The Slipperiest Theological Slope - James A. Lindsay Responds on God and infinity
The other day, I posted a reivew by a Christian to James A. Lindsay's book Dot, Dot, Dot: Infinity Plus God Equals Folly. James has kindly taken time to respond to the claims made. It now follows.
The Slipperiest Theological Slope
Theology is a peculiar artform. It posits itself as an intellectual investigation of the properties of a perfect Deity that it alone assumes exists. The theological effort thus reveals itself for what it is. Theology is not an effort in seeking truth; it is an effort in extracting as much believability out of an apparently wondrous idea as the reality of the world will allow.
The alleged greatness of God is a typical theological topic that gets repeatedly restrained by the inconvenient truths of life in the world, particularly those that fall under the banner of the so-called “rock of atheism,” the various Problems of Evil. For millennia, human beings have struggled to make sense of how a great, powerful, and good Deity can possibly have created, ordered, or allowed a world such as our own, and for millennia, all responses have proved lacking in this regard.
As a result, making the Deity more and more capable—ranging far beyond what humans can imagine—has been a particularly enticing slope on Mt. Theology, despite how slippery it proves. The Deity must be omnipotent, omniscient, and morally perfect, many theological schools argue, because it simply seems impossible to account for how it could have ordered the world so and yet retained its own inherent goodness without such attributes. God must just know that much better than we do.
If one is ready to accept an immature theology, the thorny problem seems solved, and no doubt, hordes among the masses seem perfectly content with such a vague definition of perfection. Obvious questions, however, remain, and they would plague many other believers even if there were no skeptics to ask them so doggedly. Still, the famous theologian and Church Father Anselm applied enough linguistic magic to give one of the most satisfying answers in theological history: the Deity is that than which nothing greater can be conceived.
Anselm's response bears every mark of being the result of a lengthy arms race of ideas concerning the potency of deities in a highly mythological ancient world, one in which greater gods assured victory over defeat, plenty in harvest, and security from disease and calamity. Our God, says Anselm, drawing a snow-white rabbit from his hat, is the one God because there can be nothing greater than He that is that than which nothing greater can be conceived. He didn't mention how such an idea isn't just what it is, however: an idea.
In 2013, I wrote a book, Dot, Dot, Dot: Infinity Plus God Equals Folly, investigating the common divine attribute of infinitude, which lies at the bottom of that very slippery theological slope concerning the greatness of the Deity. Anselm doesn't demand infinitude in his conception, but it follows all too easily, and many, many theologians have accepted an infinite God rather uncritically since. I wrote,
As the Peano Axioms that underlie number theory essentially suggest that infinity needs to exist (as an abstraction) and yet cannot show that it actually must, a finite God never provides security against the problem of simply arguing for a bigger God. Thus, eventually, they go infinite with it, even if doing so amounts only to a rhetorical trick. That they make their deity (necessarily?) abstract in the process apparently falls between the cracks. (pp. 86–87)
This issue is contentious, though, and however slick the theological slope set up by Anselm's characterization, one need not necessarily slide all the way to bottom. Some theologians, in fact, seem to maintain their footing, preferring to wrestle with other theological inconsistencies. Recently, a Christian reader of Dot, Dot, Dot, D. Newton, raised this concern in the midst of giving the book a kind and thoughtful review on Amazon. That review was showcased by my friend and publisher, Jonathan MS Pearce, on this blog. Pearce particularly highlighted a paragraph of Newton's review pertaining to the objection that perhaps the Deity need not be infinite to be the Deity, wanting to hear my response. He wrote,
I would be interested to see what Lindsay himself has to say about this, particularly the paragraph:
I find no problem with the concept of God being finite since I do not believe that anything that can be assigned a numerical value (e.g. a real number) can be infinite. At minimum all we require is that God has properties/potentials that are “large enough” to satisfy our concept of what God can do or be. If any property is a trillion times larger than the “large enough value” then that value is large enough for most people to be suitably impressed with that aspect of the magnitude of God. There is no need to get bogged down by infinity. Moreover, by not having to worry about infinities one can talk more logically about one aspect of God being greater than some other aspect of God.
How would this work with the sort of Anselmian understanding of being able to imagine, at least conceptually, of an entity one unit more X than previously imagined? Can this idea, as the reviewer seems to think, that a sufficiently large enough God is, well, sufficiently large enough?
Before addressing these questions more directly, I want to point out a few paragraphs from Dot, Dot, Dot itself, where I say something, at least, about this matter. First,
There are a few possibilities that we should consider [to avoid the problems associated with assuming God is infinite]. One is that God has limited faculties; he exists devoid of omni properties. This avoids the problem of omni incoherence, although it also seems to contradict the premises of the ontological argument because one could easily imagine a being with at least one unlimited faculty. We are forced to reject either this position or the argument itself, then. (p. 178)
Granted, this paragraph isn't an adequate response to the challenge at hand because I said that we can easily imagine (and should have italicized that word) a being with at least one unlimited faculty. Perhaps we cannot imagine anything infinite, though, and can merely fool ourselves that we can. A few pages later, however, I get nearer to the heart of the matter, striking much nearer to Newton's objection.
The potential objection here is that on strict finitism, or ultrafinitism, there simply aren't infinitely many natural numbers—the natural numbers only represent a potential infinity that can never be realized. This strikes me as a good rebuttal for temporal and local entities like ourselves, but it seems quite weak for an omniscient and eternal deity not limited in any worldly way. If most of what we call the natural numbers have no meaning, then surely an omniscient deity knows which of those numbers is largest. Yet if the deity knows which is largest, surely he can make sense of the square of that number, that number multiplied by itself, which has clear meaning as the number of unit squares composing a square with side of that largest meaningful length. Of course, this product is strictly larger than the one claimed to be largest (unless it is one, which is patently ridiculous). (p. 181)
There's a way around this problem still, and I mentioned it in the pages in between.
