In this post, I aim to address in my own fashion the question Evangelical apologist William Lane Craig was asked in his Q&A #325, "Infinity and God" on his Reasonable Faith website. Recall that Craig's Q&A #325 was to be background reading for the discussion with Pete's class.
The first two posts in this series aimed to answer questions that I had posed for background thinking about the topics of numbers and infinity. They can be accessed here:
Notes about infinity I: About Numbers.
Notes about infinity II: Is infinity quality or quantity?
The formatting here, since there are three different voices presented, will have my words in black, Craig's questioner's words offset in block quotes and presented in dark green, and Craig's own words offset in block quotes and and presented in purple. I'll respond as I go through various parts of the question.
We'll start immediately by looking at the question Craig was asked by a reader named Hargus from South Africa.
Dear Dr CraigImmediately, then, to do this justice, we should take a moment to examine Craig's Q&A #323, "The Concept of God." In the interest of brevity, I will only address what I considered to be relevant to the topic of infinity and God from Q&A #323. Craig is writing here, addressing a questioner named Ian.
I am a devout Christian and I read your Q&A every week. I have read Q#323 today and I was totally surprised by it because it is basically exactly what I have been thinking about for the last couple of days.
So far so good; now let’s look at your example: the set of natural numbers, which has an actually infinite number of members. You assert, “we know that no possible being can know all of the elements of this set, since it's uncountable.” This inference is mistaken, Ian. First, the set of natural numbers is countable! It’s what mathematicians call a denumerably infinite collection. For an example of a non-denumerable collection, you should have picked, e.g., the set of real numbers, which is so numerous that it can’t even be counted.I have to take issue immediately. Here we have either a misunderstanding by Craig or a despicable misuse of semantics. Mathematicians call the set of natural numbers “countable” or “denumerable,” but none imply that they could actually all be counted. Indeed, Craig himself makes this point (and does so in the response we’re investigating, numbered Q&A #325, as we will see in the next post in this series).
His questioner, Ian, is using “uncountable” in a colloquial, non-precise way, and Craig has turned to a precise mathematical term. Craig is correct to identify the set of real numbers as “non-denumerable,” or “uncountable,” but the way in which he points this out reveals that he’s a novice, not an expert, with this topic. A simple note that mathematicians call that size "countable," yet another lamentable term, and that he hopes to avoid confusion by making that clear is all that is necessary.
Never mind that minor point, however. Why can’t someone know all the real numbers? There’s nothing incoherent about the universally quantified claim that for any person S and any number n, if n is a real number, then n is known by S. Certainly, we may agree that “if one being knows or can know more natural numbers than another, then that being is more knowledgeable or capable, respectively, in this area.” It follows that a being who knew all the real numbers would be maximally knowledgeable or capable in that area because there aren’t any more real numbers than that. So this is a nice illustration of maximal greatness in this respect.While Craig is accurate in saying that knowing all of the real numbers would exhibit "maximal greatness in this respect," it's worth noting that it is a poor illustration of "maximal greatness," since "this respect" is itself not maximal. I do not point this out to say that Craig is in error or has misspoken, but I do wish to make it clear that the use of "maximal" here is local to a given context, that of a particular set of potential knowledge. It's easy to misread this to think that he's implying maximal greatness in more generality.
To quickly elaborate, Georg Cantor, who is known to Craig in this context, proved that given any infinite collection, one can create a larger infinite collection from it (known as its power set). Thus, the non-denumerable (uncountable) infinite cardinality of the real numbers is not maximal but rather just another size of infinity that is certainly larger than the so-called “countable” infinity that enumerates the natural numbers. This is a poor illustration of “maximal greatness,” then, as I pointed out in God Doesn't; We Do and again in Dot, Dot, Dot--there is no such thing because given any level of greatness, in this respect, a greater level can be conceived of immediately.
Can someone know any real number?
Now, more importantly, I’m not sure about this claim that any person S can know any (arbitrary) real number n. Without at least implicitly assuming infinities exist, it may not be correct to say that anyone can know the wide majority of irrational numbers, and the vast bulk of real numbers are irrational. How could we without infinity? Each has an infinitely long decimal expansion that does not repeat periodically, and most exhibit no discernible pattern in their decimal digits. Again, we see inexpert thinking on these matters.
