include a horse with infinite legs! My arguments would show exactly why such a proof is specious. So the comparison is not only invidious; it betrays an utter lack of understanding of what's going on. Instead of trying to portray me as some lower-than-a-creationist whack job using mathematical trickery to wish away what I don't like into the cornfield, Den Beste might have paid more attention to the philosophical gist of the argument.