The only potential escape here is to fall back on a purely qualitative understanding of the infinite. Combining Anselm's argument with Gödel's, for instance, could allow “Most High” to be defined as the accrual of every conceivable positive property with no negative properties. This, though, is the definition of the Platonic ideal of goodness—an abstraction. If believers want it that way, so be it, but we know that they do not.
Additionally, this argument also equivocates, now on the number of positive properties. Positive properties are also abstractly defined concepts, and so we have no reason to accept that there cannot be an infinite number of them, especially if the God that
they're supposed to define is an eternal God. Indeed, Gödel's first axiom states that any property entailed by a positive property is also a positive property. It quickly becomes easy to imagine, then, that each positive property creates a cascade of others. Indeed, I suspect it is the case that God defined in this way must exhibit an infinite number of positive properties, which has interesting consequences. (pp. 179–180)
When I say “a cascade of others” here, I really mean a big cascade, an infinite one (because it is a cascade of abstractions). I present the argument in three cases, and the first two of these follow. Case 2 is very near to Newton's objection.
Case 1 has us consider the statements “N is a prime number” for every natural number N . Each of these statements is unambiguously either true or false for every natural number N, of which most mathematicians agree there are an infinite number (finitists claim a potential infinite). For one specific, we know that every natural number that ends with a zero in our number system is not prime, returning potentially infinitely many known, immediately identifiable examples of numbers for which this statement is false. An omniscient being should know this answer for every one of these numbers, and as knowing each of those is entailed by omniscience, each is a positive property. Thus, there are infinitely many positive properties contained in being God-like.
Case 2 faces the situation in which an apologist might argue that an omniscient God need only know the answers to these questions for every natural number that non-omniscient beings will ever investigate. As discussed previously, apologists like William Lane Craig assert that there are only potential infinities at play above, in part because only finitely many numbers will ever be examined by any real mind.
Consider, then, the follow-up statements “N is a natural number that has been or will be examined on the question of being prime,” for every natural number N. From the perspective of an omniscient deity, according to the above defense, this question will also have an unambiguous, known answer. Importantly, the truth value will be known for every natural number N by such an omniscient deity. Since knowing these truth values is entailed by omniscience, knowing each truth value is a positive property. Thus, there are infinitely many positive properties. (pp. 181–182)
That said, the core of Newton's objection lies in,
At minimum all we require is that God has properties/potentials that are “large enough” to satisfy our concept of what God can do or be. If any property is a trillion times larger than the “large enough value” then that value is large enough for most people to be suitably impressed with that aspect of the magnitude of God. There is no need to get bogged down by infinity.
First, let me point out that Newton is saying exactly what I said about theology: the goal is simply to satisfy a believer's credulity concerning that a concept of a God can do or be. This is because, as I wrote in my newer book, Everybody Is Wrong About God, God isn't just an idea, it's an idea maintained directly in service to various psychological and social needs. All deity needs to do in order to satisfy those needs is seem convincing enough to maintain belief.
The obvious, and equally silly, rebuttal here, however, is simply to ask what happens if someone, say an Islamic Mullah, argues that the Deity at the center of his religion has that same property in two trillion times the “good enough” value. That should be suitably more impressive to anyone superstitious enough to believe it, right? And why couldn't he say so? Nothing on Earth prevents anyone from saying something of that kind since the concepts at hand are all abstract ideas and theology is seeking the maximum in credulity about those ideas, not truth about them, if such “truths” could even exist in principle. (My Deity, in fact, would have that property at ten trillion with ten trillion up-arrows [link: https://en.wikipedia.org/wiki/Knuth's_up-arrow_notation], just because nothing could stop me, although that still renders my Deity smaller than most—Newton could beat me at a stroke of “plus one!”)
See how silly this all is? It isn't that there's a need to get bogged down in infinity; it's that infinity is at the bottom of this slipperiest of theological slopes. It may, in fact, take more mental gymnastics to stay above it than it does to try to reconcile the nonsense within it. Worse, getting bogged down in infinity doesn't even free us from the silly problem, as I argue in Dot, Dot, Dot, because accepting one infinite cardinality immediately implies infinitely many larger, just like with numbers.
earce cuts through this fog by asking directly about being able to imagine, at least conceptually, of an entity one unit more X than anything previously imagined, and that's rather the problem. Given that deities are not ideas grounded in reality, but rather concepts defined above or separate from it, I cannot see a single reason why we cannot imagine something just that much bigger to edge out any “good enough” deity. Potential believability doesn't seem to diminish even by exponential multiplication, to say nothing of incremental enhancements. Worse, as the argument above goes, whatever is meant by omniscience of the deity should probably include an assessment of how much that deity knows. Such a deity seems trapped into having to know the first prime number he doesn't know, for instance, or into understanding the definition of the first number too big to be defined. Direct paradoxes like this cast a pretty grim shadow over any concept equipped with such properties.
These kinds of answers miss the heart of Newton's objection, though. We could simply take, even on some kind of theism, that Anselm was wrong, that there is a Deity, and yet something greater very well could be conceived of in the mind. This is a Deity as Superman, and I agree with Newton that for most people, such a concept would be suitable to maintain belief. Of course, deities or no, given that we can imagine a better world for ourselves quite easily, that belief will be exactly as it so often presents itself now: doubt-riddled or blindly ignorant. In the end, though, this is just theology doing theology—the effort of extracting as much credulity as possible out of certain wishful and yet outlandish ideas.