Given some of his other arguments, take a moment to note that Craig explicitly admits that knowing more propositional knowledge confers a degree of greatness of a kind upon a more knowledgeable being.
What about all of them?
While I suspect Craig is merely using this as an example, I'd like to point out that knowing all of the real numbers is hardly short of preposterous. It is very difficult to convey how large the set of real numbers is, but it is so large that when asked this about the natural numbers, my response is "I don't think so," whereas when asked it about the reals, my immediate reaction is incredulous laughter with a loud "no."
To get a sense of it, think about a number with lots of decimal places. Realize that each of those decimal places admits ten possible digits. If there are only a few decimal places, say twelve, that gives us a trillion possibilities. If there are hundreds of decimal places, well, that's a lot of possible real numbers. Now realize that every real number can be expressed with a decimal expansion that contains infinitely many decimal places, each of which can take ten possible digits. Change any one of those anywhere, and the result is a new real number.
Of course, we should wonder, since he brought it up, if God could know all of the real numbers and what it would mean in both cases. Craig wants to deny that God knows infinitely many things, as we'll see, so we must also ask: If not, why not?
I think your confusion is that you’re conflating something’s having no upper bound with a property’s not having a maximal degree. Hence, you say, “Since there are certain areas pertaining to knowledge and power which do not have an upper-bound, it is necessarily the case that any possible being's net knowledge and capabilities are limited and imperfect.” As we’ve seen, that doesn’t follow. A property can have a highest degree, e.g., knowing all the natural numbers, even though there is no highest number.”First, I don't think Ian is confusing anything. He admits that he feels that there are certain areas, noting knowledge and power, that do not appear to be bounded above. Imagine, for instance, lifting a weight. It could weigh 100 pounds, 200 pounds, or more. At what point would we say that even in the imagination the weight could not weigh any more than that? I'd contend that we can never say that and be both honest and correct about it. We could also wonder how many natural or real numbers someone can know. There is no upper bound to this, and saying that knowing all of them means 'highest degree' doesn't get around that fact.
Additionally, there are some tricky points here. One is the nearest to a sticking point for salvaging the “God’s maximal greatness” argument (due to Anselm, 11th c.). Before addressing it, notice that Craig has directly implied that it may be possible to know all of the natural numbers. This is interesting because (a) he later argues even God doesn’t know them all, and (b) that God’s knowledge doesn’t work propositionally like this anyway. Why make this argument, then?
The big sticky point here, for what it’s worth, is on whether we can conceive of the idea of God as being the sum-total of all possible goodness (possessing all positive properties), and whether this list is finite or not, as possessing all of them implies maximality. This argument is the crux of Alvin Plantinga's ontological argument, which he appears to have adapted from Kurt Gödel. Here, the issue of infinitude of God’s nature can be side-stepped, which may be desirable for apologists since I assert that if God is infinite then God is also abstract (i.e. not real).
The main issue with this “maximal (though finitely) good” definition of God is that it seems highly implausible given our observation of the world (Cf. Problem of Evil). Still, even without appealing to that, it is somewhat problematic because it raises the question of why an all-powerful, all-knowing, perfectly benevolent God isn’t able to conceive of or create a single additional good thing in the universe. Clearly, this question lays bright highlights all over the Problem of Evil, once we go back to it--particularly, our universe is filled with much obvious suffering and what appear to be sub-optimal circumstances for identifying a universal state of perfect goodness.
In effect, taking the position that God is maximally good, though finitely so, also ties God’s hands. In this conception, God has so ordered the universe such that were he to change anything, it would be worse off for his doing so. Obviously, this contradicts or renders pointless other theological objects (like intercessory prayer, for one).
Can someone know all the natural numbers?
I don't think so. The first property of the natural numbers most people should put in their minds in this context is that there is no largest natural number. It is entirely unclear how someone could know a collection of information that is infinite in scope.
We could wonder about God, though, since God is hypothesized to have no normal limitations, like a physical brain with only so much storage space, and is said to be omniscient. If God can know all of the natural numbers, then two things follow: first, God actualizes infinity in his knowledge set, and second, God has propositional knowledge ("n is a natural number" is a proposition). I don't immediately see a problem with these consequences, but Craig denies both of them. The reason I don't see a problem with them, of course, is because I think God is also an abstraction we have imagined, not a real thing.
