Here is Part IV of James A. Lindsay’s series on infinity. The first three can be found here, here and here. I edited his fantastic book, Dot, Dot, Dot: Infinity Plus God Equals Folly, which is available form the sidebar over there. Over to James:
As previously noted, I’m writing a series of blog posts that are adapted from notes I made as preparation to talk with philosopher and author Peter Boghossian’s Atheism class at Portland State on November 19, 2013. This is the fourth and final post in this series. I visited his class to address infinity and God, following from the theme presented in my new book, Dot, Dot, Dot: Infinity Plus God Equals Folly.
In this post, I aim to address the reply given by Evangelical apologist William Lane Craig 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. The third post provides my answers to the questions Craig is asked in Q&A #325. They can be accessed here:
Notes about infinity I: About Numbers.
Notes about infinity II: Is infinity quality or quantity?
Notes about infinity III: Answering for William Lane Craig.
The formatting here, since there are two different voices presented, will have my words in black and Craig’s own words offset in block quotes and and presented in purple (to be consistent with the usage in Part III of this series). I’ll respond as I go through various parts of Craig’s reply.
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I’ll start off by immediately getting into Craig’s reply. All emphasis in Craig’s words are his.
These sorts of questions can keep you awake at night, can’t they, Hardus? Let me address each one.
1. Could God have counted all the natural numbers? It’s fairly widely agreed that God could not begin at 0 and successively count all the natural numbers.
Widely agreed? By who? Anyone that knows? This kind of thing should be considered blasphemy, but it passes for sophisticated apologetics–which is exactly why I call it the art of “making stuff up.”
It is worth recalling that in Part III of this series, we saw Craig (in his Q&A #323, referenced in the question) discussing what would be implied by someone knowing all of the real numbers, which are even more numerous than the natural numbers. He does not categorically deny that such a thing would be possible, which is a bit odd given his stance here.
Of course, for fairness, this question asks if God could count the natural numbers, which is a kind of activity, and not what God might know. And for more fairness, I also suggest that counting all of the natural numbers is impossible and thus would agree with his assessment–if (a) it had a clear enough definition of God so that it meant anything and (b) didn’t assume that people know things they don’t, including and especially about God.
On the other hand, we might wonder if God, being omniscient, knows the largest natural number that he could count to if he tried, seeing as we’re agreeing that he cannot count all of them. As previously noted, an omniscient God perhaps should know this largest quantity and yet would then immediately also know its successor, which is a logical contradiction.
Here it’s helpful to distinguish between a potential infinite and an actual infinite. A potential infinite is a series which has a beginning and is growing indefinitely; infinity serves merely as an ideal limit of the series which it never reaches. An actual infinite is a collection which includes an infinite number of members, that is, the numbers of members in the collection exceeds any natural number.
Craig is correct about the difference between a potential and actual infinity, and it’s worth pausing to make this idea clear (and also to distinguish it from a physical infinite). We’ve discussed this pointpreviously, so I will not elaborate here.
His definition of an actual infinite is essentially correct but contains an element that is a bit off in that he is talking about a “number of members,” which is fine when not being careful or otherwise speaking loosely but is technically both wrong and misleading. I’m pointing this out only for clarity for my readers and not to pick a nit, and I’m fairly certain Craig knows the distinction. None of the infinite cardinals is a number, even if they represent something quantitative.
Counting generates a potential infinite. To say that someone could count to infinity is to say that a potential infinite could be converted into an actual infinite by adding one member at a time. That’s impossible, since for any natural number n, n+1 is always a finite number.
The property that Craig is referencing here is that the infinite cardinals are each what are called limit cardinals, meaning that you cannot get to them from smaller cardinals by repeated successorship (adding). The infinity that represents the size of the natural numbers is actually a strong limit cardinal, which means one cannot get there from smaller cardinals even by the more powerful operation called exponentiation. Countable infinity is the only example of a strong limit cardinal usually defined or discussed. Each larger infinite cardinal is reached from the smaller ones via exponentiation of the “number to the infinity power” kind. Craig’s assessment here is essentially correct.
The question is, could someone count all the natural numbers one at a time by never beginning but ending at 0?
What does this even mean? By “never beginning”?! And what does it have to do with the question asked?
