Some Mathematical Reflections on Communion

Mark Harris wrote a post on his blog the other day about Bishop David Bena’s recent move from The Episcopal Church to the Convocation for Anglicans in North America (CANA). Bishop Bena is recently retired from the ultraconservative Diocese of Albany. CANA is claimed by the Church of Nigeria (Anglican Communion) to be “a mission of the Church of Nigeria.” Mark raises various questions about who is in communion with whom and what is the nature of the “transfer” that has taken place. When I read Mark’s post, it had already attracted 23 comments from visitors, and it is probably fair to say that all this analysis has raised rather more questions than it has answered.

Let me assure you at the outset that I do not have the answers that would clarify all the issues involved. The situation makes it clear that the informality of the Anglican Communion can be something of a problem in unfamiliar and challenging circumstances. Although I am not enthusiastic about any proposal for an Anglican covenant—the much talked about idea is really intended as a mechanism to keep The Episcopal Church out of the twenty-first century—some agreement that spelled out how the Anglican Communion is supposed to work could actually be helpful if it avoided anything touching upon doctrine.

Even before the 2003 General Convention voted to consecrate Gene Robinson, there was some doubt about what churches were in communion with what other churches, since the ordination of women had already put strains on our Anglican fellowship. It is becoming increasingly clear that “being in communion” does not have a simple, universally accepted meaning, and, in practice, the concept does not always imply what we might expect. What is required for one church to be in communion with another?

Consider a particular case. The Episcopal and Nigerian churches are in the Anglican Communion and in communion with Canterbury, which might lead one to think that they should be in communion with one another. Nigeria claims not to be in communion with The Episcopal Church, however, which, as far as I can tell, has not declared itself out of communion with the Church of Nigeria. This could have the consequence—I cannot say whether it actually does—that a bishop could transfer “lawfully” from The Episcopal Church to the Church of Nigeria but could not transfer back. (Since, of course, the Anglican Communion really has no enforcement mechanism, anyone can get away with anything that doesn’t cause too much commotion.) This is, of course, goofy. It’s certainly goofy if it’s true, and it’s also goofy that we don’t know whether it’s true or not! Of course, it may be that the Episcopal/Nigerian church relationship cannot be what the Nigerians say it is.

As a mathematician, I find it impossible not to view the matter of communion in terms of relations. A mathematical relation is a formal description that is used to capture and reason about relationships between objects. For example, “equality” is a relation defined on, to keep matters simple, integers. If x and y represent arbitrary integers, we write x = y, meaning that x and y have the same value. In our case, our variables represent churches, and we can define a relation C such that x C y means that church x is in communion with church y. Mathematically, C is given by a set of ordered pairs, such that, (x, y) is contained in the set C if and only if x C y.

The eyes of most readers are, most likely, glazed over by now, but there is a point to this. Relations provide a useful set of named concepts and operations that allow efficient discussion of relationships like “being in communion with.” Whatever this relationship is in practice, it surely involves rules about exchange of clergy and allowing members of the related churches to take communion in one another’s houses of worship. (More definition on the part of the Anglican Communion would likely be helpful.) Now we get mathematical, but only trivially so. We can say that, for any church x, x C x. That is, every church is in communion with itself. This provides no big insight, but it shows that C is, as a mathematician would say, reflexive.

Now for a more interesting mathematical question: is the relation C symmetric? If x C y, is, necessarily, y C x? If being in communion is reciprocal, then the relation should be symmetric. Is it not crazy to suggest that we can be in communion with the Church of Nigeria, but the Church of Nigeria is not in communion with The Episcopal Church? Several observations are in order here. First, how does one “break communion,” i.e., how does church x, in communion with church y, change x C y to not x C y? The answer to this is something else that might go into the aforementioned Anglican covenant. I won’t deal with it here, but I will offer the thought that C should be symmetric because it borders on nonsensical for it not to be. If we all agreed that “being in communion with” is a symmetric relation, then, if the Church of Nigeria declares itself to be out of communion with us in some effective way—whatever that may mean—we do not have to declare what some would think uncharitable, namely, that The Episcopal Church is not in communion with the Church of Nigeria. Instead, our not being in communion with Nigeria would be automatic. Mathematically, C’s being symmetric means that, if (x, y) is not in the relation, then, necessarily, neither is (y, x).

One church in the Anglican Communion is special, namely, the Church of England, represented by the Archbishop of Canterbury. Peter Akinola, Archbishop of Nigeria, has suggested throwing the Church of England out of the Anglican Communion. This, however, is a nonsensical idea, as, at the very least, the Communion has always been seen as churches in communion with the Church of England. One imagines that the Church of England can no more be thrown out of the Anglican Communion than, say, the U.S. could be expelled from the nations of North America.

From this point on, we will consider the relation C defined only on churches of the Anglican Communion. Thus, we will not talk about, say, the Evangelical Lutheran Church in America. The Episcopal Church is in communion with the Lutherans, but this relation may not be the same as the one we have designated by C. In any case, no one thinks the Evangelical Lutheran Church in America is in the Anglican Communion, so no variable z that might be of interest can represent that church.

Now consider C defined for Anglican Communion members (or possible members). We see that one object (as the mathematicians would say) is special. It is called the Church of England, and we will represent it as e. Thus, we must have e C e. If the members of the Communion are necessarily churches with which the Archbishop of Canterbury is in communion, then, for any church x in the communion, we must have e C x. If we assume that C is symmetric, which surely seems reasonable, then we can also say that x C e.

Now, should not “being in communion with” within the Anglican Communion be transitive? That is, if x C y and y C z, then should we not be guaranteed that x C z? If C is transitive and symmetric, since every church in the Communion is in communion with the Church of England (and vice versa), then every church in the Communion is necessarily in communion with the other churches of the Communion. Surely this is as it should be.

Now, permit me to offer a definition of the members of the Anglican Communion. First, define the communion relationships of the Anglican Communion as the reflexive, symmetric, transitive closure of C. In other words, if we begin with e C x for each church x that Canterbury is in communion with, we get all of C by adding all ordered pairs needed to make C reflexive, symmetric, and transitive. Don’t worry about the mathematics; this effectively means that all churches in the communion are in communion with all other churches in the communion. As I said, this is clearly how things should be. Seemingly, however, it is not how things are, because, for example, the Church of Nigeria is not in communion with The Episcopal Church.

If you define the communion as I have suggested, however, what happens when the Church of Nigeria declares itself not in communion with our church? Mathematically, this is very simple. Let t represent The Episcopal Church and n represent the Church of Nigeria. Since it is now not the case that n C t, we cannot have t C n because, if we did, it would violate our symmetry requirement. Automatically, The Episcopal Church is not in communion with the Church of Nigeria, and we did not have to “declare” anything to make it so. By transitivity, however, since t C e and e C n, transitivity would imply that t C n. Since this is not in the relation—I’ll skip a logically rigorous argument here—either t or n must never appear in one of our ordered pairs. It would be perverse to suggest that one church that is not the Church of England could unilaterally throw another church out of the Communion, so we must conclude that, if Nigeria breaks communion with The Episcopal Church, the only way to maintain the definition of the Communion is to conclude that Nigeria breaks all relationships with all other Communion churches. In other words, it removes itself from the communion.

Well, admittedly, that was a roundabout way of getting where I finally ended up, but I really do like my mathematical definition of the Communion. After all, an Anglican Communion out of communion with itself just makes no sense. For that part of the Communion that yearns for “clarity” and “unity,” my definition delivers.