Now we can move on to Q&A #325, the object of discussion.
I do not aim to tackle Craig's 325th response in this post but rather will answer his questioner's questions in my own way. Hargus writes,
Infinity is a concept that has been driving me nuts as I cannot completely fathom it. A few comments I want to make:I feel this requires an important note from the outset. The first question, could God count all of the natural numbers, actually doesn’t make sense since we have no clear notion of what “God” means.
1. Is it possible that God could have counted the whole set of natural numbers? The answer according to me is no. Why? Because the set of natural numbers is infinite. You will never stop counting. So even though God is almighty, He cannot do things like these. It simply doesn't make sense. It's like saying God can create a rock that He cannot lift. It's a logical incoherent statement.
This sounds pedantic, but I think it's a core problem with dealing with apologists, theologians, and theology in general: we are simply to assume that the apologist means something that is clearly understood when speaking of God, and that's simply not the case. If we suppose that he means an omni-capable intelligence of some kind, then the question could possibly be addressed. If, instead, we assume only the uncaused first cause Craig defends in the Kalam Argument, I don't think it has any meaning. If we assume Yahweh of the Old Testament, the answer is probably not, seeing as this Yahweh is frequently surprised and angered to find out things he didn't know when there were only two to four people living on earth in the special garden he allegedly created for them. If we mean the laws of physics or a grand sense of universal oneness in the universe, the question is literally nonsense. Can you see how it is a problem? It's a very big one that apologists all expect us to gloss over so that they can make their arguments.
More directly to Hargus's question, that the infinity of the natural numbers cannot be completed by succession (counting) is accurate, so it shouldn’t be the case that any being or entity could “count the whole set of natural numbers.” In other words, I think Hargus is correct in his assessment here. Further, it is my opinion that this creates a problem for an omniscient deity who seemingly must know what is the last number he would ever count, if he tried, and yet then would automatically know the next number by the simple rules of succession (a logical contradiction).
2. Regarding the existence of abstract objects: One thing that has bugged me about this is the whole thing about infinity. Does it exist? I have read some of your comments on this, namely that you say that infinity is a mere concept of the mind that doesn't exist in the real word. This makes sense to me. So if there are an infinite amount of natural numbers, how could they all exist? Well I believe numbers exist in two to three ways:This is complex--the ontology of mathematical objects is its own rich philosophical field with many unsettled debates that I’m not an expert in. I generally fall into the camp that numbers are abstract objects that count other objects, real or abstract. This is closest to Hargus's situation that he labels (a). It would be helpful to read Part I in this series if you want more information about what numbers are and hopefully addresses Hargus's (b).
a. Since numbers are mere concepts, they exist in our minds as we think about them.
b. Numbers exist conceptually as real world objects represent them. I.e. the number of planets in the solar system.
c. They exist conceptually as symbols on a piece of paper (or computer screen).
Because only a finite amount of thoughts, objects and written symbols exist, it follows that only a finite amount of real numbers exist in the real world.
In that sense, things are still tricky to get right because while it seems clear that there are only finitely many objects (say atoms, protons, or what-have-we) in the observable universe, they can be combined and recombined in various ways, and the combinations thereof can be combined and recombined in various ways again, ad infinitum, providing not only abstract structures of arbitrary size but also ones rooted in physically identifiable structures, though these are eventually quite convoluted. Some physicists suggest that the universe itself may actually be infinite in scope, perhaps containing infinitely many objects (e.g. stars), though I’m not entirely clear on what this even means or how we could find it out.
As to infinity, I’d say that we don’t know if infinity is physically actualized, that we lack reason to claim that it is physically actualized, and that as an abstraction there is no good reason for it not to be “actualized” (as mental “stuff”). I give no real ontology to abstractions, though, and do not identify mental “stuff” as “stuff” at all--it’s just ideas. Indeed, a rich, interesting, and useful abstract framework has been built up around the notion of an actualized (but abstract) infinite.
Finitists, a small fringe of people in the philosophy of mathematics, reject the actualization of infinity even as abstractions, and they’ve found that effectively all mathematics can be done without it, but the cost is relatively heavy. Though it may not be satisfying to think of it this way, infinity may very well be a useful fiction of a kind that makes certain kinds of math (like calculus) considerably easier and more accessible. Reading Part II in this series might be of some help here.