Observe, though, that Craig’s sloppiness shows up in considering infinity to be something like a number (as opposed to a more general quantity), and this is a source of confusion. This question commits a category error, misinterpreting infinity as being number-like because it is a cardinality. The problem with Craig’s hypothesis is that there literally is no starting point, so as he notes, the task could never even be begun.
To my mind that is just as impossible as the first task.
Technically, it is more impossible (if that means anything), since it is also guilty of a category error and “never begins.” The advice of Samwise Gamgee’s old gaffer comes to mind: “the job that takes longest to finish is the one that’s never started.”
If you can’t count to infinity, how could you count down from infinity?
Again: category error. Nonsense. Note that this particular nonsense is a usual line for Craig when defending the Kalam, which is, of course, why he brought it up in the first place.
If I’m right about this, then the series of past events cannot be infinite, either potentially or actually—potentially because it is not growing in a backward direction toward infinity, nor actual because you can’t get through an actually infinite number of items one at a time. So the series of past events must have had a beginning.
Now he reveals where he’s going–he wants to slide in his argument for an ultimate beginning that he can credit to (or blame on?) God. He’s not right about this, though. He’s thinking via a category error that is worth exploring more deeply.
Particularly, Craig is essentially doing what he says can’t be done: jumping over the vastness of the infinitude of the natural numbers to start at a stopping place that doesn’t exist. He is treating infinity like it is a number, claiming it cannot be “gotten to,” and then jumping to it to say you can’t get back (implied: without jumping) once you’re there.
To say that the series of past events must have a beginning by this argument is to assume that there is a beginning that occurs either some finite amount of time ago or infinitely long ago. Craig dismisses the second possibility and concludes the first, but he misses the nuance of the very argument he makes: the series of past events could also be a potential infinite, and epistemically we can say no more. What that would mean is that however far back we look, we could conceivably look back further. This is why Craig talks about the impossibility of a task that is never begun: he’s comparing it against the existence of the Universe for which he has assumed a beginning.
Physics, of course, has some things to say about this, but not enough to nail things down. This is why Craig used to hang his hat on the Hawking-Penrose cosmology and no longer does now that he’s realized that the Hawking-Penrose cosmology is not a quantum theory and thus cannot account for what happened before a certain time in what we call “the early universe.” We should think of the early universe, before it cooled enough to be transparent, as a place we don’t understand, not a de factobeginning or a period containing a necessary starting place for everything.
2. Does an infinite number of numbers exist? My answer to that question is no, not because I think that the number of numbers is finite but because I think that there are no such things as numbers! Numbers are just useful fictions, like the Equator or the center of mass of the solar system. Do numbers exist in any of the three ways you suggest? Not really.
Perhaps surprisingly, I agree to an extent. I’d prefer the term “abstractions” to “fictions,” although after some fuzzy point it may make sense to make the semantic switch. I’ll add to his list of other useful fictions, or abstractions, if he prefers, though: God. Does it exist in the way he suggests? I’d say, “Not really.”
Incidentally, we can actually find the Equator and the center of mass of the solar system… not so with God.
Consider your alternatives:
a. Certainly I have the idea of the number 2, for example. But that thought is not the number 2 itself. The nineteenth century mathematician Gottlob Frege called the view that numbers are ideas in our minds “psychologism” and subjected it mercilessly to criticism. That 2 and my idea of 2 are not identical is evident in that they have different properties; for example, my idea of 2 comes and goes, but 2 itself, if it exists, doesn’t depend upon my thinking about it!
It may be a matter of semantics here, but I prefer to think of “two” as the abstract entity that describes the property of “twoness”: Any set of objects with the property of “twoness” will be said to be enumerated by the value “two.” This property is apparently eternal and immutable, as mathematical properties and other abstractions are, but I doubt it has meaning without minds that have the idea of it. Perhaps in this sense, talking about the teleology of mathematical objects is more appropriate than about their ontology.
b. Certainly, things that exist in the world can be numbered; for example, Mars has two moons. But this adjectival use of number terms doesn’t require that numbers themselves exist. As Frege showed, we can express that Mars has two moons without using any number terms at all by saying that there is some entity x which is a moon of Mars and some entity y which is a moon of Mars and x is not identical to y, and for any other object z, if z is a moon of mars, then either x is identical to z or y is identical to z. (In logical notation: (∃x) (∃y) (Mx & My & x≠y ((∀z) (Mz ⊃ z=x ∨ z=y)).) Pretty slick, eh?
