As a freshman in college, I was taught that parallel lines never cross. This is the mathematical layman’s way of summarizing the fifth postulate of Euclid’s Elements of Geometry. Euclid’s plane geometry, in which space has no curvature, is the standard geometry that we use to explain the world as we experience it. In a Euclidean world, all triangles are 180 degrees, for instance. Euclidean geometry is beautiful and regular; its elegance and symmetry helps to explain our everyday experience of the world, and for many centuries it was assumed that Euclidean geometry was an absolute explanation of reality.

Mathematicians, however, including Euclid himself, were not happy with the fifth postulate because it was neither self-evident nor provable. This led, in the 19th century, to the development of non-Euclidean geometries, which reject the fifth postulate. As a senior in college, I was introduced to Lobachevsky’s non-Euclidean geometry, which allows for parallel lines to cross at some point in the infinite distance. According to the Britannica Concise Encyclopedia, “Mathematicians were forced to abandon the idea of a single correct geometry; it became their task not to discover mathematical systems but to create them by selecting consistent axioms and studying the theorems that could be derived from them. The development of these alternative geometries had a profound impact on the notion of space and paved the way for the theory of relativity.”

I was reminded of the differences between Euclidean and non-Euclidean or Lobachevskian geometry as I read “Communion, Covenant, and Our Anglican Future,” the recent statement by archbishop of Canterbury Rowan Williams. In it he writes:

22….[T]here is at least the possibility of a twofold ecclesial reality in view in the middle distance: that is, a ‘covenanted’ Anglican global body, fully sharing certain aspects of a vision of how the Church should be and behave, able to take part as a body in ecumenical and interfaith dialogue; and, related to this body, but in less formal ways with fewer formal expectations, there may be associated local churches in various kinds of mutual partnership and solidarity with one another and with ‘covenanted’ provinces.

23. This has been called a ‘two-tier’ model, or, more disparagingly, a first- and second-class structure. But perhaps we are faced with the possibility rather of a ‘two-track’ model, two ways of witnessing to the Anglican heritage, one of which had decided that local autonomy had to be the prevailing value and so had in good faith declined a covenantal structure. If those who elect this model do not take official roles in the ecumenical interchanges and processes in which the ‘covenanted’ body participates, this is simply because within these processes there has to be clarity about who has the authority to speak for whom.

24. It helps to be clear about these possible futures, however much we think them less than ideal, and to speak about them not in apocalyptic terms of schism and excommunication but plainly as what they are – two styles of being Anglican, whose mutual relation will certainly need working out but which would not exclude co-operation in mission and service of the kind now shared in the Communion. It should not need to be said that a competitive hostility between the two would be one of the worst possible outcomes, and needs to be clearly repudiated. The ideal is that both ‘tracks’ should be able to pursue what they believe God is calling them to be as Church, with greater integrity and consistency. It is right to hope for and work for the best kinds of shared networks and institutions of common interest that could be maintained as between different visions of the Anglican heritage. And if the prospect of greater structural distance is unwelcome, we must look seriously at what might yet make it less likely.

What got me thinking of geometry was the archbishop’s helpful suggestion that we think of these two emerging ways of being Anglican as being two tracks. The question that arose in my mind was: are these two tracks parallel to each other in a Euclidean sense or a non-Euclidean sense?

In a Euclidean world, these two tracks would be parallel but never converge, not in the time and space of our everyday ecclesial existence, not in the space of the imagination, and not in the infinitude of God. In a Euclidean world, ACNA and TEC might be parallel but always in competition and hostility, like two siblings in perpetual reactive rivalry. But in a non-Euclidean world, these two tracks might converge, where and when one cannot say, but at least there is the possibility. At the very least, we might concede that it is possible that both tracks have the potential of converging in the Kingdom of God. We might even say that they converge in the realm of the One Holy Catholic and Apostolic Church in the here-and-now, though those who hold to the Euclidean notion that ACNA and TEC, for instance, are preaching “separate gospels” could not accept such a non-Euclidean postulate. In a Euclidean world, parallel tracks never meet. This is the fifth postulate of an ecclesiology where schism is the necessary result of unacceptable differentiation.

By contast, the archbishop seems to be setting out a rather non-Euclidean vision for the Church, because he does not see this sort of “structural differentiation,” as he terms it in the final paragraph, as necessarily destructive of the Church, but leading, rather, to greater clarity and integrity on the part of each “track.” Indeed, as Williams writes, “All of this is to do with becoming the Church God wants us to be, for the better proclamation of the liberating gospel of Jesus Christ.”

I’m glad the archbishop reminds us of the main thing. This should be the main thing whether our ecclesiology is Euclidean or non-Euclidean. All too often, however, it gets lost in the equation.

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Just so you know, all of Hilbert’s axioms, which form the basis for ‘neutral’ geometry (geometry without any ‘parallel line’ axioms) are not provable.

Euclicd’s parallel postulate is roughly stated, “Given line l and point a not on l, there is one unique line through a parallel to l” (there are may statesments which are logically equivalent to this). One can construct consistent geometries with different assumptions such as “no lines through a parallel to l” (which gives rise to elliptical geometry, the basis of relativity therory) or ” an infinate number of lines passing through a parallel to l” (which gives rise to hyperbolic geometry).

(just showing off)

Fr. N.J.A. Humphrey - 30 July 2009 02:01 PM

As a freshman in college, I was taught that parallel lines never cross. (very large snip)
What got me thinking of geometry was the archbishop’s helpful suggestion that we think of these two emerging ways of being Anglican as being two tracks. The question that arose in my mind was: are these two tracks parallel to each other in a Euclidean sense or a non-Euclidean sense?