As for the third way Hargus feels that numbers may exist, this is an interesting aside (that I won't explore) into the difference between numbers and numerals. Craig actually addresses it in his response, so keep your eyes open for Part IV where I'll present that.
3. Abstract objects aren't limited to numbers. It includes things like music, literature and other forms of art. Now there are an infinite amount of songs that can be composed. Do they all exist? Again the same line of reasoning goes. A tune lives in the mind of the one that composed it and the ones who hear it. It is a concept, an idea which didn't exist before someone thought about it. So there are only a finite number of art forms in the real world and always will be even though the number increases all the time.This is an interesting question, isn’t it? In the sense that a song could be defined as a concatenation of notes, or a piece of literature a concatenation of words, etc., we could say that at least potentially, yes, the possibilities are infinite. Because these things take time to create, though, ultimately only a very small number of the possible pieces of music or works of literature will ever be made. Interestingly, we could define "rational" songs (which periodically repeat a loop of notes unchangingly forever or that stop eventually) and "irrational" songs (which do not satisfy that definition) in this way, analogously to rational and irrational numbers.
For fun, we might wonder about a machine that, hypothetically, runs indefinitely into the future, choosing notes at random within a particular key (or not within one), creating an unending "song," but this is mostly just something to muse about. By considering the number of ways that each note could occur, we can quickly start to realize that the potential variation in such songs is infinite, although even if they were all explored, at no point would an infinite number of songs have been composed in this way.
4. Now I want to say something about the future. God as we know can foresee the future. This is what makes prophecy possible. We know that the world has a beginning, and it will have an end in this age as the Bible predicts, but the age to come will have no end. So the number of future events counts up to infinity. Now is it possible for God to know everything in the future? Again as explained above this is impossible. One cannot know an infinite amount of things. It is simply an illogical thing to ask from someone like God, even though He is almighty, and I am not trying to limit Him in any way. So God knows the future. How much and how far? Well as much as He chooses or as far He deems necessary.First, this “as we know” claptrap about God is nonsense that really grates me (and no doubt it grates Boghossian as well). No, you do not know.
We also don't know that the world has a beginning, unless we're talking about something so local as the planet Earth or the human species. The question about whether or not the universe has a beginning is not answered, and no amount of theology-led philosophy on the part of apologists like Craig can make it known.
We also don't know that the world will have an end "in this age" (or even what defines "this age" in this context), again unless we're very local about things, like knowing that the Earth will be destroyed by the sun in a few billion years--which humanity could potentially survive, yet another thing we don't know. The Bible predicting something does not confer knowledge except about whether or not the Bible contains that idea or doesn't, which does not transfer to claims about the real world on the grounds that the Bible is scripture in some religion.
We also don't know that "the age to come" means anything or that it is endless, although that point is discussed in Part II of this series. Hargus goes on, though, to make the same point I do about the idea of an endless future: it seems to commit God to knowing an actually infinite quantity of ideas, which Craig rejects.
The rest of this reveals rather how inane and incoherent the concept of God is, doesn’t it? God is put to an impossible thing by what is considered to be his apparently obvious nature. It looks also as if the concept of God is frightening, otherwise such honest questions seem not to require qualifiers about not wanting to "limit Him in any way."
These are my comments. I therefore believe God is almighty, but He cannot do illogical things, or know things that do not exist. Thanks for reading and please let me know if you agree with my views. Also please tell me of articles I can read which might give me more insight into these things.I've only included the closing for completeness and don't wish to comment upon it beyond drawing attention to his use of "therefore," after which he makes the right point but only after again reminding us that he believes God is "almighty," for whatever set of reasons he holds in that regard.
Finally I want to thank you for the work you're doing. I think you're making a big difference in many people's lives.
There's still more to come! In the next, and final, post in this series, I plan to address Craig's answers to Hargus, which can be found at Craig's Q&A #325, "Infinity and God".
If you like how this reads or think this kind of subject matter is interesting, it's very likely that you'll enjoy reading Dot, Dot, Dot, so please pick it up.
Edit: The fourth and final post in this series is available here.