Slick? I guess. Observe that in the slick part we still identifiably have a set of things (x and y) that together exhibit the property of twoness. As often is the case, things have progressed substantially since the nineteenth century, where Craig finds an apparently surprising number of his arguments for God (perhaps because God was easier to defend then than now?).
The hotly disputed question is whether the use of number words as referring terms commits us ontologically to the existence of numbers. For example, the statement “Two is the number of Mars’ moons” is thought by many philosophers to commit its user to the reality of the number 2. This strikes me as perverse. Metaphysics is surely a lot more difficult than that! These sorts of ontological commitments happen, as Wittgenstein said, when “language goes on holiday.” You can’t read metaphysical commitments off language—at least so I think.
This is why I lament that the term for “existence” and “existence in the abstract sense” happen to be the same word. I do wonder what commits Craig to the reality of God if he finds this kind of thing a perversion of metaphysics, though.
c. Obviously, we make marks on paper in doing mathematics. But here we need to distinguish between numbers and numerals. There are many different ways to represent the number two: 2, II, úú , and so on. But these are numerals, not numbers. There are many numerals but only one number 2, if such a thing exists. So numbers do not exist on paper or computer screens. So what you should say, I think, is that there is at most only a finite number of numerals in the world, not that there is a finite number of numbers in the world. There are no numbers anywhere.
A rather fun “philosophical” question that rolls around in some mathematics departments asks the question “Does 2=2?” where the two numerals are written in different fonts.
It’s an interesting matter as well to marvel at the confidence with which Craig can assert “there are no numbers anywhere” while maintaining an adamant belief in God. I mean, I do agree with him about the numbers, but I am currently looking at a collection of four pencils on my desk and have an immediate sense of fourness but no matter where I look, I have no sense of deity.
3. What about other kinds of abstract objects? You are absolutely right that philosophers who believe in the existence of abstract objects think that novels, plays, musical compositions, fictional characters, and so forth, exist as abstract objects. What is disputed is whether these are created by their writers and composers or whether these people just happened to stumble upon these pre-existing objects. Many people feel quite uncomfortable in saying, for example, that Leo Tolstoy did not createAnna Karenina but just found it. This view seems to seriously depreciate the creative genius of authors and composers. So many want to say that people created these abstract objects. Still, it’s hard to see why, once you grant that such abstract objects exist, these collections of words or notes did not pre-exist their discovery by these folks. I think it’s better to just deny that such abstract entities exist and maintain that our ability to talk truthfully about them (e.g., “Sherlock Holmes is the most famous detective in English fiction”) doesn’t entail their existence.
This topic may be interesting within philosophy, but the only commentary I have for it at present is a remark by Richard Dawkins in The God Delusion: “The God of the Old Testament is arguably the most unpleasant character in all fiction….”
4. Does God have complete foreknowledge of the future? Yes, why not? Your statement that “the number of future events counts up to infinity” is ambiguous. We’ve already agreed that it’s impossible to “count up” to infinity. So the series of future events “counts up to infinity” only in the sense of a potential infinite: infinity is the limit to which the series of events strives but never reaches. There will never be an actually infinite number of events. From any point in time that you pick the number of events future from that point is always finite and always increasing. If you pick the present event as your point of reference, the number of future events is 0! That’s because temporal becoming is a real and objective feature of the world.
Again! A question about God, and Craig’s answer is an absolutely solid and unqualified “yes,” as if heknows. This point can hardly be overstated. He doesn’t know; in fact, he can’t. He’s making it up–with all the confidence of an apologist.
This segment of his reply kind of goes all over the place, but his point is solid, though I suspect it misses the mark. His point is that at no point in the future, from now or any reference point, will an infinite number of events have taken place. I’d say that’s correct.
I got the impression, however, that Hargus’s question was about whether or not God’s mind actualizes this potential infinite. Craig does not address this question, and he is wise not to. To actualize infinity in the mind of God is to render God abstract by Craig’s own argument (or is it a petard, ready to hoist?).
The question I suspect Hargus meant presses, though, particularly since one of the properties attributed to God is being eternal and standing outside of time. Is God bound up with the unfolding of time in our universe or not? If so, in what way is God eternal or outside of time? If not, then why shouldn’t God be able to see the totality of the past, present, and future and have proper knowledge of it? And why shouldn’t this constitute an actual infinity if time presents a potential infinity–again, per Christian theology, at least in the “world to come”?