ISTM that before attempting to determine whether the tracks are parallel in a Euclidean or in a non-Euclidean sense, one needs to find out whether they are parallel at all. The ABC does not say in his reflection that the tracks are parallel, and Bp Wright (quoting from his “unpacking” of the ABC’s reflection) says it is possible that they are not:

”... the ‘two-track’ option is not intended as an indifferentist, shoulder-shrugging thing ...To say ‘two-tier’, as some have done at earlier stages in the discussion, implies that the two are still ‘tiers’ of the same thing, whereas ‘tracks’ may be going in quite different directions.”

Acknowledging that the tracks aren’t parallel doesn’t mean that only one of them will be able “to pursue what they believe God is calling them to be as Church, with greater integrity and consistency”, it means that those who choose a non-covenant track (and IMO this group is likely to include TEC) do not see the Covenant as part of what God is calling them to be. Nor is there any reason, even in Euclidean geometry, that a diverging track could not at some future time curve around to a converging course and eventually meet up again with the other.  Maybe TEC (or other non-signers) will eventually elect to adopt the Covenant, even if not at the first opportunity.

In a Euclidean world, these two tracks would be parallel but never converge, not in the time and space of our everyday ecclesial existence, not in the space of the imagination, and not in the infinitude of God. In a Euclidean world, ACNA and TEC might be parallel but always in competition and hostility, like two siblings in perpetual reactive rivalry. But in a non-Euclidean world, these two tracks might converge, where and when one cannot say, but at least there is the possibility. At the very least, we might concede that it is possible that both tracks have the potential of converging in the Kingdom of God. We might even say that they converge in the realm of the One Holy Catholic and Apostolic Church in the here-and-now, though those who hold to the Euclidean notion that ACNA and TEC, for instance, are preaching “separate gospels” could not accept such a non-Euclidean postulate. In a Euclidean world, parallel tracks never meet. This is the fifth postulate of an ecclesiology where schism is the necessary result of unacceptable differentiation.

It is still not clear to me what you suggest as an alternative to schism in cases where one portion of the church differentiates itself in a way which is unacceptable to another. What should the portion of the church which finds the innovation unacceptable do instead of dividing from the innovators? Is the two-track model better, in your opinion, or is it just schism under a different name? As was pointed out on another thread, and indeed included in the ABC’s reflection after GC 06, the second track would relate to the first more or less as the Methodists relate to the C of E, which I think means we’re friendly and listen to what each other have to say, we acknowledge our common ecclesial ancestry, but we don’t have interchangeable clergy and our jurisdictions overlap. That is not the “sibling rivalry” you describe that might arise between TEC and ACNA, but it isn’t full communion either.

By contast, the archbishop seems to be setting out a rather non-Euclidean vision for the Church, because he does not see this sort of “structural differentiation,” as he terms it in the final paragraph, as necessarily destructive of the Church, but leading, rather, to greater clarity and integrity on the part of each “track.” Indeed, as Williams writes, “All of this is to do with becoming the Church God wants us to be, for the better proclamation of the liberating gospel of Jesus Christ.”

By contast, the archbishop seems to be setting out a rather non-Euclidean vision for the Church, because he does not see this sort of “structural differentiation,” as he terms it in the final paragraph, as necessarily destructive of the Church, but leading, rather, to greater clarity and integrity on the part of each “track.” Indeed, as Williams writes, “All of this is to do with becoming the Church God wants us to be, for the better proclamation of the liberating gospel of Jesus Christ.”

I’m glad the archbishop reminds us of the main thing. This should be the main thing whether our ecclesiology is Euclidean or non-Euclidean. All too often, however, it gets lost in the equation.

I hope to goodness that the ABC is thinking in Euclidean terms! It is in the here-and-now that the communion is in danger of coming apart around our ears, and in the here-and-now, Euclidean is the only geometry that applies. I know it’s impossible to build a house or a highway with non-Euclidean geometry, and I suspect it’s impossible to build a Communion, based on the assumption that, despite all appearances to the contrary, TEC is actually on a parallel track with the rest of the Communion. Even accepting at face value TEC’s assertion that D025 and C056 are only descriptive, what they describe is a Province which is at this time inconsistent with the agreed-on teaching of the rest of the Communion respecting ordination of non-celibate gay persons, and committed to “gathering resources” to become inconsistent on blessing of same-sex unions.

Karen

Bishop Tom Wright’s observation “‘tracks’ may be going in quite different directions” begs for elaboration, especially in the context of what Fr. Humphrey’s calls “The Archbishop’s Non-Euclidean Ecclesiology.”

The revisionist track leads eventually to eternal derailment. Nevertheless, a crossing may well appear at various points to allow ones on the revisionist track to repent, amend their ways, and take a feeder line back to the track of apostolic, catholic Christianity.

Those grounded in biblical truth and 2000 years of church tradition will recognize that a massive train wreck is somewhere ahead as an unrepentant TEC and its allies gather speed going downhill with defective brakes (called a “hot box” in the old days of railroading). No amount of ecclesiastical verbiage can obscure this looming tragedy.

So will the TEC narrative play out like a Greek tragedy rather than an axiom of Greek mathematics? I suggest that when considering the direction that TEC and its allies are headed, we have more to learn from Aeschylus, Euripides, and Sophocles (and even William Shakespeare)—or certainly from the Deuteronomist and St. John the Revelator—than from Euclid and Einstein.

Or, using another analogy, will a paradigm shift occur that ushers in this dramatic change: a TEC restored to biblical orthodoxy and full Communion participation?

We can pray and work to that end.

Dick Wire