There is an interesting little book, which used to be quite popular among philosophy majors, called Flatland: A Romance of Many Dimensions, published by Edwin Abbot in 1884. It tells the story of a two-dimensional world populated by characters of various shapes: women are thin, straight lines, men are polygons with numbers of sides corresponding to social status, and the hero, A. Square, has adventures in strange realms called Spaceland, Lineland, and Pointland. He gets into trouble when he starts to think about a land of more than three dimensions, and he must return to his own world. (As an aside, Ian Stewart has published a quasi-sequel to Flatland called Flatterland: Like Flatland, Only More So [Perseus, 2001].)
I was reminded of this book today in a tutorial with a student who has been working on St. Anselm. We were reading On the Procession of the Holy Spirit, in which the logical relations of the Persons of the Trinity are explored in some detail. Thinking about these logical relations reminded me of Flatland because the difficulties reminded me that there may be certain sorts of things for which we have no cognitive categories, thus making it quite difficult, if not impossible, to conceptualize what it is we're talking about.
Take this simple example. If you were a citizen of Flatland and lived in a universe of only two dimensions, then if you found yourself inside a circle, and somebody said to you: "Exit the circle without passing through the line that defines it," you would be at a loss as to how to do it. However, those of us who live in the three-dimensional world can easily see that all you need to do to escape a circle without passing through the line that defines it is to jump over the line, that is, move into the third spacial dimension. In a world in which there are only two dimensions, of course, it would be impossible to conceive of such a thing. To see why, imagine this: suppose you are not living in Flatland, but here and now, in the 3D world. If you find youself inside a sphere, what would you do if someone said to you "Exit the sphere without passing through the wall that defines it"? It just doesn't seem possible, but if we were to reason on a par with the previous example, we would say "Simple: just move into the fourth dimension and jump beyond the wall that way," just as we recommended to the two-dimensional person in the circle. But we can't conceive of what it even means to move into the fourth dimension in order to exit a sphere without passing through its wall, because our minds conceive of space in three dimensions, not in four. But logically, of course, it should be possible.
The Trinity is a very knotty problem for anybody who wants to live only with things that are logically possible. It's tempting to say that the set of things that are logically possible contains more items than the set of things that are actual, since there seem to be plenty of things that are at least logically possible but that could never appear in the real world, and yet it at least seems to be the case that everything that is actually in the world is also something that is logically possible. For example, it's at least logically possible for there to be a lump of gold that is larger than the known universe, but it just isn't going to happen. At least not until Bill Gates sells his shares of Microsoft. But just because there are more items in the set of things that are logically possible, must we also accept it as a given that the set of things that are actual is a proper subset of the things that are logically possible? Is there, in short, a kind of actuality that is something other than a merely logical possibility?
It is tempting to say no. But if that's what you're tempted to say, it seems that you are committing yourself to denying the Trinity, since it is not very difficult to show that the Trinity is not logically possible. One of the best known proofs of this in recent times is Richard Cartwright's "On the Logical Problem of the Trinity," which was originally given as a public lecture in 1978 at North Carolina State University, but which has since been published in his Philosophical Essays (MIT Press, 1987, pp. 187-200).
The argument rests on the text of the Athanasian Creed. According to that Creed, the Father is God, the Son is God, and the Holy Spirit is God, but there are not three gods, but One God; further, the Father is not the Son, the Father is not the Holy Spirit, and the Son is not the Holy Spirit. The problem is one of identity. If the Three Persons of the Trinity are all identical to God, and if God is a single essence, then logically at least the Three Persons of the Trinity ought to be identical to each other, but they aren't, they are distinct insofar as they are not all one Person.
The argument relies on a conception of identity that is sometimes called the Indiscernability of Identicals. The basic idea is that all objects which have every property in common (that is, are indiscernable from each other) are identical. For example, we call the morning star Phosphorus, the evening star Hesperus, but if you examing Phosphorus and Hesperus, you will find that they have every property in common: brightness, diameter, position relative to the sun, etc. The reason, of course, is because "Phosphorus" and "Hesperus" are just two different names for the planet Venus. The words have different meanings--one means "morning star", the other means "evening star"--but they have the same reference.
The upshot of the Indiscernability of Identicals is that whenever any objects fail to share even one property, they are not identical. The Father has the property of begetting the Son, but the Son does not have that property, instead he has the property of being begotten by the Father. So the Father is not identical to the Son. So far so good.
The trouble begins when we consider a common intuition about identity: it is supposed to be transitive. That is, if x is identical to y, and y is identical to z, then x ought to be identical to z. For example, if (2 + 7) is identical to (3 + 6), and (3 + 6) is identical to (4 + 5), then (2 + 7) ought to be identical to (4 + 5). And sure enough, all three equations are identical to 9. We can write this as (2 + 7) = (3 + 6) = (4 + 5) = 9.
So, God is identical to the Father, and God is identical to the Son. Let x = "The Father", y = "God", and z = "The Son", and you get x = y = z. From this, it ought to follow that x = z. But according to the Athanasian Creed, the Father is not identical to the Son, and that is a violation of our logical intuitions about identity.
If we were to base all of our beliefs only on those things that are logically possible, then, we would not be able to believe in the Athanasian Creed. That would be too bad, since the Creed teaches us that the only way to be saved is to believe what it teaches. Perhaps it is false, but that would not help the orthodox Christian, for whom there are many other magisterial sources of what the Athanasian Creed teaches, and you simply cannot escape them and remain a Christian of any stripe.
One way out, perhaps, is to redefine the notion of identity. Or of metaphysical unity. This way out seems rather extreme, and would give rise to further puzzles, many of them non-theological. It's better to keep out intutions about these things as they are.
The other way out is to grant that it is not possible to make logical sense of the Trinity, but argue that logical sense is only one way of conceiving of reality, and only of one part of reality at that--the logically conceivable part. But just as the two dimensional man has no cognitive capacity for logically imagining what it would mean to jump over the line of a circle, it may be case that our logic just is not up to the task of giving us the equipment needed to conceive of how Three distinct Persons can all be One Thing, namely God.
St. Anselm's solution to this puzzle, loosely based on St. Augustine's argument in the De Trinitate, is quite different, and I will have more to say about it later.