The only escape from this pen for Craig is to reject some Christian theology. It could be about the hereafter, and perhaps the reality of cosmology, depending on how that goes. Maybe he could claim that at some point in the future, God ends everything that is a kind of universe in which there is knowledge of events. Such a claim contradicts the idea of eternally keeping one’s personality in heaven, at the least, though.
You say that it’s impossible for God to know everything in the future. That doesn’t follow from anything we’ve said.
Except apparently it does. Craig has claimed that an actual infinity is an abstract thing only, and yet if God were to know everything in a potentially infinite future, which is what Hargus asked, then God should know an actually infinite number of things–even in an astoundingly simple universe. Craig can’t have God knowing an actual infinity of things because that would render God an abstraction as well, so it would seem to follow directly from Craig’s rejection of the actual infinity that God cannot know an actually infinite set of things.
To get an objectionable, actually infinite number of things out of this, you have to think that God’s knowledge is broken up into propositional bits that actually exist. But such a view of God’s knowledge is not obligatory for the theist (and traditionally has been denied by theists). Suppose God’s knowledge of reality, including the future, is non-propositional in nature, and we finite cognizers represent what God knows non-propositionally by breaking it up into propositional bits. (For an analogy think of your unbroken visual field, which someone could represent by breaking it up into pixels.) Then there is no actual infinity of ideas, thoughts, propositions, or what have you.
This part gets a fairly big “uh, what?” attached to it. God’s knowledge is not propositional in nature? Not only does this appear to contradict what Craig himself wrote in Q&A #323 (referred to by the questioner and noted above), it seems to contradict essentially everything people believe about God. For example, here’s a proposition: “Charles murdered Jane.” This is precisely the kind of knowledge required for God to judge Charles (and perhaps Jane).
Perhaps more relevantly, “Charles thought x (at 11:46) and then thought y (at 11:48),” which is also precisely the kind of knowledge that the Abrahamic religions are so utterly concerned with having God judging. If time runs ever on and on with sentient minds under God’s dominion all throughout, once they began, then someone is going to be thinking something at every moment in the potentially infinite future, and God traditionally is taken to be cognizant of all of it.
This strikes me, then, as being intellectual gerrymandering because it insists that this perfect and omniscient being is, indeed, required to be ignorant of simple propositional knowledge for no better reason than that apparently otherwise it would be too hard to defend the being from philosophical analysis.
To pick a nit, with regard to Craig’s analogy about the visual field, someone should perhaps tell him that there are only finitely many rods and cones in the retinas of the eyes, and so our “unbroken visual field” is indeed pixelated in a sense before the brain interprets it. It is not yet known if reality is continuously smooth or discrete at very small scales. If discrete, his analogy is even more utterly bogus. If continuous, infinity is actualized physically everywhere, contradicting his claim that infinity isn’t actualized.
So do not limit God by denying His complete foreknowledge of the future. There is no good reason to adopt such a view and it impugns God’s greatness.
Does this not feel like talking out of both sides of his mouth now? Perhaps this is how God’s greatness is kept unimpugned: by saying one thing about God now and another thing later. God can’t, but must, know infinitely much, apparently, but that shall not be said in the same sentence.
It is interesting and worth commenting on that Craig chides Hargus for limiting God after Hargus expressed clear discomfort and unwillingness to do that. Perhaps this also is how God’s alleged greatness is maintained: utter fear of challenging the idea even accidentally and sidelong.
The most pressing commentary to make here, though, I think is on the nature of these kinds of questions in the first place. Hargus admits that he is a committed and devout believer, but these are questions for doubters, questions that exist at the fraying edges of the fabric of belief. It is incredibly unlikely that Craig will ever convince a skeptic to believe in God based upon this kind of waffling about infinity and highly abstract philosophy, though I suppose he can reassure the devout and faltering by folding under the loose edges.
This finishes the series!
If you have enjoyed reading it or find this kind of material interesting, I think it is very likely that you’ll enjoy Dot, Dot, Dot: Infinity Plus God Equals Folly, so do pick it up.
More importantly, do yourself a big favor by joining me in thanking Peter Boghossian for initiating this entire series by picking up his Manual For Creating Atheists.
